Stability of Overshoots of Markov Additive Processes
aa r X i v : . [ m a t h . P R ] F e b Stability of Overshoots of Markov Additive Processes
Leif Döring ∗ Lukas Trottner ∗ , † Abstract
We prove precise stability results for overshoots of Markov additive processes (MAPs)with finite modulating space. Our approach is based on the Markovian nature of overshootsof MAPs whose mixing and ergodic properties are investigated in terms of the characteristicsof the MAP. On our way we extend fluctuation theory of MAPs, contributing among othersto the understanding of the Wiener–Hopf factorization for MAPs by generalizing Vigon’séquations amicales inversés known for Lévy processes. Using the Lamperti transformationthe results can be applied to self-similar Markov processes. Among many possible applica-tions, we study the mixing behavior of stable processes sampled at first hitting times as aconcrete example.
1. Introduction
Overshoots of a Lévy process 𝜉 , defined by O 𝑥 = 𝜉 𝑇 𝑥 − 𝑥, 𝑥 ≥ , on { 𝑇 𝑥 < ∞} , where 𝑇 𝑥 : = inf { 𝑡 ≥ 𝜉 𝑡 > 𝑥 } , are classical objects in the study of Lévy processes.Their asymptotic analysis is essentially rooted in renewal theory for random walks and hasgained a lot of interest in the past two decades starting with the observation in [6] that classicallimit theorems for the residual time chain of renewal processes have a natural analogue inweak convergence of overshoots of subordinators to a non-trivial limiting distribution. Besidesapplications and extensions in ruin theory for insurance risk processes driven by Lévy pro-cesses (see [27, 35, 47]), this observation was used to explain the entrance behavior of positiveself-similar Markov processes (pssMps) at the origin. Using the Lamperti transformation fortransient pssMps one can show that a pssMp can be started from the origin if and only if theovershoots of the underlying Lévy process converge weakly as the overshoot level 𝑥 divergesto +∞ (see [10, 16]). This was generalized in [21] to the question of how to start real self-similar Markov processes (rssMps) from the origin. Methods for rssMps are similar to thosefor pssMps replacing the Lévy processes 𝜉 in the Lamperti transformation by Markov additiveprocesses ( 𝜉 , 𝐽 ) , MAPs in the following, with finite modulating space {− , } . The correspondingtransformation is usually called Lamperti–Kiu transform. MAPs ( 𝜉 , 𝐽 ) are also called Markovmodulated Lévy processes, due to the ordinator 𝜉 behaving as a Lévy process in between jumpsof a modulating chain 𝐽 , with the Lévy triplet of 𝜉 being determined by the current state of 𝐽 . Thelimiting behavior of overshoots of MAPs, defined by ( O 𝑥 , J 𝑥 ) = (cid:0) 𝜉 𝑇 𝑥 − 𝑥, 𝐽 𝑇 𝑥 (cid:1) , 𝑥 ≥ , ∗ University of Mannheim, Institute of Mathematics, B6 26, 68159 Mannheim, Germany.Email: [email protected]/[email protected] † Supported by the Research Training Group ”Statistical Modeling of Complex Systems” funded by the GermanScience Foundation. { 𝑇 𝑥 < ∞} , where 𝑇 𝑥 : = inf { 𝑡 ≥ 𝜉 𝑡 > 𝑥 } , then plays the same role for the entrance law at 0of rssMps, as do overshoots of Lévy processes for pssMps.The aim of this article is to explore in detail mixing and ergodicity of overshoots of MAPs.We study the convergence in total variation norm, including conditions for polynomial andexponential rates of convergence. Based on fluctuation theory of MAPs developed in [21] wewill use the Meyn and Tweedie approach to stability of continuous time Markov processes (seefor instance [41, 44, 45, 53]) to demonstrate that overshoot convergence can be much more finelyanalyzed once we take the perspective on overshoots as a Markov process, where the subsequentspatial levels that are passed by the ordinator 𝜉 serve as time index for the overshoot process ( O , J ) = ( O 𝑡 , J 𝑡 ) 𝑡 ≥ . This idea is inspired by the observation that for the overshoot process ofa Lévy subordinator 𝜎 , inverse local time at 0 is given by 𝜎 itself [8]. For this special case,this opens the door to powerful results of excursion theory for general Markov processes andallows, among others, to derive explicit formulas for the invariant measure and resolvent ofthe overshoot process of a Lévy subordinator in terms of its triplet [13, 26]. We generalizethese findings to the MAP situation and consequently make use of the analytical tractabilityof overshoots to analyze their ergodic behavior. For the particular case of Lévy processes, theresults can be interpreted as a natural continuous time generalization of results on ergodicityand exponential convergence of the residual time chain belonging to a renewal process, whichcan be found in the standard references on stability of discrete time Markov chains, Meyn andTweedie [42] and Nummelin [46].Our fine analysis of overshoot stability of MAPs is not only inspired by a theoretical desire tounderstand their asymptotics, but also by a practical need to develop statistical and numericalprocedures to get hold of the ascending ladder height process ( 𝐻 + , 𝐽 + ) of a given MAP ( 𝜉 , 𝐽 ) . Thisprocess is one of the cornerstones of fluctuation theory of MAPs and is theoretically accessibleby means of the Wiener–Hopf factorization. However, its explicit analytical characteristics arein general unknown, with a notable exception being the factorization of the MAP associated toan 𝛼 -stable Lévy process via the Lamperti–Kiu transform, which was found in [37]. Due to itsintimate connection with the running supremum of the MAP, observing ( 𝜉 , 𝐽 ) at first hittingtimes offers all information needed to determine ( 𝐻 + , 𝐽 + ) in numerical or statistical procedures.The results of the present article have applications in optimal control problems based onMAPs, see e.g. the recent article [19] for the more particular case of a Lévy driven impulsecontrol problem. There, the generator of the ascending ladder height process is decisive fordetermining optimal threshold times of a desired reflection strategy. Thus, under uncertaintyconcerning the underlying Lévy process, efficient statistical estimation of the ascending ladderheight process is needed. This will be the issue of future work. Moreover, parametric estimationbecomes feasible for the Lévy system of MAPs – which encodes the jumps of a MAP in analogyto the Lévy measure of a Lévy process – with explicit overshoot distributions based on theMAP observed at first hitting times ( 𝑇 𝑛 Δ ) 𝑛 ∈ ℕ for some step size Δ >
0. Such observationscheme can be described as stochastic low frequency scheme as opposed to deterministic low andhigh frequency schemes usually encountered in parametric inference of stochastic processes(see [5] for an overview in the context of Lévy processes) or the stochastic high-frequencyscheme analyzed in [48] for Lévy processes. Furthermore, nonparametric statistical estimationprocedures for the ascending ladder height characteristics can be developed based on ourobservation that under some natural conditions, the overshoot process is exponentially 𝛽 - mixing . This property, describing rigorously asymptotic independence of the past and thefuture of a Markov process, is earmarked in [22] as a central building block to nonparametricstatistical analysis of non-reversible ergodic Markov processes. Hence, our results indicate howto include MAPs (which are non-ergodic) in an ergodic statistical setting by considering thespace-time transform introduced in form of overshoots.Due to recent applications of MAPs we also expect applications of our mixing estimatesin other fields of probability theory such as planar maps (see for instance [9]). We highlight2his point by making use of the the Lamperti–Kiu transform to translate the mixing behaviorof MAPs into mixing bounds for self-similar Markov processes sampled at first hitting times.Further applications to non-parametric statistical estimation for MAPs, Lévy processes andequivalently self-similar Markov processes will be subject to future research. We start in Section 2 with formally introducing Markov additive processes and summarizingsome results belonging to their fluctuation theory as given in [21]. We then proceed in Section3 with the stability analysis of MAP overshoots, starting with the rigorous description oftheir Markovian nature and then studying important concepts from the theory of stability forMarkov processes such as Harris recurrence, invariant measures and petite sets. For the readerunfamiliar with these concepts, we have devoted Appendix A to a brief summary of stabilityof Markov processes in the sense of Meyn and Tweedie, additionally clarifying some resultsin the literature and developing a new technique for deriving invariant measures of Markovprocesses based on a limiting argument involving the resolvent of the process in PropositionA.1. Moreover, some general terms for Markov processes, such as Borel right processes, theFeller property and resolvents are introduced in Appendix A without further explanations inthe main body of the text. With this setup we come to our primary goal, the ergodicity analysisof overshoots. Our main results in this respect, taking also account of the developments inSection 4 described below, can be informally summarized as follows.
Theorem.
Suppose that the MAP ( 𝜉 , 𝐽 ) is upward regular, 𝐽 is irreducible and the ascending ladder heightMAP ( 𝐻 + , 𝐽 + ) has a finite first moment. Under mild assumptions on the Lévy system of ( 𝜉 , 𝐽 ) , ( O 𝑡 , J 𝑡 ) 𝑡 ≥ converges in total variation to a unique stationary distribution, which encodes the characteristics of theascending ladder height MAP. If moreover the jump measures associated to the MAP’s Lévy systempossess a common (exponential) moment, then the convergence takes place at (exponential) polynomialspeed and overshoots are (exponentially) polynomially 𝛽 -mixing. This will be made precise in a sequence of theorems in Section 3. In Theorem 3.18 we establishconditions on either the creeping probabilities of the subordinators associated to the ascendingladder height MAP or its Lévy system that guarantee total variation convergence of overshoots.Theorem 3.21 and Theorem 3.24 build on this result, giving exponential/polynomial ergodicityand the exponential/polynomial 𝛽 -mixing property, respectively.Section 4 is devoted to finding conditions on the Lévy system of the parent MAP, whichimply the required assumptions on ( 𝐻 + , 𝐽 + ) for the ergodic results of the previous section,thus enhancing significantly our understanding of asymptotics of MAP overshoots. The toolwe develop for this purpose is an extension of Vigon‘s équations amicales inversés for Lévyprocesses given in [54] to MAPs. These equations analytically relate the Lévy systems of ( 𝜉 , 𝐽 ) and ( 𝐻 + , 𝐽 + ) , which makes inference of distributional properties of the ascending ladder heightprocess based on the characteristics of the parent MAP possible.Finally, in Section 5 we apply our 𝛽 -mixing result for MAPs to real self-similar Markovprocesses sampled at first hitting times by exploiting the Lamperti–Kiu transform, which bridgesthese two classes of processes. As an even more specific application, we then consider the mixingbehavior of 𝛼 -stable Lévy processes and ergodicity of overshoots of the associated Lamperti-stable MAP. For a given space X we will denote by B ( X ) its Borel 𝜎 -algebra and by B + ( X ) and B 𝑏 ( X ) thespace of positive, resp. bounded real-valued functions on X . If X is locally compact, then C ( X ) denotes the space of continuous, real-valued functions on X vanishing at infinity. If X = ℝ 𝜇 is a measure on ( ℝ , B ( ℝ )) , we let 𝜇 ( 𝑦 ) ≔ 𝜇 (( 𝑦, ∞)) , 𝑦 ∈ ℝ , be its tail measure. Leb ( d 𝑥 ) denotes the Lebesgue measure on ℝ and Leb + ( d 𝑥 ) is its restriction to ℝ + = [ , ∞) .
2. Markov additive processes and their fluctuation theory
We start with introducing Markov additive processes with finite modulating space. For thegeneral theory of Markov additive processes the reader may consult the landmark papers ofÇinlar [14, 15], a good start for the particular case of finite modulating space is [2, Chapter XI],and a focus on fluctuation theory is given in [21]. Let
Θ = { , . . . , 𝑛 } be a finite set and ( ℝ × Θ ) 𝜗 be the Alexandrov one-point compactification of ℝ × Θ with some isolated state 𝜗 = (∞ , 𝜛 ) .Throughout we will always extend a function 𝑓 ∈ B ( ℝ × Θ ) to a function in B (( ℝ × Θ ) 𝜗 ) by setting 𝑓 ( 𝜗 ) =
0, which will make notation more convenient. A (killed) Markov additiveprocess (MAP) ( 𝜉 , 𝐽 ) with finite modulating space Θ is defined as a Feller process with statespace ℝ × Θ and cemetery state 𝜗 , having a possibly finite lifetime 𝜁 and underlying stochasticbase ( Ω , F , 𝔽 = ( F 𝑡 ) 𝑡 ≥ , ( ℙ 𝑥,𝑖 ) ( 𝑥,𝑖 )∈( ℝ × Θ ) 𝜗 ) and which moreover has the characteristic propertythat given 𝑠 , 𝑡 ≥ , ( 𝑥, 𝑖 ) ∈ ℝ × Θ and 𝑓 ∈ B 𝑏 (( ℝ × Θ ) 𝜗 ) it holds that 𝔼 𝑥,𝑖 (cid:2) 𝑓 ( 𝜉 𝑡 + 𝑠 − 𝜉 𝑡 , 𝐽 𝑡 + 𝑠 ) { 𝑡 <𝜁 } | F 𝑡 (cid:3) = 𝔼 ,𝐽 𝑡 [ 𝑓 ( 𝜉 𝑠 , 𝐽 𝑠 )] { 𝑡 <𝜁 } , ℙ 𝑥,𝑖 -a.s.In other words, conditionally on { 𝐽 𝑡 = 𝑖 } and no killing before time 𝑡 ≥
0, the pair ( 𝜉 𝑡 + 𝑠 − 𝜉 𝑡 , 𝐽 𝑡 + 𝑠 ) 𝑠 ≥ is independent of the past and has the same distribution as ( 𝜉 𝑠 , 𝐽 𝑠 ) 𝑠 ≥ under ℙ ,𝑖 ,which is an equivalent definition for MAPs with finite modulating space often encountered inthe literature such as [21]. A straightforward consequence of this property is conditional spatialhomogeneity of the process, i.e. 𝔼 𝑥,𝑖 [ 𝑓 ( 𝜉 , 𝐽 )] = 𝔼 ,𝑖 [ 𝑓 ( 𝜉 + 𝑥, 𝐽 )] holds for any measurable 𝑓 on the Skorokhod space D ( ℝ × Θ ) of càdlàg functions mappingfrom ℝ + = [ , ∞) to ℝ × Θ equipped with its Borel 𝜎 -algebra (here and for the rest of the paperwe implicitly assume that ( 𝜉 , 𝐽 ) has exclusively càdlàg paths, which can be easily achieved byeither constructing the process as the canonical coordinate process on the Skorokhod space orby a reduction of the probability space and the facts that, by definition, Feller processes havecàdlàg paths almost surely and F is complete). Moreover, ( 𝐽 𝑡 ) 𝑡 ≥ is a continuous time Markovchain, whose transition function is independent of the initial distribution of 𝜉 . Conditionalindependence of increments and spatial homogeneity of the ordinator 𝜉 already teases anintimate relation of MAPs and Lévy processes. In fact, any MAP can be decomposed into anindependent sequence of Lévy processes, whose characteristic triplet depends on the currentstate of the modulating Markov chain 𝐽 .More precisely, we suppose that the measurable space ( Ω , F ) is rich enough to support aprobability measure ℙ such that ℙ 𝑥,𝑖 = ℙ (·| 𝜉 = 𝑥, 𝐽 = 𝑖 ) , i.e. the probabilities underlying theMarkov process ( 𝜉 , 𝐽 ) are given as regular conditional probabilites of ℙ . Then, Proposition 2 in[21] (see also [29, Proposition 2.5] or [15, Theorem 2.23]) gives the following characterizationof a MAP, showing that in between jumps of 𝐽 , 𝜉 behaves as a Lévy process with characteristictriplet determined by the current state of 𝐽 and every jump of 𝐽 potentially triggers an additionaljump of 𝜉 . Proposition 2.1.
A process ( 𝜉 , 𝐽 ) is an unkilled MAP if and only if there exist sequences of • (killed) Lévy processes ( 𝜉 𝑛,𝑖 ) 𝑛 ∈ ℕ , i.i.d. under ℙ for fixed 𝑖 ∈ Θ , • real random variables ( Δ 𝑛𝑖,𝑗 ) 𝑛 ∈ ℕ , i.i.d. under ℙ for fixed and distinct 𝑖, 𝑗 ∈ Θ , ndependent of 𝐽 and of each other under ℙ , such that if 𝜎 𝑛 is the 𝑛 -th jump time of 𝐽 , then under ℙ 𝑥,𝑖 , 𝜉 can be written almost surely as 𝜉 𝑡 = 𝑥 + 𝜉 ,𝑖𝑡 , 𝑡 ∈ [ , 𝜎 ) , 𝜉 𝜎 𝑛 − + Δ 𝑛𝐽 𝜎 𝑛 − ,𝐽 𝜎 𝑛 + 𝜉 𝑛,𝐽 𝜎 𝑛 𝑡 − 𝜎 𝑛 , 𝑡 ∈ [ 𝜎 𝑛 , 𝜎 𝑛 + ) , 𝑡 < 𝜁 , 𝜉 𝑡 = ∞ , 𝑡 ≥ 𝜁 , where the lifetime 𝜁 is the first time one of the appearing Lévy processes is killed: 𝜁 = inf (cid:8) 𝑡 > ∃ 𝑛 ∈ ℕ , 𝜎 𝑛 ≤ 𝑡 such that 𝜉 𝑛,𝐽 𝜎 𝑛 is killed at time 𝑡 − 𝜎 𝑛 (cid:9) . In this paper, we will only deal with MAPs ( 𝜉 , 𝐽 ) with infinite lifetime , i.e. 𝜁 = ∞ , ℙ 𝑥,𝑖 -a.s. for all ( 𝑥, 𝑖 ) ∈ ℝ × Θ . However, killing is relevant for fluctuation theory of MAPs as described below.Let us define ( 𝜉 ( 𝑖 ) ) 𝑖 ∈ Θ as Lévy processes with characteristic triplets ( 𝑎 𝑖 , 𝑏 𝑖 , Π 𝑖 ) that have the samelaw as ( 𝜉 ,𝑖 ) 𝑖 ∈ Θ and ( Δ 𝑖,𝑗 ) 𝑖,𝑗 ∈ Θ as random variables sharing the same law as the corresponding ( Δ 𝑖,𝑗 ) 𝑖,𝑗 ∈ Θ , with Δ 𝑖,𝑖 ≔ 𝑖 ∈ Θ . Moreover, let 𝐹 𝑖,𝑗 be the law of Δ 𝑖,𝑗 . Then, ( 𝜉 , 𝐽 ) canbe uniquely characterized by the Lévy–Khintchine exponents Ψ 𝑖 ( 𝜃 ) = log 𝔼 [ exp ( i 𝜃𝜉 ( 𝑖 ) )] , 𝑖 ∈ Θ ,the transition rate matrix 𝑸 = ( 𝑞 𝑖,𝑗 ) 𝑖,𝑗 ∈ Θ of 𝐽 and the Fourier transforms of Δ 𝑖,𝑗 denoted by 𝐺 𝑖,𝑗 ( 𝜃 ) = 𝔼 [ exp ( i Δ 𝑖,𝑗 )] , 𝑖, 𝑗 ∈ Θ . For convenience we assume Δ 𝑖,𝑗 = 𝑞 𝑖,𝑗 =
0, whichis without loss of generality because Proposition 2.1 shows that these transitional jumps neveroccur. If we now define the characteristic matrix exponent 𝚿 ( 𝜃 ) ≔ diag ( Ψ ( 𝜃 ) , . . . , Ψ 𝑛 ( 𝜃 )) + 𝑸 ⊙ 𝑮 ( 𝜃 ) , as an analogue to the Lévy–Khintchine exponent of a Lévy process, then 𝔼 ,𝑖 h e i 𝜃𝜉 𝑡 ; 𝐽 𝑡 = 𝑗 i = (cid:0) e 𝑡 𝚿 ( 𝜃 ) (cid:1) 𝑖,𝑗 , 𝑖, 𝑗 ∈ Θ , 𝜃 ∈ ℝ . Here, 𝑮 ( 𝜃 ) = ( 𝐺 𝑖,𝑗 ( 𝜃 )) 𝑖,𝑗 ∈ Θ and ⊙ denotes the Hadamard product, i.e. pointwise multiplicationof matrices of the same dimension. Note that since Δ 𝑖,𝑖 = 𝐺 𝑖,𝑖 ( 𝜃 ) = 𝑖 ∈ Θ andhence ( 𝑸 ⊙ 𝑮 ( 𝜃 )) 𝑖,𝑖 = − 𝑞 𝑖,𝑖 . Let us also define the family of potential measures ( 𝑈 𝑖,𝑗 ) 𝑖,𝑗 ∈ Θ givenby 𝑈 𝑖,𝑗 ( d 𝑥 ) = 𝔼 ,𝑖 h ∫ ∞ { 𝜉 𝑡 ∈ d 𝑥,𝐽 𝑡 = 𝑗 } d 𝑡 i = ∫ ∞ ℙ ,𝑖 ( 𝜉 𝑡 ∈ d 𝑥, 𝐽 𝑡 = 𝑗 ) d 𝑡 , 𝑥 ∈ ℝ , 𝑖, 𝑗 ∈ Θ , i.e., 𝑈 𝑖,𝑗 ( 𝐴 ) measures the time 𝜉 spends in 𝐴 when started in 𝑖 , while the modulator 𝐽 is in state 𝑗 . Another important concept in the theory of (general state space) Markov additive processesis the existence of a Lévy system , see Çinlar [14], which generalizes the notion of a Lévy measureand becomes explicit for MAPs with finite modulating space thanks to the path decompositiongiven in Proposition 2.1. We say that ( 𝚷 , 𝐴 ) , where 𝚷 is a kernel on ( Θ , B ( ℝ × Θ )) satisfying 𝚷 ( 𝑖, {( , 𝑖 )}) = , ∫ ℝ (cid:0) ∧ | 𝑦 | (cid:1) 𝚷 ( 𝑖, d 𝑦 × { 𝑖 }) < ∞ , 𝑖 ∈ Θ , and 𝐴 is an increasing continuous additive functional of ( 𝜉 , 𝐽 ) such that for any 𝑓 ∈ B + ( Θ × ℝ × Θ )) and ( 𝑥, 𝑖 ) ∈ ℝ × Θ , 𝔼 ,𝑖 h Õ 𝑠 ≤ 𝑡 𝑓 ( 𝐽 𝑠 − , Δ 𝜉 𝑠 , 𝐽 𝑠 ) { Δ 𝜉 𝑠 ≠ 𝐽 𝑠 − ≠ 𝐽 𝑠 } i = 𝔼 ,𝑖 h ∫ 𝑡 𝐴 𝑠 ∫ ℝ × Θ 𝚷 ( 𝐽 𝑠 , d 𝑥, d 𝑦 ) 𝑓 ( 𝐽 𝑠 , 𝑥, 𝑦 ) i , (2.1)is a Lévy system for ( 𝜉 , 𝐽 ) . Using Proposition 2.1 and results on expectations of functionals ofPoisson random measures, see e.g. Theorem 2.7 in [38], one can demonstrate that 𝐴 𝑡 = 𝑡 ∧ 𝜁 and 𝚷 ( 𝑖, d 𝑦 × { 𝑗 }) = { 𝑖 = 𝑗 } Π 𝑖 + { 𝑖 ≠ 𝑗 } 𝑞 𝑖,𝑗 𝐹 𝑖,𝑗 , 𝑖, 𝑗 ∈ Θ , 𝑖 ∈ Θ , 𝔼 ,𝑖 h Õ 𝑠 ≤ 𝑡 𝑓 ( 𝐽 𝑠 − , Δ 𝜉 𝑠 , 𝐽 𝑠 ) { Δ 𝜉 𝑠 ≠ 𝐽 𝑠 − ≠ 𝐽 𝑠 } i = 𝑛 Õ 𝑘 = (cid:16) 𝔼 ,𝑖 h ∫ 𝑡 ∫ ℝ \{ } 𝑓 ( 𝑘, 𝑥, 𝑘 ) { 𝐽 𝑠 = 𝑘 } Π 𝑘 ( d 𝑥 ) d 𝑠 i + Õ 𝑗 ≠ 𝑘 𝑞 𝑘,𝑗 𝔼 ,𝑖 h ∫ 𝑡 ∫ ℝ 𝑓 ( 𝑘, 𝑥, 𝑗 ) { 𝐽 𝑠 = 𝑘 } 𝐹 𝑘,𝑗 ( d 𝑥 ) d 𝑠 i (cid:17) = 𝑛 Õ 𝑘 = ∫ 𝑡 ℙ ,𝑖 ( 𝐽 𝑠 = 𝑘 ) d 𝑠 (cid:16) ∫ ℝ \{ } 𝑓 ( 𝑘, 𝑥, 𝑘 ) Π 𝑘 ( d 𝑥 )+ Õ 𝑗 ≠ 𝑘 𝑞 𝑘,𝑗 ∫ ℝ 𝑓 ( 𝑘, 𝑥, 𝑗 ) 𝐹 𝑘,𝑗 ( d 𝑥 ) (cid:17) . (2.2)Since 𝐴 is simply the uniform motion, we will also refer to just 𝚷 as the Lévy system for theremainder of this article. As remarked in [39], this can be generalized to the following identityfor any predictable process ( 𝑍 𝑡 ) 𝑡 ≥ and 𝑔 ∈ B + ( Θ × ℝ × ℝ × Θ ) : 𝔼 ,𝑖 h Õ 𝑠 ≤ 𝑡 𝑍 𝑠 𝑔 ( 𝐽 𝑠 − , 𝜉 𝑠 − , 𝜉 𝑠 , 𝐽 𝑠 ) { Δ 𝜉 𝑠 ≠ 𝐽 𝑠 − ≠ 𝐽 𝑠 } i = 𝑛 Õ 𝑘 = (cid:16) 𝔼 ,𝑖 h ∫ 𝑡 d 𝑠 𝑍 𝑠 { 𝐽 𝑠 = 𝑘 } ∫ ℝ \{ } Π 𝑘 ( d 𝑥 ) 𝑔 ( 𝑘, 𝜉 𝑠 , 𝜉 𝑠 + 𝑥, 𝑘 ) i + Õ 𝑗 ≠ 𝑘 𝑞 𝑘,𝑗 𝔼 ,𝑖 h ∫ 𝑡 d 𝑠 𝑍 𝑠 { 𝐽 𝑠 = 𝑘 } ∫ ℝ 𝐹 𝑘,𝑗 ( d 𝑥 ) 𝑔 ( 𝑘, 𝜉 𝑠 , 𝜉 𝑠 + 𝑥, 𝑗 ) i (cid:17) . (2.3)Let us now dive into fluctuation theory of MAPs, which in the form suited to our needswas developed in [21]. An essential tool for our upcoming analysis of the overshoots is the ascending ladder MAP ( 𝐻 + 𝑡 , 𝐽 + 𝑡 ) 𝑡 ≥ , which is defined as follows (see the appendix of [21] for moredetails). Let ( L ( 𝑖 ) 𝑡 ) 𝑡 ≥ be a version of local time at the point ( , 𝑖 ) for the strong Markov process ( 𝜉 𝑡 − 𝜉 𝑡 , 𝐽 𝑡 ) 𝑡 ≥ , where 𝜉 𝑡 ≔ sup 𝑠 ≤ 𝑡 𝜉 𝑠 . Define then L 𝑡 ≔ Í 𝑛𝑖 = L ( 𝑖 ) 𝑡 , which is a continuous additivefunctional of ( 𝜉 𝑡 − 𝜉 𝑡 , 𝐽 𝑡 ) 𝑡 ≥ , increasing almost surely on the set of times when 𝜉 attains a newmaximum.With this at hand we define the ladder height process ( 𝐻 + , 𝐽 + ) by the time change (cid:0) 𝐻 + 𝑡 , 𝐽 + 𝑡 (cid:1) = (cid:26) (cid:0) 𝜉 L − 𝑡 , 𝐽 L − 𝑡 (cid:1) , ≤ 𝑡 < L ∞ , 𝜗 = (∞ , 𝜛 ) , 𝑡 ≥ L ∞ , where L − 𝑡 ≔ inf { 𝑠 ≥ L 𝑠 > 𝑡 } is the right-continuous inverse of L . It can be shown that ( 𝐻 + , 𝐽 + ) is a Markov additive subordinator with lifetime L ∞ , i.e. a Markov additive process suchthat the ordinator 𝐻 + has increasing paths almost surely before killing. Moreover, ( L − 𝑡 ) ≤ 𝑡 < ∞ almost surely equals the ordered set of times, when 𝜉 reaches a maximum and hence the closureof the range of 𝐻 + up to its lifetime is identical to that of the supremum process 𝜉 almost surely.Denote by 𝐻 + , ( 𝑖 ) the Lévy subordinators appearing in the decomposition of ( 𝐻 + , 𝐽 + ) in the spiritof Proposition 2.1. The respective drifts and Lévy measures are denoted by 𝑑 + 𝑖 and Π + 𝑖 , theintensity matrix of 𝐽 + by Q + = ( 𝑞 + 𝑖,𝑗 ) 𝑖,𝑗 ∈ Θ and the killing rates of 𝐻 + , ( 𝑖 ) by † + 𝑖 , i.e., when † + 𝑖 > 𝜁 + 𝑖 of 𝐻 + , ( 𝑖 ) is exponentially distributed with mean 1 /† + 𝑖 and otherwise, for † + 𝑖 = 𝜁 + 𝑖 = ∞ almost surely. Note that the MAP subordinator ( 𝐻 + , 𝐽 + ) is then uniquely characterizedby its Laplace exponent, given as follows: 𝚽 + ( 𝜃 ) ≔ diag (cid:0) Φ + ( 𝜃 ) , . . . , Φ + 𝑛 ( 𝜃 ) (cid:1) − 𝑸 + ⊙ 𝑮 + ( 𝜃 ) , 𝜃 ≥ , (2.4)6here Φ + 𝑖 is the Laplace exponent of 𝐻 + , ( 𝑖 ) and 𝑮 + ( 𝜃 ) = ( 𝐺 + 𝑖,𝑗 ( 𝜃 )) 𝑖,𝑗 ∈ Θ = ( 𝔼 [ exp (− 𝜃 Δ + 𝑖,𝑗 )]) 𝑖,𝑗 ∈ Θ .It then holds that 𝔼 ,𝑖 (cid:2) exp (− 𝜃 𝐻 + 𝑡 ) ; 𝐽 + 𝑡 = 𝑗 (cid:3) = (cid:0) e − 𝚽 + ( 𝜃 ) 𝑡 (cid:1) 𝑖,𝑗 , 𝑡 ≥ , 𝜃 ≥ , 𝑖, 𝑗 ∈ Θ . Let us also denote the family of potential measures of ( 𝐻 + , 𝐽 + ) by ( 𝑈 + 𝑖,𝑗 ) 𝑖,𝑗 ∈ Θ . In analogy to the case for Lévy processes we also need the ascending ladder height process ofthe dual of the MAP ( 𝜉 , 𝐽 ) , i.e. a MAP which has the same law as the time reversed MAP ( 𝜉 , 𝐽 ) .As remarked in [21] the construction of the dual MAP is slightly more elaborate comparedto the Lévy case, where the dual process is simply the negative of the original Lévy process,because we have to take care of time reversion of the ordinator 𝐽 . Suppose that 𝐽 is irreducible– and hence ergodic thanks to its finite state space – and denote its invariant distribution by 𝝅 = ( 𝜋 ( 𝑖 )) 𝑖 ∈ Θ . Moreover, let b 𝑞 𝑖,𝑗 = 𝜋 ( 𝑗 ) 𝜋 ( 𝑖 ) 𝑞 𝑗,𝑖 , 𝑖, 𝑗 ∈ Θ , which are the intensities of the time reversed modulating Markov chain 𝐽 and let b 𝑸 = ( b 𝑞 𝑖,𝑗 ) 𝑖,𝑗 ∈ Θ .Now let ( b ℙ 𝑥,𝑖 ) ( 𝑥,𝑖 )∈ ℝ × Θ be a family of probability measures such that ( 𝜉 , 𝐽 ) has characteristicmatrix exponent given by b 𝚿 ( 𝜃 ) = (cid:0)b 𝔼 ,𝑖 (cid:2) exp ( i 𝜃𝜉 ) ; 𝐽 = 𝑗 (cid:3) (cid:1) 𝑖,𝑗 ∈ Θ = diag ( 𝜓 (− 𝜃 ) , . . . , 𝜓 𝑛 (− 𝜃 )) + b 𝑸 ⊙ 𝑮 (− 𝜃 ) ⊤ , 𝜃 ∈ ℝ . Then indeed, under ℙ , 𝝅 ≔ Í 𝑛𝑖 = 𝜋 ( 𝑖 ) ℙ ,𝑖 , it holds that the time reversed process ( 𝜉 ( 𝑡 − 𝑠 )− − 𝜉 𝑡 , 𝐽 ( 𝑡 − 𝑠 )− ) ≤ 𝑠 ≤ 𝑡 is equal in law to ( 𝜉 𝑠 , 𝐽 𝑠 ) 𝑠 ≤ 𝑡 under b ℙ , 𝝅 , see Lemma 21 in [21]. Let 𝚫 𝝅 ≔ diag ( 𝝅 ) and denote the matrix Laplace exponent of the ascending ladder height process of the dualprocess of ( 𝜉 , 𝐽 ) by b 𝚽 + and also the objects belonging to its Lévy system in the obvious way. The key result for fluctuation theory of MAPs is the (spatial) Wiener–Hopf factorization givenin Theorem 26 of [21], which states that up to pre-multiplication by a positive diagonal matrixcorresponding to the scaling of local time at the supremum, − 𝚿 ( 𝜃 ) = 𝚫 − 𝝅 b 𝚽 + ( i 𝜃 ) ⊤ 𝚫 𝝅 𝚽 + (− i 𝜃 ) = 𝚫 − 𝝅 b 𝚿 + (− 𝜃 ) ⊤ 𝚫 𝝅 𝚿 + ( 𝜃 ) , 𝜃 ∈ ℝ , (2.5)and thus gives a factorization of the characteristic matrix exponent 𝚿 of ( 𝜉 , 𝐽 ) in terms of thecharacteristic exponents 𝚿 + and b 𝚿 + of the ascending ladder height processes of ( 𝜉 , 𝐽 ) and itsdual, respectively. This identity is the key for understanding the interplay between the parentMAP 𝜉 and the ladder height processes, which we will further explore in Section 4.
3. Stability analysis of overshoots of MAPs
In this section, we assume that the lifetime 𝜁 of ( 𝜉 , 𝐽 ) is equal to ∞ on all of Ω . For 𝑡 ≥ 𝜉 first hitting time 𝑇 𝑡 of the set ( 𝑡 , ∞) by 𝑇 𝑡 ≔ inf { 𝑠 ≥ 𝜉 𝑠 > 𝑡 } . Note that by right-continuous paths of the process and right-continuity of the filtration ( F 𝑡 ) 𝑡 ≥ underlying ( 𝜉 , 𝐽 ) this is a stopping time for the MAP. Set also 𝜉 ∞ ≔ sup ≤ 𝑡 < ∞ 𝜉 𝑡 . A word of caution at this point: b 𝚽 + is not the matrix exponent of the dual of the ascending ladder height MAP ( 𝐻 + , 𝐽 + ) . To not confuse the reader we will therefore withhold the temptation to denote the ascending ladderheight process of the dual of ( 𝜉 , 𝐽 ) by ( b 𝐻 + , b 𝐽 + ) .
7e now define the process ( O 𝑡 , J 𝑡 ) 𝑡 ≥ by ( O 𝑡 , J 𝑡 ) = (cid:26) (cid:0) 𝜉 𝑇 𝑡 − 𝑡 , 𝐽 𝑇 𝑡 ) , if 𝑡 < 𝜉 ∞ , 𝜗 , if 𝑡 ≥ 𝜉 ∞ , 𝑡 ≥ , i.e. if the level 𝑡 is smaller than the supremum of the process over its entire lifetime, then O 𝑡 corresponds to the overshoot of 𝜉 over 𝑡 and J 𝑡 is equal to the state of the modulator at firstpassage of 𝑡 , whereas for 𝑡 ≥ 𝜉 ∞ the process is sent to the cemetery state 𝜗 . An essentialobservation for our analysis is that ( O 𝑡 , J 𝑡 ) 𝑡 ≥ is indistinguishable with respect to the familyof probability measures ( ℙ 𝑥,𝑖 ) ( 𝑥,𝑖 )∈( ℝ + × Θ ) 𝜗 from the process ( O + 𝑡 , J + 𝑡 ) 𝑡 ≥ corresponding to theascending ladder MAP ( 𝐻 + , 𝐽 + ) , and hence is given by ( O + 𝑡 , J + 𝑡 ) = ( (cid:0) 𝐻 + 𝑇 + 𝑡 − 𝑡 , 𝐽 + 𝑇 + 𝑡 ) , if 𝑡 < 𝐻 +∞ , 𝜗 , if 𝑡 ≥ 𝐻 +∞ , 𝑡 ≥ , where ( 𝑇 + 𝑡 ) 𝑡 ≥ is the first passage process of 𝐻 + , which by increasing paths of 𝐻 + is equal toits right-continuous inverse. Indistinguishability of the processes follows immediately fromthe fact that on [ , L ∞ ) , the range of the increasing process ( L − 𝑡 ) 𝑡 ≥ almost surely equals theset of times when 𝜉 reaches a maximum. Using this relationship, (2.3) and arguing as in theclassical proof for the law of the undershoot/overshoot distribution for Lévy processes (see [38,Theorem 5.6]), we obtain the following formula for the marginal distribution of the overshootprocess ℙ 𝑥,𝑖 ( O 𝑡 ∈ d 𝑦, J 𝑡 = 𝑗 ) = ℙ ,𝑖 ( O + 𝑡 − 𝑥 ∈ d 𝑦, J + 𝑡 = 𝑗 ) = ∫ [ ,𝑡 − 𝑥 ) Π + 𝑗 ( 𝑢 + d 𝑦 ) 𝑈 + 𝑖,𝑗 ( 𝑡 − 𝑥 − d 𝑢 )+ Õ 𝑘 ≠ 𝑗 𝑞 + 𝑘,𝑗 ∫ [ ,𝑡 − 𝑥 ) 𝐹 + 𝑘,𝑗 ( 𝑢 + d 𝑦 ) 𝑈 + 𝑖,𝑘 ( 𝑡 − 𝑥 − d 𝑢 ) , 𝑖, 𝑗 ∈ Θ , 𝑥 ∈ [ , 𝑡 ) , 𝑦 ≥ , (3.1)and 𝔼 𝑥,𝑖 [ 𝑓 ( O 𝑡 , J 𝑡 )] = 𝑓 ( 𝑥 − 𝑡 , 𝑖 ) , 𝑥 ∈ [ 𝑡 , ∞) , 𝑖 ∈ Θ , 𝑦 ≥ , (3.2)provided that ℙ ,𝑖 ( 𝑇 + = ) =
1. Assumption ( 𝒜
0) introduced below will ensure this property.Equation (3.2) describes the characteristic behavior of the overshoot process away from 0 inthe sense that if O 𝑡 ( 𝜔 ) = 𝑦 > O 𝑠 ( 𝜔 ) = 𝑦 − ( 𝑠 − 𝑡 ) for 𝑠 ∈ [ 𝑡 , 𝑡 + 𝑦 ] , i.e. the originis approached at unit speed. This characteristic path structure of the overshoot process isvisualized in Figure 3.1 for the case of a compound Poisson subordinator 𝜎 with positive drift,and is the reason why for such Lévy subordinators the overshoot process is also known assawtooth process, cf. Chapter II.3 in [13]. We will therefore also refer to it as the sawtoothstructure for MAP overshoots.Let G 𝑡 ≔ F 𝑇 𝑡 for 𝑡 ≥ 𝔾 ≔ ( G 𝑡 ) 𝑡 ≥ . The following technical resultshold. Lemma 3.1. 𝔾 is right-continuous.Proof. First note that F 𝑇 𝑡 = F 𝑇 𝑡 + , with F 𝑇 𝑡 + ≔ (cid:8) Λ ∈ F : Λ ∩ { 𝑇 𝑡 < 𝑠 } ∈ F 𝑠 for all 𝑠 ≥ (cid:9) since the latter can be shown to be equal to (cid:8) Λ ∈ F : Λ ∩ { 𝑇 𝑡 ≤ 𝑠 } ∈ F 𝑠 + for all 𝑠 ≥ (cid:9) , O 𝜎 𝑡 𝜎 𝑠 , 𝑡 𝑇 𝜎 𝑡 O 𝜎 𝑡 = Δ 𝜎 𝑇 𝜎 𝑡 𝑇 𝜎 𝑡 O 𝜎 𝑡 = Δ 𝜎 𝑇 𝜎 𝑡 𝑇 𝜎 𝑡 O 𝜎 𝑡 = Δ 𝜎 𝑇 𝜎 𝑡 ( 𝜎 𝑠 ) 𝑠 ≥ 𝑇 𝜎 𝑡 O 𝜎 𝑡 = Δ 𝜎 𝑇 𝜎 𝑡 𝜎 𝑇 𝜎 𝑡 − = 𝑡 𝜎 𝑇 𝜎 𝑡 − = 𝑡 𝜎 𝑇 𝜎 𝑡 − = 𝑡 𝜎 𝑇 𝜎 𝑡 − = 𝑡 ( O 𝜎 𝑡 = 𝜎 𝑇 𝜎 𝑡 − 𝑡 ) 𝑡 ≥ Figure 3.1: Path of a compound Poisson subordinator with drift, 𝜎 , and associated overshootprocess O 𝜎 which in turn equals F 𝑇 𝑡 thanks to right-continuity of 𝔽 . Letting Λ ∈ G 𝑡 + = Ñ 𝑛 ∈ ℕ F 𝑇 𝑡 + / 𝑛 weobtain by right-continuity of 𝑡 ↦→ 𝑇 𝑡 that for any 𝑠 ≥ Λ ∩ { 𝑇 𝑡 < 𝑠 } = Ø 𝑛 ∈ ℕ Λ ∩ (cid:8) 𝑇 𝑡 + 𝑛 < 𝑠 (cid:9) ∈ G 𝑠 , since any set in the right-hand union belongs to G 𝑠 thanks to F 𝑇 𝑡 + / 𝑛 = F 𝑇 ( 𝑡 + / 𝑛 ) + . It follows that Λ ∈ F 𝑇 𝑡 + = F 𝑇 𝑡 = G 𝑡 , which proves right-continuity of 𝔾 . (cid:4) Corollary 3.2.
For any ≤ 𝑠 ≤ ∞ the running supremum 𝜉 𝑠 is a stopping time with respect to 𝔾 . Inparticular, the lifetime 𝜉 ∞ of ( O 𝑡 , J 𝑡 ) 𝑡 ≥ is a 𝔾 -stopping time.Proof. Let 𝑠 ∈ [ , ∞] . For any 𝑡 ≥ { 𝜉 𝑠 < 𝑡 } = { 𝑇 𝑡 > 𝑠 } ∈ F 𝑇 𝑡 = G 𝑡 , which implies { 𝜉 𝑠 ≤ 𝑡 } ∈ G 𝑡 + and since G 𝑡 + = G 𝑡 by Lemma 3.1 we conclude { 𝜉 𝑠 ≤ 𝑡 } ∈ G 𝑡 . (cid:4) We now show that under a technical assumption, the overshoot process given by the quintuple ( Ω , F , 𝔾 , ( O 𝑡 , J 𝑡 ) 𝑡 ≥ , ( ℙ 𝑥,𝑖 ) ( 𝑥,𝑖 )∈( ℝ + × Θ ) 𝜗 ) determines a Feller process and therefore also a Borelright process. The technical assumption under which we will be working throughout the restof the paper without further mention, is the following.( 𝒜
0) The MAP ( 𝜉 , 𝐽 ) is upward regular, i.e. for any 𝑖 ∈ Θ it holds that ℙ ,𝑖 ( 𝑇 = ) = ( 𝜉 , 𝐽 ) is upward regular if, independently of the starting point of the modulator 𝐽 , 𝜉 started from 0 immediately hits the upper half line. By the path decomposition given inProposition 2.1, this is the case if and only if the underlying Lévy processes 𝜉 ( 𝑖 ) are regularupward for any 𝑖 ∈ Θ . Upward regularity for Lévy processes is completely understood, seethe full characterization given in Theorem 6.5 of [38], and hence upward regularity of the MAPcan be characterized by properties of its underlying Lévy processes. Moreover, by the generaltheory on local times of Markov processes, see e.g. Chapter 4 in Bertoin [7] or the landmark9aper Blumenthal and Getoor [11], it follows that upward regularity implies that for each 𝑖 ∈ Θ ,the local time L ( 𝑖 ) of ( 𝜉 − 𝜉 , 𝐽 ) at ( , 𝑖 ) is almost surely continuous and hence L = Í 𝑛𝑖 = L ( 𝑖 ) isalmost surely continuous as well. Hence, the right-continuous inverse ( L − 𝑡 ) 𝑡 ≥ , correspondingto the set of times when a new maximum of 𝜉 is reached, is strictly increasing on [ , L ∞ ) almostsurely and it follows that 𝐻 + is strictly increasing up to its lifetime. This property is essentialfor ( O , J ) being a Feller process, as the proof of the following proposition shows. Proposition 3.3. ( O , J ) is a càdlàg Feller process with lifetime 𝜉 ∞ .Proof. Càdlàg paths of the process are a direct consequence of càdlàg paths of ( 𝜉 , 𝐽 ) and the factthat 𝑡 ↦→ 𝑇 𝑡 is right-continuous on [ , ∞) and increasing on [ , 𝜉 ∞ ) . Let now 𝑓 ∈ B 𝑏 (( ℝ + × Θ ) 𝜗 ) and ( 𝑥, 𝑖 ) ∈ ( ℝ + × Θ ) 𝜗 . Recalling that 𝜉 ∞ is a 𝔾 -stopping time and using 𝑇 𝑡 + 𝑠 = 𝑇 𝑡 + 𝑇 𝑡 + 𝑠 ◦ 𝜃 𝑇 𝑡 , on { 𝑇 𝑡 < ∞} , where ( 𝜃 𝑡 ) 𝑡 ≥ are the transition opertors of ( 𝜉 , 𝐽 ) , it follows that ℙ 𝑥,𝑖 -a.s. 𝔼 𝑥,𝑖 [ 𝑓 ( O 𝑡 + 𝑠 , J 𝑡 + 𝑠 )| G 𝑡 ] = 𝔼 𝑥,𝑖 (cid:2) 𝑓 (cid:0) 𝜉 𝑇 𝑡 + 𝑠 − ( 𝑡 + 𝑠 ) , 𝐽 𝑇 𝑡 + 𝑠 (cid:1) ◦ 𝜃 𝑇 𝑡 | F 𝑇 𝑡 (cid:3) { 𝑡 <𝜉 ∞ } + 𝑓 ( 𝜗 ) { 𝑡 ≥ 𝜉 ∞ } = 𝔼 𝜉 𝑇𝑡 ,𝐽 𝑇𝑡 (cid:2) 𝑓 (cid:0) 𝜉 𝑇 𝑡 + 𝑠 − ( 𝑡 + 𝑠 ) , 𝐽 𝑇 𝑡 + 𝑠 (cid:1) (cid:3) { 𝑡 <𝜉 ∞ } + 𝑓 ( 𝜗 ) { 𝑡 ≥ 𝜉 ∞ } = 𝔼 𝜉 𝑇𝑡 − 𝑡,𝐽 𝑇𝑡 (cid:2) 𝑓 (cid:0) 𝜉 𝑇 𝑠 − 𝑠 , 𝐽 𝑇 𝑠 (cid:1) (cid:3) { 𝑡 <𝜉 ∞ } + 𝑓 ( 𝜗 ) { 𝑡 ≥ 𝜉 ∞ } = 𝔼 O 𝑡 , J 𝑡 (cid:2) 𝑓 (cid:0) O 𝑠 , J 𝑠 (cid:1) (cid:3) . Here, we used the strong Markov property of ( 𝜉 , 𝐽 ) for the second and spatial homogeneity of 𝜉 for the third equality. This proves the Markov property of ( O , J ) . Moreover, for 𝑥 > 𝑖 ∈ Θ we have ℙ 𝑥,𝑖 ( 𝑇 = ) = 𝜉 we also have ℙ ,𝑖 ( 𝑇 = ) =
1. Thus, ℙ 𝑥,𝑖 ( O , J ) = ( 𝑥, 𝑖 ) for any ( 𝑥, 𝑖 ) ∈ ( ℝ + × Θ ) 𝜗 , i.e. the process is a normal Markov process andits lifetime is given by 𝜉 ∞ by construction. Let ( P 𝑡 ) 𝑡 ≥ be its sub-Markov transition semigroup,i.e. P 𝑡 𝑓 ( 𝑥, 𝑖 ) = 𝔼 𝑥,𝑖 [ 𝑓 ( O 𝑡 , J 𝑡 ) ; 𝑡 < 𝜉 ∞ ] , ( 𝑥, 𝑖 ) ∈ ( ℝ + × Θ ) 𝜗 , 𝑓 ∈ B 𝑏 (( ℝ + × Θ ) 𝜗 ) . Let us check the Feller property. Let 𝑓 ∈ C ( ℝ + × Θ ) . Since Θ is finite and recalling ourconvention that 𝑓 ( 𝜗 ) =
0, it suffices to show for fixed 𝑖 ∈ Θ that 𝑥 ↦→ P 𝑡 𝑓 ( 𝑥, 𝑖 ) = 𝔼 𝑥,𝑖 [ 𝑓 ( O 𝑡 , J 𝑡 )] iscontinuous to prove that ( 𝑥, 𝑖 ) ↦→ 𝔼 𝑥,𝑖 [ 𝑓 ( O 𝑡 , J 𝑡 )] is continuous. If 𝑥 > 𝑡 this is obvious. For 𝑥 ≤ 𝑡 let first 𝑦 ↑ 𝑥 . By right-continuity of 𝑡 ↦→ ( O 𝑡 , J 𝑡 ) , continuity and boundedness of 𝑓 , dominatedconvergence and conditional spatial homogeneity of ( 𝜉 , 𝐽 ) , it follows thatlim 𝑦 ↑ 𝑥 𝔼 𝑦,𝑖 [ 𝑓 ( O 𝑡 , J 𝑡 )] = lim 𝑦 ↑ 𝑥 𝔼 ,𝑖 [ 𝑓 ( O 𝑡 − 𝑦 , J 𝑡 − 𝑦 )] = 𝔼 ,𝑖 [ 𝑓 ( O 𝑡 − 𝑥 , J 𝑡 − 𝑥 )] = 𝔼 𝑥,𝑖 [ 𝑓 ( O 𝑡 , J 𝑡 )] , showing left-continuity of 𝑥 ↦→ 𝔼 𝑥,𝑖 [ 𝑓 ( O 𝑡 , J 𝑡 )] . To show right-continuity, note that for 𝑦 ↓ 𝑥 itholds that 𝑇 + 𝑡 − 𝑦 increases to inf { 𝑠 ≥ 𝐻 + 𝑠 ≥ 𝑡 − 𝑥 } on { 𝑇 + 𝑡 − 𝑥 < ∞} and since 𝐻 + is strictlyincreasing up to its lifetime by upward regularity of 𝜉 , it follows that the latter hitting time isalmost surely equal to 𝑇 + 𝑡 − 𝑥 . Since ( 𝐻 + , 𝐽 + ) as a Feller process is quasi-left-continuous, it thereforefollows that on { 𝑇 + 𝑡 − 𝑥 < ∞} ,lim 𝑦 ↓ 𝑥 (cid:16) 𝐻 + 𝑇 + 𝑡 − 𝑦 , 𝐽 + 𝑇 + 𝑡 − 𝑦 (cid:17) = (cid:16) 𝐻 + 𝑇 + 𝑡 − 𝑥 , 𝐽 + 𝑇 + 𝑡 − 𝑥 (cid:17) , ℙ ,𝑖 -a.s.By indistinguishability of ( O + , J + ) and ( O , J ) we therefore obtainlim 𝑦 ↓ 𝑥 𝔼 𝑦,𝑖 [ 𝑓 ( O 𝑡 , J 𝑡 )] = lim 𝑦 ↓ 𝑥 𝔼 ,𝑖 [ 𝑓 ( O + 𝑡 − 𝑦 , J + 𝑡 − 𝑦 )] = 𝔼 ,𝑖 [ 𝑓 ( O + 𝑡 − 𝑥 , J + 𝑡 − 𝑥 )] = 𝔼 𝑥,𝑖 [ 𝑓 ( O 𝑡 , J 𝑡 )] , proving also right-continuity of ( 𝑥, 𝑖 ) ↦→ P 𝑡 𝑓 ( 𝑥, 𝑖 ) . Since moreover Θ is compact and for fixed 𝑖 ∈ Θ , lim 𝑥 →∞ P 𝑡 𝑓 ( 𝑥, 𝑖 ) = lim 𝑥 →∞ 𝑓 ( 𝑥 − 𝑡 , 𝑖 ) = 𝑓 ∈ C ( ℝ + × Θ ) , we conclude that P 𝑡 C ( ℝ + × Θ ) ⊂ C ( ℝ + × Θ ) . Finally, for fixed ( 𝑥, 𝑖 ) ∈ ℝ + × Θ (again applying to upward regularity in case 𝑥 =
0) it follows from 𝑇 𝑡 → 𝑡 ↓ P 𝑡 𝑓 ( 𝑥, 𝑖 ) → P 𝑓 ( 𝑥, 𝑖 ) = 𝑓 ( 𝑥, 𝑖 ) . This is enough to showthat ( P 𝑡 ) 𝑡 ≥ is a Feller semigroup, as discussed in Appendix A.It remains to check right-continuity and completeness of 𝔾 . Right-continuity was shown inLemma 3.1. Moreover, it can be easily seen that the ℙ 𝑥,𝑖 -augmentation of F 𝑇 𝑡 is equal to F 𝑇 𝑡 itself, since 𝔽 is ℙ 𝑥,𝑖 -augmented already, see also p.36 of [12]. This finishes the proof. (cid:4) Having established the Markovian nature of the overshoot process, we now proceed byinvestigating its stability properties and long-time behavior. We must therefore restrict to thecase, when the overshoot process is almost surely unkilled, which is the case if and only ifsup ≤ 𝑠 < ∞ 𝜉 𝑠 = ∞ , ℙ ,𝑖 -a.s. for all 𝑖 ∈ Θ . As for Lévy processes, there is a dichotomy concerningthe long-time behavior of the ordinator 𝜉 , namely that exactly one of the following cases canoccur:(a) for any ( 𝑥, 𝑖 ) ∈ ℝ × Θ , lim sup 𝑡 →∞ 𝜉 𝑡 = ∞ , ℙ 𝑥,𝑖 -almost surely, and in this case eitherlim inf 𝑡 →∞ 𝜉 𝑡 = −∞ or lim 𝑡 →∞ 𝜉 𝑡 = ∞ , ℙ 𝑥,𝑖 -a.s.;(b) for any ( 𝑥, 𝑖 ) ∈ ℝ × Θ , lim 𝑡 →∞ 𝜉 𝑡 = −∞ , ℙ 𝑥,𝑖 -almost surely.When 𝐽 is irreducible and the MAP’s ordinator possesses an exponential moment, which ofthese cases occurs for a given MAP is determined by a Perron–Frobenius type eigenvalue of theMAP’s Laplace exponent, see Asmussen [2, Proposition XII.2.10]. We will therefore henceforthwork under the following additional assumption, which guarantees that ( O , J ) is an unkilledBorel right Markov process and therefore gives us access to the theory of stability for Markovprocesses teased in Appendix A.( 𝒜
1) For any ( 𝑥, 𝑖 ) ∈ ( ℝ × Θ ) it ℙ 𝑥,𝑖 -almost surely holds lim sup 𝑡 →∞ 𝜉 𝑡 = ∞ . Let us give the following definition.
Definition 3.4.
Let 𝑨 = ( 𝑎 𝑖,𝑗 ) 𝑖,𝑗 = ,...,𝑛 ∈ ℝ 𝑛 × 𝑛 be a matrix with 𝑎 𝑖,𝑗 ≥ 𝑖 ≠ 𝑗 . We say that 𝑨 is irreducible , if for any 𝑖 ≠ 𝑗 there exists ( 𝑎 𝑖 𝑘 ,𝑖 𝑘 + ) 𝑘 = ,...,𝑚 − for some 𝑚 ∈ ℕ with 𝑖 = 𝑖, 𝑖 𝑚 = 𝑗 suchthat Î 𝑚 − 𝑘 = 𝑎 𝑖 𝑘 ,𝑖 𝑘 + > . A matrix e 𝑨 = ( e 𝑎 𝑖,𝑗 ) 𝑖,𝑗 = ,...,𝑛 such that diag ( 𝑨 ) = diag ( e 𝑨 ) and e 𝑎 𝑖,𝑗 ∈ { 𝑎 𝑖,𝑗 , } for any 𝑖 ≠ 𝑗 is said to be a minimal irreducible version of an irreducible matrix 𝑨 , if any matrixobtained from e 𝐴 by setting some off-diagonal element to 0 is not irreducible anymore.If we visualize a matrix 𝑨 as in the definition above as a directed graph with vertices 𝑉 = { , . . . , 𝑛 } representing the on-diagonal elements of 𝑨 and edges 𝐸 = {( 𝑖, 𝑗 ) : 𝑎 𝑖,𝑗 > } representing the non-zero off-diagonal elements of 𝑨 , irreducibility of 𝑨 is equivalent to con-nectedness of the graph of 𝑨 . The graph of a minimal irreducible version e 𝑨 of an irreduciblematrix 𝑨 is therefore a minimal connected subgraph of the graph of 𝑨 with e 𝑉 = 𝑉 and e 𝐸 ⊂ 𝐸 .Also note that a continuous time Markov chain is irreducible if and only if its 𝑄 -matrix isirreducible.As a minimal requirement for stability we need to ensure irreducibility of the Markov process ( O , J ) . We therefore introduce the following assumption.( 𝒜
2) The modulator 𝐽 + of the ascending ladder MAP is irreducible, i.e., 𝑸 + is an irreduciblematrix.For general MAPs irreducibility of 𝐽 does not necessarily entail irreducibility of 𝐽 + , with thelatter property implying that 𝜉 can reach a maximum in any phase of 𝐽 . E.g., if one of the Lévycomponents 𝜉 ( 𝑖 ) is a negative subordinator and Δ 𝑗,𝑖 < 𝑗 ∈ Θ , 𝐽 + is not irreducible since 𝜉 never reaches a new maximum when its phase is 𝑖 . However, the following result showsthat irreducibility of 𝐽 + is given for a wide range of MAPs with irreducible modulator 𝐽 . To11ive one particular example covered by Proposition 3.5 below, suppose that for any 𝑗 ∈ Θ theLévy component 𝜉 ( 𝑗 ) is neither a negative subordinator nor spectrally negative with boundedvariation, or, when this fails for some 𝑗 ∈ Θ this is compensated for by some unboundedtransitional jump of 𝜉 when 𝐽 switches to 𝑗 . Then, 𝐽 + is irreducible whenever 𝐽 is irreducibleand Assumption ( 𝒜
1) is in place. We emphasize that upward regularity ( 𝒜
0) is not neededfor the statement of Proposition 3.5. Recall that for any measure 𝜈 on ( ℝ , B ( ℝ )) , the supportsupp ( 𝜈 ) is defined as the set of points 𝑥 ∈ ℝ such that for any open neighborhood 𝑈 𝑥 of 𝑥 itholds 𝜈 ( 𝑈 𝑥 ) > Proposition 3.5.
Suppose that 𝐽 is irreducible and ( 𝒜 ) holds.(i) Introduce the following conditions for 𝑗 ∈ Θ :( ℋ ( 𝑗 )) 𝜉 ( 𝑗 ) is of unbounded variation or supp ( Π 𝑗 ) ∩ ( , ∞) ≠ ∅ ;( ℐ ( 𝑗 )) there exists 𝑘 ≠ 𝑗 such that supp ( 𝑞 𝑘,𝑗 𝐹 𝑘,𝑗 ) is unbounded from above.Let Λ ≔ { 𝑗 ∈ Θ : ( ℋ ( 𝑗 ) ) or ( ℐ ( 𝑗 ) ) holds } and Λ ≔ { 𝑗 ∈ Θ \ Λ : ∃ 𝑘 ∈ Λ s.t. supp ( 𝑞 𝑘,𝑗 𝐹 𝑘,𝑗 ) ∩ ( , ∞) ≠ ∅} . Then, 𝐽 + is irreducible if Λ ∪ Λ = Θ .(ii) Let e 𝑸 be a minimal irreducible version of 𝑸 . If {( 𝑖, 𝑗 ) ∈ Θ \ {( 𝑖, 𝑖 ) : 𝑖 ∈ Θ } : e 𝑞 𝑖,𝑗 > }⊂ {( 𝑖, 𝑗 ) ∈ Θ \ {( 𝑖, 𝑖 ) : 𝑖 ∈ Θ } : ( ℋ ( 𝑗 ) ) holds or supp ( 𝑞 𝑖,𝑗 𝐹 𝑖,𝑗 ) ∩ ( , ∞) ≠ ∅} , then 𝐽 + is irreducible.Proof. (i) Fix 𝑖, 𝑗 ∈ Θ with 𝑖 ≠ 𝑗 . We have to show that ℙ ,𝑖 ( 𝜏 + ( 𝑗 ) < ∞) >
0, where 𝜏 + ( 𝑗 ) ≔ inf { 𝑡 ≥ 𝐽 + 𝑡 = 𝑗 } . Recall that ( 𝜎 𝑛 ) 𝑛 ∈ ℕ denote the jump times of the modulatingchain 𝐽 . Let 𝑛 ∈ ℕ such that ℙ ,𝑖 ( 𝐽 𝜎 𝑛 = 𝑗 ) >
0, which exists by irreducibility of 𝐽 .Let 𝐺 𝑡 ≔ sup { ≤ 𝑠 < 𝑡 : 𝜉 𝑠 = 𝜉 𝑠 } be the last time before 𝑡 > 𝜉 attains itssupremum. By construction of local time at the supremum L , the range of ( L − 𝑡 ) 𝑡 ≥ almostsurely equals the set of times, when 𝜉 reaches a maximum. Thus, we have ℙ ,𝑖 ( 𝜏 + ( 𝑗 ) < ∞) ≥ ℙ ,𝑖 ( 𝐺 𝜎 𝑛 + ≥ 𝜎 𝑛 , 𝐽 𝜎 𝑛 = 𝑗 )≥ max (cid:8) ℙ ,𝑖 ( 𝜉 𝜎 𝑛 ≥ 𝜉 𝜎 𝑛 − , 𝐽 𝜎 𝑛 = 𝑗 ) , ℙ ,𝑖 ( 𝐺 𝜎 𝑛 + > 𝜎 𝑛 , 𝐽 𝜎 𝑛 = 𝑗 ) (cid:9) . (3.3)Suppose first that ( ℋ ( 𝑗 ) ) holds. By the path decomposition of ( 𝜉 , 𝐽 ) from Proposition 2.1,we obtain ℙ ,𝑖 ( 𝐺 𝜎 𝑛 + > 𝜎 𝑛 , 𝐽 𝜎 𝑛 = 𝑗 ) = ℙ ,𝑖 (cid:0) { 𝐽 𝜎 𝑛 = 𝑗 } ∩ {∃ 𝑡 ∈ ( , 𝜎 𝑛 + − 𝜎 𝑛 ) : 𝜉 𝑡 + 𝜎 𝑛 − 𝜉 𝜎 𝑛 ≥ 𝜉 𝜎 𝑛 − 𝜉 𝜎 𝑛 } (cid:1) = 𝔼 ,𝑖 (cid:2) ℙ ,𝑗 (∃ 𝑡 ∈ ( , 𝜎 ) : 𝜉 𝑡 ≥ 𝑥 )| 𝑥 = 𝜉 𝜎 𝑛 − 𝜉 𝜎 𝑛 ; 𝐽 𝜎 𝑛 = 𝑗 (cid:3) ≥ 𝔼 ,𝑖 (cid:2) ℙ ,𝑗 ( 𝜉 𝜎 / ≥ 𝑥 )| 𝑥 = 𝜉 𝜎 𝑛 − 𝜉 𝜎 𝑛 ; 𝐽 𝜎 𝑛 = 𝑗 (cid:3) = 𝔼 ,𝑖 h (cid:16) ∫ ∞ − 𝑞 𝑗,𝑗 e 𝑞 𝑗,𝑗 𝑡 ℙ ( 𝜉 ( 𝑗 ) 𝑡 ≥ 𝑥 ) d 𝑡 (cid:17)(cid:12)(cid:12)(cid:12) 𝑥 = 𝜉 𝜎 𝑛 − 𝜉 𝜎 𝑛 ; 𝐽 𝜎 𝑛 = 𝑗 i > . (3.4)To argue that the last inequality holds, note that since ( ℋ ( 𝑗 ) ) was assumed, Theorem 24.7in [50] yields that for any 𝑡 >
0, supp ( ℙ ( 𝜉 ( 𝑗 ) 𝑡 ∈ ·)) is not bounded from above. Thus, ℙ ( 𝜉 ( 𝑗 ) 𝑡 ≥ 𝑥 ) > 𝑥 ∈ ℝ and hence (cid:16) ∫ ∞ − 𝑞 𝑗,𝑗 e 𝑞 𝑗,𝑗 𝑡 ℙ ( 𝜉 ( 𝑗 ) 𝑡 ≥ 𝑥 ) d 𝑡 (cid:17)(cid:12)(cid:12)(cid:12) 𝑥 = 𝜉 𝜎 − 𝜉 𝜎 > , ℙ ,𝑖 -a.s. . ℙ ,𝑖 ( 𝐽 𝜎 𝑛 = 𝑗 ) > 𝑛 ∈ ℕ , the inequality follows.Suppose now that ( ℐ ( 𝑗 ) ) holds, i.e., supp ( 𝐹 𝑘,𝑗 ) is unbounded from above for some 𝑘 ≠ 𝑗 s.t. 𝑞 𝑘,𝑗 >
0. Let 𝑚 ∈ ℕ such that ℙ ,𝑖 ( 𝐽 𝜎 𝑚 − = 𝑘, 𝐽 𝜎 𝑚 = 𝑗 ) >
0, which exists by irreducibilityof 𝐽 and 𝑞 𝑘,𝑗 >
0. Then, again by Proposition 2.1, ℙ ,𝑖 ( 𝜉 𝜎 𝑚 ≥ 𝜉 𝜎 𝑚 − , 𝐽 𝜎 𝑚 = 𝑗 ) = Õ 𝑘 ≠ 𝑗 𝔼 ,𝑖 (cid:2) ℙ ( Δ 𝑘,𝑗 ≥ 𝑥 )]| 𝑥 = 𝜉 𝜎 𝑚 − − 𝜉 𝜎 𝑚 − ; 𝐽 𝜎 𝑚 − = 𝑘, 𝐽 𝜎 𝑚 = 𝑗 (cid:3) > , where the inequality follows from ℙ ( Δ 𝑘,𝑗 ≥ 𝑥 )| 𝑥 = 𝜉 𝜎 𝑚 − − 𝜉 𝜎 𝑚 − > , ℙ ,𝑖 -a.s. , thanks to assumed unboundedness of the support of 𝐹 𝑘,𝑗 . We therefore conclude with(3.3) that ℙ ,𝑖 ( 𝜏 + ( 𝑗 ) < ∞) > 𝑗 ∈ Λ . Suppose now that 𝑗 ∈ Λ , i.e., there exists 𝑘 ∈ Λ s.t. supp ( 𝑞 𝑘,𝑗 𝐹 𝑘,𝑗 ) ∩ ( , ∞) ≠ ∅ . Then, by Lemma 4.5, 𝑞 + 𝑘,𝑗 > 𝑘 ∈ Λ , itfollows from above that ℙ ,𝑖 ( 𝜏 + ( 𝑘 ) < ∞) >
0. Combining these observations yields again ℙ ,𝑖 ( 𝜏 + ( 𝑗 ) < ∞) >
0. Thus, the assumption Λ ∪ Λ = Θ implies ℙ ,𝑖 ( 𝜏 + ( 𝑗 ) < ∞) > 𝑗 ≠ 𝑖 , as desired.(ii) Let ( 𝑖, 𝑗 ) ∈ Θ with 𝑖 ≠ 𝑗 s.t. e 𝑞 𝑖,𝑗 >
0. Suppose first that ( ℋ ( 𝑗 ) ) holds. Then, 𝑞 + 𝑖,𝑗 > ℙ ,𝑖 ( 𝐺 𝜎 > 𝜎 , 𝐽 𝜎 = 𝑗 ) >
0. This is an immediate consequence of(3.4) with 𝑛 = 𝑞 𝑖,𝑗 = e 𝑞 𝑖,𝑗 > ℙ ,𝑖 ( 𝐽 𝜎 = 𝑗 ) = − 𝑞 𝑖,𝑗 / 𝑞 𝑖,𝑖 >
0. Suppose nowthat supp ( 𝑞 𝑖,𝑗 𝐹 𝑖,𝑗 ) ∩ ( , ∞) ≠ ∅ . Then, again by Lemma 4.5, 𝑞 + 𝑖,𝑗 > {( 𝑖, 𝑗 ) ∈ Θ \ {( 𝑖, 𝑖 ) : 𝑖 ∈ Θ } : e 𝑞 𝑖,𝑗 > } ⊂ {( 𝑖, 𝑗 ) ∈ Θ \ {( 𝑖, 𝑖 ) : 𝑖 ∈ Θ } : 𝑞 + 𝑖,𝑗 > } , and irreducibility of 𝑸 + follows from irreducibility of 𝑸 . (cid:4) Assume for the rest of this section that ( 𝒜
2) is satisfied and denote by 𝝅 + = ( 𝜋 + ( ) , . . . , 𝜋 + ( 𝑛 )) the invariant distribution of 𝐽 + . Our main goal is to understand the asymptotic behaviorof overshoots. As a natural extension of the well-known limiting distributional behavior ofovershoots of Lévy processes, cf. [6], it is shown in Theorem 28 of [21] that under assumptions( 𝒜
1) and ( 𝒜
2) the overshoot process converges weakly to the limiting distribution 𝜌 ( d 𝑦, { 𝑖 }) ≔ 𝔼 , 𝝅 + [ 𝐻 + ] (cid:18) 𝜋 + ( 𝑖 ) 𝑑 + 𝑖 𝛿 ( d 𝑦 ) + ( , ∞) ( 𝑦 ) (cid:16) 𝜋 + ( 𝑖 ) Π + 𝑖 ( 𝑦 ) + Õ 𝑗 ≠ 𝑖 𝜋 + ( 𝑗 ) 𝑞 + 𝑗,𝑖 𝐹 + 𝑗,𝑖 ( 𝑦 ) (cid:17) d 𝑦 (cid:19) , ( 𝑦, 𝑖 ) ∈ ℝ + × Θ , if and only if 𝔼 , 𝝅 + [ 𝐻 + ] < ∞ . The Feller property of the overshoot processguarantees that in this case 𝜌 is also an invariant measure. We will show that deleting the scalingfactor 𝔼 , 𝝅 + [ 𝐻 + ] − yields the essentially unique invariant measure of the overshoot process andhence a stationary distribution coinciding with 𝜌 exists iff overshoots are tight. Moreover, wewill dig deeper into the mode of convergence, establishing conditions ensuring convergence inthe total variation norm and exponential or polynomial speed of convergence, which also givesnew results for the special case of Lévy process overshoots. Here we made a correction to [21], since in the authors’ statement the limiting distribution of the parents modulator 𝐽 , 𝝅 , appears instead of 𝝅 + . As argued before, irreducibility of 𝐽 does not necessarily imply irreducibility of 𝐽 + and even when 𝐽 + is irreducible, 𝝅 and 𝝅 + are not the same, see [39, Proposition 2.19]. Our analysis will showhowever that stationarity of the ascending ladder height’s modulator and its stationary distribution are decisivefor tight overshoots.
13n analytical tool of central importance to us is the resolvent of the overshoot process.Let ( P 𝑡 ) 𝑡 ≥ be the transition function of ( O , J ) defined by P 𝑡 𝑓 ( 𝑥, 𝑖 ) = 𝔼 𝑥,𝑖 [ 𝑓 ( O 𝑡 , J 𝑡 )] for any 𝑓 ∈ B 𝑏 ( ℝ + × Θ ) ∪ B + ( ℝ + × Θ ) and ( U 𝜆 ) 𝜆> be the associated resolvent given by U 𝜆 𝑓 ( 𝑥, 𝑖 ) = ∫ ∞ e − 𝜆 𝑡 P 𝑡 𝑓 ( 𝑥, 𝑖 ) d 𝑡 , for any 𝜆 >
0. Our proof for the explicit formula of the resolvent is close in spirit to the prooffor the overshoot process of a Lévy subordinator in Blumenthal [13], which in turn is a specialcase of a general result by It ¯o for Markov processes possessing a local time at a specific point ofthe state space, see [28, Theorem 2.5.5]. The detailed proof is quite long and can be found inAppendix B.
Theorem 3.6.
For any 𝑓 ∈ B + ( ℝ + × Θ ) ∪ B 𝑏 ( ℝ + × Θ ) and 𝑥 ∈ ℝ + it holds that ( U 𝜆 𝑓 ( 𝑥, 𝑖 )) ⊤ 𝑖 = ,...,𝑛 = ( 𝑄 𝜆 𝑓 ( 𝑥, 𝑖 )) ⊤ 𝑖 = ,...𝑛 + e − 𝜆 𝑥 𝚽 + ( 𝜆 ) − · 𝝍 ( 𝑓 , 𝜆 ) , (3.5) where 𝝍 ( 𝑓 , 𝜆 ) = © « 𝑑 + 𝑖 𝑓 ( , 𝑖 ) + ∫ ∞ 𝑄 𝜆 𝑓 ( 𝑥, 𝑖 ) Π + 𝑖 ( d 𝑥 ) + Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 𝔼 [ 𝑄 𝜆 𝑓 ( Δ + 𝑖,𝑗 , 𝑗 )] ª®¬ ⊤ 𝑖 = ,...,𝑛 and 𝑄 𝜆 𝑓 ( 𝑥, 𝑖 ) = ∫ 𝑥 e − 𝜆 𝑡 𝑓 ( 𝑥 − 𝑡 , 𝑖 ) d 𝑡 , ( 𝑥, 𝑖 ) ∈ ℝ + × Θ . The resolvent formula has far reaching consequences for understanding the behavior of theMAP at first passage. A first neat observation is the strong Feller property of the resolventoperator, which implies that ( O , J ) is a 𝑇 -process. Corollary 3.7.
For any 𝜆 > the resolvent U 𝜆 has the strong Feller property. In particular the overshootprocess ( O , J ) is a 𝑇 -process.Proof. Let 𝜆 > 𝑓 ∈ B 𝑏 ( ℝ + × Θ ) . Since we can write 𝑄 𝜆 𝑓 ( 𝑥, 𝑖 ) = e − 𝜆 𝑥 ∫ 𝑥 e 𝜆 𝑡 𝑓 ( 𝑡 , 𝑖 ) d 𝑡 , itfollows that ( 𝑥, 𝑖 ) ↦→ 𝑄 𝜆 𝑓 ( 𝑥, 𝑖 ) is continuous and hence ( 𝑥, 𝑖 ) ↦→ U 𝜆 𝑓 ( 𝑥, 𝑖 ) is clearly continuous.Moreover, U 𝜆 𝑓 is bounded and thus, U 𝜆 B 𝑏 ( ℝ + × Θ ) ⊂ C 𝑏 ( ℝ + × Θ ) follows, i.e. U 𝜆 has the strongFeller property. Hence, the resolvent kernel R 𝜆 ≔ 𝜆 U 𝜆 is a continuous component for itself,implying that ( O , J ) is a 𝑇 -process. (cid:4) We will also use the resolvent formula combined with Proposition A.1 to determine aninvariant measure for the overshoot process. To show its essential uniqueness, we need toestablish Harris recurrence first, which is taken care of in the following proposition.
Proposition 3.8.
The overshoot process ( O , J ) is Harris recurrent.Proof. Let 𝑗 ∈ Θ be arbitrarily chosen and let 𝜇 ≔ 𝛿 ⊗ 𝛿 𝑗 . Fix ( 𝑥, 𝑖 ) ∈ ℝ + × Θ and let 𝐵 ∈ B ( ℝ + × Θ ) such that 𝜇 ( 𝐵 ) >
0, i.e. { } × { 𝑗 } ∈ 𝐵 . Since 𝐽 + is irreducible and 𝑡 ↦→ 𝑇 + 𝑡 is continuous andincreases to ∞ as 𝑡 → ∞ , it follows that ℙ 𝑥,𝑖 ( 𝔱 + ( 𝑗 ) < ∞) >
0, where 𝔱 + ( 𝑗 ) ≔ inf { 𝑡 > J + 𝑡 = 𝑗 } isthe first hitting time of { 𝑗 } of J + . Let 𝑇 Λ = inf { 𝑡 ≥ ( O 𝑡 , J 𝑡 ) ∈ Λ } be the first hitting time ofa set Λ ∈ ℝ + × Θ by ( O , J ) and denote by 𝑇 + Λ the first hitting time of ( O + , J + ) . By the sawtoothstructure of O + we have 𝑇 +{ }×{ 𝑗 } = 𝑥 , ℙ 𝑥,𝑗 -a.s.. Since 𝔱 + ( 𝑗 ) ≤ 𝑇 +{ }×{ 𝑗 } it therefore follows by thestrong Markov property of ( O + , J + ) that ℙ 𝑥,𝑖 ( 𝑇 𝐵 < ∞) ≥ ℙ 𝑥,𝑖 (cid:0) 𝑇 { }×{ 𝑗 } < ∞ (cid:1) = ℙ 𝑥,𝑖 (cid:0) 𝑇 +{ }×{ 𝑗 } < ∞ (cid:1) = 𝔼 𝑥,𝑖 h ℙ O + 𝔱 +( 𝑗 ) , J + 𝔱 +( 𝑗 ) (cid:0) 𝑇 +{ }×{ 𝑗 } < ∞ (cid:1) { 𝔱 + ( 𝑗 ) < ∞} i ℙ 𝑥,𝑖 ( 𝔱 + ( 𝑗 ) < ∞) > , where we used for the last equality that J + 𝔱 + ( 𝑗 ) = 𝑗 and O + 𝔱 + ( 𝑗 ) < ∞ almost surely. It now followsfrom Proposition 2.1 in [44] that ( O , J ) is irreducible with irreducibility measure R 𝜇 ( d 𝑦 ) ≔ ∫ ℝ + × Θ R ( 𝑥, d 𝑦 ) 𝜇 ( d 𝑥 ) = R (( , 𝑗 ) , d 𝑦 ) , 𝑦 ∈ ℝ + × Θ . Moreover, ( O , J ) is a 𝑇 -process by Corollary 3.7. Hence, if we can argue that the process isnon-evanescent, i.e. that there exists a compact set 𝐾 such that ( O , J ) returns to 𝐾 at arbitrarilylarge times, it will follow from Theorem 3.2 in [44] that ( O , J ) is Harris recurrent. But non-evanescence is a direct consequence of the sawtooth structure of the overshoot process, sincefor the compact set 𝐾 ≔ { } × Θ we have for any ( 𝑥, 𝑖 ) ∈ ℝ + × Θ and 𝑡 > ℙ 𝑥,𝑖 ( inf { 𝑠 ≥ 𝑡 : ( O 𝑠 , J 𝑠 ) ∈ { } × Θ } < ∞) = 𝔼 𝑥,𝑖 [ ℙ O 𝑡 , J 𝑡 ( 𝑇 { }× Θ < ∞)] = , where we used that 𝑇 { }× Θ = 𝑦 , ℙ 𝑦,𝑗 -a.s. for any ( 𝑦, 𝑗 ) ∈ ℝ + × Θ and O 𝑡 < ∞ almost surely.Hence, ( O , J ) is non-evanescent and the assertion follows. (cid:4) As a consequence of irreducibility implied by Harris recurrence and ( O , J ) being a 𝑇 -process,we obtain that every compact set is petite, which will be useful for our proof of exponentialconvergence of the overshoot process later on. Corollary 3.9.
Every compact set is petite for the overshoot process.Proof.
This is an immediate consequence of Theorem 5.1 in [53] since ( O , J ) is a Harris recurrent 𝑇 -process under the given assumptions and Harris recurrence implies irreducibility. (cid:4) Let us now determine the essential unique invariant measure of ( O , J ) and also derive anecessary and sufficient condition for the existence of a unique stationary distribution, whichis the same condition needed for weak convergence of overshoots. Theorem 3.10.
The overshoot process ( O , J ) has an essentially unique invariant measure given by 𝜒 ( d 𝑦, { 𝑖 }) = 𝜋 + ( 𝑖 ) 𝑑 + 𝑖 𝛿 ( d 𝑦 ) + ( , ∞) ( 𝑦 ) (cid:16) 𝜋 + ( 𝑖 ) Π + 𝑖 ( 𝑦 ) + Õ 𝑗 ≠ 𝑖 𝜋 + ( 𝑗 ) 𝑞 + 𝑗,𝑖 𝐹 + 𝑗,𝑖 ( 𝑦 ) (cid:17) d 𝑦, ( 𝑦, 𝑖 ) ∈ ℝ + × Θ . (3.6) In particular, a stationary distribution for ( O , J ) exists if and only if 𝔼 , 𝝅 + [ 𝐻 + ] ≔ 𝑛 Õ 𝑖 = 𝜋 + ( 𝑖 ) 𝔼 ,𝑖 [ 𝐻 + ] < ∞ . Proof.
Define 𝜶 ( 𝜆 ) ≔ 𝝅 + · 𝚽 + ( 𝜆 ) and 𝛼 𝜆 ≔ 𝑛 Õ 𝑖 = 𝜋 + ( 𝑖 ) Φ + 𝑖 ( 𝜆 ) 𝛿 { }×{ 𝑖 } , then lim 𝜆 ↓ diag ( Φ + ( 𝜆 ) , . . . , Φ + 𝑛 ( 𝜆 )) = 𝑛 × 𝑛 implies that 𝛼 𝜆 ( ℝ + × Θ ) → 𝜆 ↓
0. Moreover,since 𝝅 + is the stationary distribution of 𝐽 + and 𝑮 + ( ) = 𝕀 𝑛 , we havelim 𝜆 ↓ 𝝅 + · ( 𝑸 + ⊙ 𝑮 + ( 𝜆 )) = 𝝅 + · 𝑸 + = × 𝑛 . U 𝛼 𝜆 𝜆 ( d 𝑥 ) ≔ ∫ ℝ + × Θ U 𝜆 ( 𝑦, d 𝑥 ) 𝛼 𝜆 ( d 𝑦 ) . Plugging into theresolvent formula from Theorem 3.6 yields for any 𝑓 ∈ B 𝑏 ( ℝ + × Θ ) ∩ B + ( ℝ + × Θ ) thatlim 𝜆 ↓ U 𝛼 𝜆 𝜆 𝑓 = lim 𝜆 ↓ 𝑛 Õ 𝑖 = 𝜋 + 𝑖 Φ + 𝑖 ( 𝜆 ) U 𝜆 𝑓 ( , 𝑖 ) = lim 𝜆 ↓ 𝜶 ( 𝜆 ) · ( U 𝜆 𝑓 ( , 𝑖 )) ⊤ 𝑖 = ,...,𝑛 = lim 𝜆 ↓ 𝝅 + · © « 𝑑 + 𝑖 𝑓 ( , 𝑖 ) + ∫ ∞ 𝑄 𝜆 𝑓 ( 𝑦, 𝑖 ) Π + 𝑖 ( d 𝑦 ) + Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 𝑄 𝜆 𝑓 ( 𝑦, 𝑗 ) 𝐹 + 𝑖,𝑗 ( d 𝑦 ) ª®¬ 𝑖 = ,...𝑛 . (3.7)By monotone convergence and an integration by parts it follows that for any measure 𝜇 on ℝ + lim 𝜆 ↓ ∫ ∞ 𝑄 𝜆 𝑓 ( 𝑦, 𝑖 ) 𝜇 ( d 𝑦 ) = ∫ ∞ ∫ 𝑦 𝑓 ( 𝑦 − 𝑡 , 𝑖 ) d 𝑡 𝜇 ( d 𝑦 ) = ∫ ∞ ∫ 𝑦 𝑓 ( 𝑡 , 𝑖 ) d 𝑡 𝜇 ( d 𝑦 ) = ∫ ∞ 𝜇 ( 𝑦 ) 𝑓 ( 𝑦, 𝑖 ) d 𝑦, where 𝜇 ( 𝑦 ) ≔ 𝜇 ( 𝑦, ∞) . Thus, we obtain from (3.7) thatlim 𝜆 ↓ U 𝛼 𝜆 𝜆 𝑓 = 𝝅 + · © « 𝑑 + 𝑖 𝑓 ( , 𝑖 ) + ∫ ∞ 𝑓 ( 𝑦, 𝑖 ) Π + 𝑖 ( 𝑦 ) d 𝑦 + Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 𝑓 ( 𝑦, 𝑗 ) 𝐹 + 𝑖,𝑗 ( 𝑦 ) d 𝑦 ª®¬ ⊤ 𝑖 = ,...,𝑛 = 𝑛 Õ 𝑖 = 𝜋 + ( 𝑖 ) © « 𝑑 + 𝑖 𝑓 ( , 𝑖 ) + ∫ ∞ 𝑓 ( 𝑦, 𝑖 ) Π + 𝑖 ( 𝑦 ) d 𝑦 + Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 ∫ ∞ 𝑓 ( 𝑦, 𝑖 ) 𝐹 + 𝑖,𝑗 ( 𝑦 ) d 𝑦 ª®¬ = 𝑛 Õ 𝑖 = © « 𝜋 + ( 𝑖 ) (cid:18) 𝑑 + 𝑖 𝑓 ( , 𝑖 ) + ∫ ∞ 𝑓 ( 𝑦, 𝑖 ) Π + 𝑖 ( 𝑦 ) d 𝑦 (cid:19) + Õ 𝑗 ≠ 𝑖 𝜋 + ( 𝑗 ) 𝑞 + 𝑗,𝑖 ∫ ∞ 𝑓 ( 𝑦, 𝑖 ) 𝐹 + 𝑗,𝑖 ( 𝑦 ) d 𝑦 ª®¬ = ∫ ℝ + × Θ 𝑓 ( 𝑦, 𝑧 ) 𝜒 ( d 𝑦 × d 𝑧 ) , where for the second to last equality we used that 𝑛 Õ 𝑖 = 𝜋 + ( 𝑖 ) Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 ∫ ∞ 𝑓 ( 𝑦, 𝑗 ) 𝐹 + 𝑖,𝑗 d 𝑦 = 𝑛 Õ 𝑗 = Õ 𝑖 ≠ 𝑗 𝑞 + 𝑖,𝑗 𝜋 + ( 𝑖 ) ∫ ∞ 𝑓 ( 𝑦, 𝑗 ) 𝐹 + 𝑖,𝑗 ( 𝑦 ) d 𝑦 = 𝑛 Õ 𝑖 = Õ 𝑗 ≠ 𝑖 𝑞 + 𝑗,𝑖 𝜋 + ( 𝑗 ) ∫ ∞ 𝑓 ( 𝑦, 𝑖 ) 𝐹 + 𝑗,𝑖 ( 𝑦 ) d 𝑦. From Proposition A.1 it now follows that 𝜒 is indeed an invariant measure for ( O , J ) . Byirreducibility of 𝐽 + , ( O , J ) is a Harris recurrent Feller process according to Propositions 3.3 and3.8 and hence Theorem 2.5 in [3] yields that 𝜒 is essentially unique.Finally, using the Laplace exponent of ( 𝐻 + , 𝐽 + ) we obtain (cid:0) 𝔼 ,𝑖 [ 𝐻 + { 𝐽 + = 𝑗 } ] (cid:1) 𝑖,𝑗 = ,...,𝑛 = ∂∂ 𝜆 𝚽 + ( 𝜆 ) (cid:12)(cid:12) 𝜆 = = diag (cid:0) (cid:0) 𝔼 [ 𝐻 + , ( 𝑖 ) ] (cid:1) (cid:1) 𝑖 ∈ Θ + 𝑸 + ⊙ (cid:0) 𝔼 [ Δ + 𝑖,𝑗 ] (cid:1) 𝑖,𝑗 = ,...,𝑛 diag (cid:16) (cid:16) 𝑑 + 𝑖 + ∫ ∞ Π + 𝑖 ( 𝑥 ) d 𝑥 (cid:17) (cid:17) 𝑖 ∈ Θ + 𝑸 + ⊙ (cid:16) ∫ ∞ 𝐹 + 𝑖,𝑗 ( 𝑥 ) d 𝑥 (cid:17) 𝑖,𝑗 = ,...,𝑛 , and hence 𝔼 ,𝑖 (cid:2) 𝐻 + (cid:3) = 𝑑 + 𝑖 + ∫ ∞ Π + 𝑖 ( 𝑥 ) d 𝑥 + Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 ∫ ∞ 𝐹 + 𝑖,𝑗 ( 𝑥 ) d 𝑥, 𝑖 ∈ Θ , which shows that 𝜒 ( ℝ + × Θ ) = 𝔼 , 𝝅 + (cid:2) 𝐻 + (cid:3) . Thus, 𝜒 can be normalized to an invariant distribution if and only if 𝔼 , 𝝅 + [ 𝐻 + ] < ∞ . (cid:4) Remark . The finite mean condition for the ascending ladder height process is exactly thesame condition, which is necessary and sufficient for stationary overshoots of MAPs in thesense of weak convergence. As shown in Theorem 35 of [21] as an extension of Theorem 8 in[23] for Lévy processes, this condition is equivalent to 𝔼 ,𝑖 [| 𝜉 |] < ∞ and either lim 𝑡 →∞ 𝜉 𝑡 = ∞ , ℙ ,𝑖 -a.s., or lim sup 𝑡 →∞ 𝜉 𝑡 = − lim inf 𝑡 →∞ 𝜉 𝑡 = ∞ , ℙ ,𝑖 -a.s., together with ∫ ∞ 𝜅 𝑥 Í 𝑛𝑖 = 𝚷 ( 𝑖, [ 𝑥, ∞) × Θ ) + ∫ 𝑥 ∫ ∞ 𝑦 Í 𝑛𝑖 = 𝚷 ( 𝑖, (−∞ , − 𝑧 ] × Θ ) d 𝑧 d 𝑦 d 𝑥 < ∞ , (3.8)for some 𝜅 > 𝜒 is a maximal Harris meaure. Corollary 3.12.
The invariant measure 𝜒 given in (3.6) is a maximal Harris measure.Remark . This could have also been shown directly by an alternative proof of Proposition3.8 based on Kaspi and Mandelbaum’s characterization of Harris recurrence in terms of almostsure finiteness of first hitting times (A.3) and the characteristic property (2.2) of the Lévy systembelonging to ( 𝐻 + , 𝐽 + ) .Having established the existence of a unique invariant distribution, we now proceed toinvestigate ergodicity of overshoots. To this end, we need to find criteria ensuring the existenceof an irreducible skeleton chain. One of these criteria will be a strictly positive creepingprobability of the MAP and we lift a sufficient criterion for this to happen from the well-knownLévy process situation. Lemma 3.14.
Suppose that 𝑑 + 𝑖 > for some 𝑖 ∈ Θ . Then, for any 𝑡 > we have ℙ ,𝑖 (cid:0) 𝜉 𝑇 𝑡 = 𝑡 , 𝐽 𝑇 𝑡 = 𝑖 (cid:1) > . Proof.
Let 𝜎 + be the first jump time of 𝐽 + . If 𝑞 + 𝑖,𝑖 =
0, then under ℙ ,𝑖 , 𝐻 + is a Lévy subordinatorwith positive drift and therefore has positive creeping probability by Theorem 5.9 in [38],implying the claim. Suppose now − 𝑞 + 𝑖,𝑖 >
0. Then, using the representation from Proposition2.1 we have ℙ ,𝑖 ( O 𝑡 = , J 𝑡 = 𝑖 ) = ℙ ,𝑖 ( O + 𝑡 = , J + 𝑡 = 𝑖 )≥ ℙ ,𝑖 (cid:16) 𝐻 + , ,𝑖𝑇 + , ,𝑖𝑡 = 𝑡 , 𝑇 + , ,𝑖𝑡 < 𝜎 + (cid:17) = ∫ ∞ ℙ (cid:16) 𝐻 + , ( 𝑖 ) 𝑇 + , ( 𝑖 ) 𝑡 = 𝑡 , 𝑇 + , ( 𝑖 ) 𝑡 < 𝑦 (cid:17) ℙ ,𝑖 ( 𝜎 + ∈ d 𝑦 ) = − 𝑞 + 𝑖,𝑖 ∫ ∞ e 𝑞 + 𝑖,𝑖 𝑦 ℙ (cid:16) 𝐻 + , ( 𝑖 ) 𝑇 + , ( 𝑖 ) 𝑡 = 𝑡 , 𝑇 + , ( 𝑖 ) 𝑡 < 𝑦 (cid:17) d 𝑦, 𝐻 + , ,𝑖 and 𝐽 + for the third equality. Since again by Theorem5.9 in [38], 𝑑 + 𝑖 > ℙ ( 𝐻 + , ( 𝑖 ) 𝑇 + , ( 𝑖 ) 𝑡 = 𝑡 ) > 𝑡 ≥ 𝑦 →∞ ℙ (cid:16) 𝐻 + , ( 𝑖 ) 𝑇 + , ( 𝑖 ) 𝑡 = 𝑡 , 𝑇 + , ( 𝑖 ) 𝑡 < 𝑦 (cid:17) = ℙ (cid:16) 𝐻 + , ( 𝑖 ) 𝑇 + , ( 𝑖 ) 𝑡 = 𝑡 (cid:17) , it follows that there is 𝑧 > ℙ (cid:0) 𝐻 + , ( 𝑖 ) 𝑇 + , ( 𝑖 ) 𝑡 = 𝑡 , 𝑇 + , ( 𝑖 ) 𝑡 < 𝑦 (cid:1) > 𝑦 ≥ 𝑧 and hence, fromabove it follows that ℙ ,𝑖 ( O 𝑡 = , J 𝑡 = 𝑖 ) ≥ − 𝑞 + 𝑖,𝑖 ∫ ∞ 𝑧 e 𝑞 + 𝑖,𝑖 𝑦 ℙ (cid:16) 𝐻 + , ( 𝑖 ) 𝑇 + , ( 𝑖 ) 𝑡 = 𝑡 , 𝑇 + , ( 𝑖 ) 𝑡 < 𝑦 (cid:17) d 𝑦 > . (cid:4) Remark . The irreducibility assumption ( 𝒜
2) is not required for this statement.Let us now state properties of the ascending ladder height process that imply existence of anirreducible skeleton of ( O , J ) . Proposition 3.16.
If(i) 𝑑 + 𝑖 > for some 𝑖 ∈ Θ , then ( O , J ) is aperiodic and any Δ -skeleton is irreducible.(ii) for some 𝑗 ∈ Θ it holds Leb | ( , ∞) ≪ Π + 𝑗 | ( , ∞) , then any Δ -skeleton ( O Δ , J Δ ) is Leb + (·∩( , ∞))⊗ 𝛿 𝑗 -irreducible.(iii) for some 𝑗 ∈ Θ there exists an interval ( 𝑎, 𝑏 ) ⊂ ℝ + such that Leb | ( 𝑎,𝑏 ) ≪ Π + 𝑗 | ( 𝑎,𝑏 ) and for any 𝑖 ∈ Θ and 𝑥 > it holds that 𝑈 + 𝑖,𝑗 ([ , 𝑥 )) > , then for any Δ ∈ ( , ( 𝑎 + 𝑏 )/ ) , the Δ -skeleton ( O Δ , J Δ ) is Leb + (· ∩ ( 𝑎, ( 𝑎 + 𝑏 )/ )) ⊗ 𝛿 𝑗 -irreducible.(iv) for some ( 𝑗, 𝑘 ) ∈ Θ with 𝑘 ≠ 𝑗 it holds Leb | ( , ∞) ≪ 𝐹 + 𝑘,𝑗 | ( , ∞) and 𝑞 + 𝑘,𝑗 > , then any Δ -skeleton ( O Δ , J Δ ) is Leb + (· ∩ ( , ∞)) ⊗ 𝛿 𝑗 -irreducible.(v) for some ( 𝑗, 𝑘 ) ∈ Θ with 𝑘 ≠ 𝑗 it holds 𝑞 + 𝑘,𝑗 > , there exists an interval ( 𝑎, 𝑏 ) ⊂ ℝ + such that Leb | ( 𝑎,𝑏 ) ≪ 𝐹 + 𝑘,𝑗 | ( 𝑎,𝑏 ) and for any 𝑖 ∈ Θ and 𝑥 > it holds that 𝑈 + 𝑖,𝑘 ([ , 𝑥 )) > , then for any Δ ∈ ( , ( 𝑎 + 𝑏 )/ ) , the Δ -skeleton ( O Δ , J Δ ) is Leb + (· ∩ ( 𝑎, ( 𝑎 + 𝑏 )/ )) ⊗ 𝛿 𝑗 -irreducible.Proof. (i) The singleton set 𝐶 = { } × { 𝑖 } is trivially small (just choose 𝜈 𝑎 = 𝑃 𝑡 (( , 𝑖 ) , ·) for 𝑎 = 𝛿 𝑡 and some 𝑡 > 𝐶 ∈ B + ( ℝ + × Θ ) since Corollary 3.12 tells us that the invariantmeasure 𝜒 is an irreducibility measure for ( O , J ) and thanks to 𝑑 + 𝑖 >
0, we have 𝜒 ( 𝐶 ) > ℙ ,𝑖 (( O 𝑡 , J 𝑡 ) ∈ 𝐶 ) = ℙ ,𝑖 ( O 𝑡 = , J 𝑡 = 𝑖 ) > 𝑡 ≥
0, which implies that ( O , J ) is aperiodic with defining singleton set 𝐶 = { } × { 𝑖 } ,which by Lemma A.2 also implies that any Δ -skeleton is irreducible.(ii) Let 𝐵 = 𝐵 × 𝐵 ∈ B ( ℝ + × Θ ) such that Leb | ( , ∞) ⊗ 𝛿 𝑗 ( 𝐵 ) >
0. Without loss of generalitywe may assume that 0 ∉ 𝐵 . Since 𝐽 + is irreducible it holds ℙ ,𝑖 ( 𝐽 + 𝑡 = 𝑗 ) > 𝑡 > 𝑖 ∈ Θ and hence by monotone convergence,lim 𝑥 →∞ 𝑈 + 𝑖,𝑗 ([ , 𝑥 )) = ∫ ∞ ℙ ,𝑖 ( 𝐽 + 𝑡 = 𝑗 ) d 𝑡 > , 𝑥 > 𝑈 + 𝑖,𝑗 ([ , 𝑥 )) > 𝑥 ≥ 𝑥 and 𝑖 ∈ Θ . Forgiven 𝑥 ≥ 𝑡 > 𝑥 + 𝑥 . Then, by the overshoot formula and Fubini it follows that forany 𝑖 ∈ Θ we have ℙ 𝑥,𝑖 ( O 𝑡 ∈ 𝐵 , J 𝑡 ∈ 𝐵 ) ≥ ∫ [ ,𝑡 − 𝑥 ) ∫ 𝐵 Π + 𝑗 ( 𝑦 + d 𝑢 ) 𝑈 + 𝑖,𝑗 ( 𝑡 − 𝑥 − d 𝑦 ) = ∫ [ ,𝑡 − 𝑥 ) Π + 𝑗 ( 𝐵 + 𝑡 − 𝑥 − 𝑦 ) 𝑈 + 𝑖,𝑗 ( d 𝑦 ) . (3.9)Since by translation invariance of the Lebesgue measure it holds Leb ( 𝐵 + 𝑧 ) > 𝑧 ≥ | ( , ∞) ≪ Π + 𝑗 | ( , ∞) by assumption, it follows that for any 𝑦 ∈ [ , 𝑡 − 𝑥 ) we have Π + 𝑗 ( 𝐵 + 𝑡 − 𝑥 − 𝑦 ) >
0. By our choice of 𝑡 it also holds that 𝑈 + 𝑖,𝑗 ([ , 𝑡 − 𝑥 )) >
0, thus (3.9)yields that ℙ 𝑥,𝑖 ( O 𝑡 ∈ 𝐵 , J 𝑡 ∈ 𝐵 ) >
0. Hence, given Δ >
0, choosing 𝑛 𝑥 ∈ ℕ large enoughsuch that 𝑛 𝑥 Δ > 𝑥 + 𝑥 , it follows that ℙ 𝑥,𝑖 (( O 𝑛 𝑥 Δ , J 𝑛 𝑥 Δ ) ∈ 𝐵 ) > 𝑖 ∈ Θ , which showsthat any Δ -skeleton is Leb + (· ∩ ( , ∞)) ⊗ 𝛿 𝑗 -irreducible.(iii) Choose 𝐵 = 𝐵 × 𝐵 ∈ B ( ℝ + × Θ ) such that Leb | ( 𝑎,𝑏 ) ⊗ 𝛿 𝑗 ( 𝐵 ) >
0. Again we may assumethat 0 ∉ 𝐵 . Let ( 𝑥, 𝑖 ) ∈ ℝ + × Θ and 𝑡 ∈ ( 𝑥, 𝑥 + ( 𝑏 − 𝑎 )/ ) . Since for any 𝑧 ≥ ( 𝐵 + 𝑧 ) ∩ ( 𝑎, 𝑏 ) = ( 𝐵 ∩ ( 𝑎 − 𝑧, 𝑏 − 𝑧 )) + 𝑧 it follows for 𝑧 ∈ ( , ( 𝑏 − 𝑎 )/ ) by translation invariance of the Lebesgue measure thatLeb (( 𝐵 + 𝑧 ) ∩ ( 𝑎, 𝑏 )) = Leb ( 𝐵 ∩ ( 𝑎 − 𝑧, 𝑏 − 𝑧 )) ≥ Leb ( 𝐵 ∩ ( 𝑎, ( 𝑎 + 𝑏 )/ )) > . By our choice of 𝑡 ∈ ( 𝑥, 𝑥 +( 𝑏 − 𝑎 )/ ) it holds that 0 < 𝑡 − 𝑥 − 𝑦 < ( 𝑏 − 𝑎 )/ 𝑦 ∈ ( , 𝑡 − 𝑥 ) and therefore Leb (( 𝐵 + 𝑡 − 𝑥 − 𝑦 )∩ ( 𝑎, 𝑏 )) >
0, which by our assumption Leb | ( 𝑎,𝑏 ) ≪ Π + 𝑗 | ( 𝑎,𝑏 ) implies that Π + 𝑗 ( 𝐵 + 𝑡 − 𝑥 − 𝑦 ) >
0. Since 𝑈 + 𝑖,𝑗 ([ , 𝑡 − 𝑥 )) > ℙ 𝑥,𝑖 (( O 𝑡 , J 𝑡 ) ∈ 𝐵 ) >
0. Hence, given Δ ∈ ( , ( 𝑏 − 𝑎 )/ ) , if we choose 𝑘 ∈ ℕ such that 𝑘 Δ ∈ ( 𝑥, 𝑥 + ( 𝑏 − 𝑎 )/ ) it follows that ℙ 𝑥,𝑖 (( O 𝑘 Δ , J 𝑘 Δ ) ∈ 𝐵 ) > Í ∞ 𝑘 = ℙ 𝑥,𝑖 (( O 𝑘 Δ , J 𝑘 Δ ) ∈ 𝐵 ) >
0. Since ( 𝑥, 𝑖 ) ∈ ℝ + × Θ was chosen arbitrarily we concludethat the Δ -skeleton is irreducible with irreducibility measure Leb + (· ∩ ( 𝑎, ( 𝑎 + 𝑏 )/ )) ⊗ 𝛿 𝑗 .Parts (iv) and (v) can be demonstrated exactly as parts (ii) and (iii) when instead of (3.9) we usethat for 𝐵 = 𝐵 × 𝐵 ∈ B ( ℝ + × Θ ) with 𝑗 ∈ 𝐵 , ( 𝑥, 𝑖 ) ∈ ℝ + × Θ and 𝑡 > 𝑥 it holds ℙ 𝑥,𝑖 ( O 𝑡 ∈ 𝐵 , J 𝑡 ∈ 𝐵 ) ≥ 𝑞 + 𝑘,𝑗 ∫ [ ,𝑡 − 𝑥 ) 𝐹 + 𝑘,𝑗 ( 𝐵 + 𝑡 − 𝑥 − 𝑦 ) 𝑈 + 𝑖,𝑘 ( d 𝑦 ) . (cid:4) Remark . The condition in part (iii) and (v) that 𝑈 + 𝑖,𝑗 ([ , 𝑥 )) > 𝑖 ≠ 𝑗 is non-redundantin general. If, e.g., 𝐹 + 𝑖,𝑗 ([ , 𝑥 )) = 𝑖 ≠ 𝑗 , then 𝑈 + 𝑖,𝑗 ([ , 𝑥 )) = ( O , J ) is ergodic. Theorem 3.18.
Suppose that 𝔼 , 𝝅 + [ 𝐻 + ] < ∞ . Then, under any of the conditions of Proposition 3.16, itholds that ( O , J ) is ergodic, i.e. ∀( 𝑥, 𝑖 ) ∈ ℝ + × Θ : lim 𝑡 →∞ k ℙ 𝑥,𝑖 (( O 𝑡 , J 𝑡 ) ∈ ·) − 𝜌 k TV = , where for ( 𝑥, 𝑖 ) ∈ ℝ + × Θ , 𝜌 ( d 𝑥, { 𝑖 }) ≔ 𝔼 , 𝝅 + [ 𝐻 + ] (cid:16) 𝜋 + ( 𝑖 ) 𝑑 + 𝑖 𝛿 ( d 𝑦 ) + ( , ∞) ( 𝑦 ) (cid:16) 𝜋 + ( 𝑖 ) Π + 𝑖 ( 𝑦 ) + Õ 𝑗 ≠ 𝑖 𝜋 + ( 𝑗 ) 𝑞 + 𝑗,𝑖 𝐹 + 𝑗,𝑖 ( 𝑦 ) (cid:17) d 𝑦 (cid:17) , (3.10) is the stationary distribution of ( O , J ) . roof. As a consequence of Proposition 3.3, Proposition 3.8 and Theorem 3.10, it follows thatunder any of the conditions of Proposition 3.16, ( O , J ) is a positive Harris recurrent Borel rightMarkov process with unique stationary distribution given in (3.10) such that some Δ -skeletonis irreducible. Thus, Theorem 6.1 in [44] yields the assertion. (cid:4) A direct implication of ergodicity is that a continuous time version of the von Neumann–Birkhoff ergodic theorem holds, see the discussion in [49].
Corollary 3.19.
Given the assumptions from Theorem 3.18, it holds for any 𝑓 ∈ 𝐿 𝑝 ( ℝ + × Θ , 𝜌 ) and ( 𝑥, 𝑖 ) ∈ ℝ + × Θ that lim 𝑇 →∞ 𝑇 ∫ 𝑇 𝑓 ( O 𝑡 , J 𝑡 ) d 𝑡 = 𝜌 ( 𝑓 ) , ℙ 𝑥,𝑖 -a.s. and in 𝐿 𝑝 ( ℙ 𝜌 ) . Once we have derived an analogue of Vigon’s équations amicales inversés in Section 4, wewill be able to express conditions on the Lévy system 𝚷 of ( 𝜉 , 𝐽 ) that guarantee one of theconditions on the Lévy system 𝚷 + of ( 𝐻 + , 𝐽 + ) required for ergodicity. For the moment wecontent ourselves with studying the drifts 𝑑 + 𝑖 of the subordinators associated to the ascendingladder height process. Lemma 3.20. If 𝐽 is irreducible, then for any 𝑖 ∈ Θ and an appropriate scaling of local time, the diffusionparameter 𝑏 𝑖 of 𝜉 ( 𝑖 ) is given by 𝑏 𝑖 = 𝑑 + 𝑖 b 𝑑 + 𝑖 . Proof.
Let 𝑖 ∈ Θ . Considering the diagonal of 𝚿 , the spatial Wiener–Hopf factorization (2.5)yields for every 𝜃 ∈ ℝ i 𝑎 𝑖 𝜃 − 𝑏 𝑖 𝜃 + ∫ ℝ (cid:0) e i 𝜃 𝑥 − − i 𝜃 𝑥 [− , ] ( 𝑥 ) (cid:1) Π 𝑖 ( d 𝑥 ) + 𝑞 𝑖,𝑖 = (cid:16)b 𝑞 + 𝑖,𝑖 − b † + 𝑖 − i b 𝑑 + 𝑖 𝜃 + ∫ ∞ (cid:0) e − i 𝜃 𝑥 − (cid:1) b Π + 𝑖 ( d 𝑥 ) (cid:17) · (cid:16) 𝑞 + 𝑖,𝑖 − † + 𝑖 + i 𝑑 + 𝑖 𝜃 + ∫ ∞ (cid:0) e i 𝜃 𝑥 − (cid:1) Π + 𝑖 ( d 𝑥 ) (cid:17) + Õ 𝑘 ≠ 𝑖 𝜋 ( 𝑘 ) 𝜋 ( 𝑖 ) b 𝑞 + 𝑘,𝑖 𝑞 + 𝑘,𝑖 b 𝐺 + 𝑘,𝑖 (− 𝜃 ) 𝐺 + 𝑘,𝑖 ( 𝜃 ) . Since lim | 𝜃 |→∞ 𝜃 ∫ ℝ (cid:0) e i 𝜃 𝑥 − − i 𝜃 𝑥 [− , ] ( 𝑥 ) (cid:1) Π 𝑖 ( d 𝑥 ) = , and lim | 𝜃 |→∞ | 𝜃 | ∫ ∞ ( e i 𝜃 𝑥 − ) Π + 𝑖 ( d 𝑥 ) = , lim | 𝜃 |→∞ | 𝜃 | ∫ ∞ ( e − i 𝜃 𝑥 − ) b Π + 𝑖 ( d 𝑥 ) = , and moreover | b 𝐺 + 𝑘,𝑖 (− 𝜃 ) 𝐺 + 𝑘,𝑖 ( 𝜃 )| ≤
1, comparing coefficients yields 𝑏 𝑖 = 𝑑 + 𝑖 b 𝑑 + 𝑖 . (cid:4) Thus, 𝑏 𝑖 > 𝑑 + 𝑖 ∧ b 𝑑 + 𝑖 > 𝜉 ( 𝑖 ) with non-zero diffusion component,convergence to the stationary overshoot distribution takes place in total variation.As a next step we show that under appropriate moment conditions on the Lévy processesand transitional jumps underlying the ascending ladder height MAP, overshoots converge withpolynomial rate and in case of existence of exponential moments even exponentially fast. Thus,the speed of convergence is reflected in the tail behavior of the jump measures associated to theLévy system 𝚷 + , with light tails giving exponential decay and moderately heavy tails resultingin polynomial decay. For the proof we yet again make use of the resolvent formula (3.5) to findLyapunov functions needed for the resolvent drift criteria (A.10) and (A.12).20 heorem 3.21. Suppose that one of the conditions of Proposition 3.16 is satisfied.(i) Suppose there exists 𝜆 > such that the exponential 𝜆 -moment exists for all 𝐻 + , ( 𝑖 ) , 𝑖 ∈ Θ , and forall Δ + 𝑖,𝑗 , 𝑖 ≠ 𝑗 , such that 𝑞 + 𝑖,𝑗 ≠ . Then, for the choice 𝑉 𝜆 ( 𝑥, 𝑖 ) = exp ( 𝜆 𝑥 ) , ( 𝑥, 𝑖 ) ∈ ℝ + × Θ , ( O , J ) is R 𝜆 𝑉 𝜆 -uniformly ergodic, i.e. sup | 𝑓 |≤ R 𝜆 𝑉 𝜆 (cid:12)(cid:12) 𝔼 𝑥,𝑖 [ 𝑓 ( O 𝑡 , J 𝑡 )] − 𝜌 ( 𝑓 ) (cid:12)(cid:12) ≤ 𝐶 R 𝜆 𝑉 𝜆 ( 𝑥, 𝑖 ) e − 𝜅 𝑡 , ( 𝑥, 𝑖 ) ∈ ℝ + × Θ , for some universal constants 𝐶 , 𝜅 > . Moreover for any 𝛿 ∈ ( , ) , it holds that k ℙ 𝑥,𝑖 (( O 𝑡 , J 𝑡 ) ∈ ·) − 𝜌 k TV ≤ ℭ ( 𝛿 ) R 𝜆 𝑉 𝜆 ( 𝑥, 𝑖 ) e − 𝑡 /( + 𝛿 ) , ( 𝑥, 𝑖 ) ∈ ℝ + × Θ , (3.11) for some constant ℭ ( 𝛿 ) > .(ii) Suppose that for some 𝜆 > the 𝜆 -moment exists for all 𝐻 + , ( 𝑖 ) , 𝑖 ∈ Θ , and for all Δ + 𝑖,𝑗 , 𝑖 ≠ 𝑗 , suchthat 𝑞 + 𝑖,𝑗 ≠ . Then, there exists e 𝐶 > such that k ℙ 𝑥,𝑖 (( O 𝑡 , J 𝑡 ) ∈ ·) − 𝜌 k TV ≤ e 𝐶 R 𝜆 e 𝑉 𝜆 ( 𝑥, 𝑖 ) 𝑡 − 𝜆 , ( 𝑥, 𝑖 ) ∈ ℝ + × Θ , where e 𝑉 𝜆 ( 𝑥, 𝑖 ) = e 𝜆 𝑥 [ , ) ( 𝑥 ) + 𝑥 𝜆 [ , ∞) ( 𝑥 ) .Proof. (i) For a matrix 𝐴 ∈ ℝ 𝑛 × 𝑛 , let k 𝐴 k ∞ ≔ max 𝑖 = ,...𝑛 Í 𝑛𝑗 = | 𝑎 𝑖𝑗 | be its matrix norm inducedby the sup-norm. Let 𝑄 𝜆 be the operator from the statement of Theorem 3.6. Then, 𝜆 𝑄 𝜆 𝑉 𝜆 ( 𝑥, 𝑖 ) = 𝜆 ∫ 𝑥 e − 𝜆 𝑡 𝑉 𝜆 ( 𝑥 − 𝑡 , 𝑖 ) d 𝑡 = (cid:0) e 𝜆 𝑥 − e − 𝜆 𝑥 (cid:1) , ( 𝑥, 𝑖 ) ∈ ℝ + × Θ . Since by Taylor expansion there exists 𝑎 > 𝜆 𝑥 − e − 𝜆 𝑥 ≤ 𝑎𝑥 for any 𝑥 ∈ ( , ) and Π + 𝑖 are Lévy subordinator measures, it follows that ∫ 𝜆 𝑄 𝜆 𝑉 𝜆 ( 𝑥, 𝑖 ) Π + 𝑖 ( d 𝑥 ) < ∞ . Moreover, by assumption 𝐻 + , ( 𝑖 ) has an exponential 𝜆 -moment, which according to Theo-rem 3.6 of [38] is equivalent to ∫ ∞ exp ( 𝜆 𝑥 ) Π + 𝑖 ( d 𝑥 ) < ∞ , implying that ∫ ∞ 𝜆 𝑄 𝜆 𝑉 𝜆 ( 𝑥, 𝑖 ) Π + 𝑖 ( d 𝑥 ) < ∞ as well and thus ∫ ∞ 𝜆 𝑄 𝜆 𝑉 𝜆 ( 𝑥, 𝑖 ) Π + 𝑖 ( d 𝑥 ) < ∞ for all 𝑖 ∈ Θ . Since additionally 𝔼 [ exp ( 𝜆 Δ + 𝑖,𝑗 )] < ∞ for any 𝑖, 𝑗 ∈ Θ such that 𝑖 ≠ 𝑗 and 𝑞 + 𝑖,𝑗 >
0, it follows that if we define 𝑏 ≔ 𝜆 k 𝚽 + ( 𝜆 ) − k ∞ 𝑛 Õ 𝑖 = © « 𝑑 + 𝑖 + ∫ ∞ 𝑄 𝜆 𝑉 𝜆 ( 𝑥, 𝑖 ) Π + 𝑖 ( d 𝑥 ) + Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 𝔼 [ 𝑄 𝜆 𝑉 𝜆 ( Δ + 𝑖,𝑗 , 𝑗 )] ª®¬ ≤ k 𝚽 + ( 𝜆 ) − k ∞ 𝑛 Õ 𝑖 = © « 𝜆 𝑑 + 𝑖 + ∫ ∞ (cid:0) e 𝜆 𝑥 − e − 𝜆 𝑥 (cid:1) Π + 𝑖 ( d 𝑥 ) + Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 𝔼 (cid:2) exp (cid:0) 𝜆 Δ + 𝑖,𝑗 (cid:1) (cid:3)ª®¬ ,
21e have 𝑏 < ∞ . Using (3.5) it therefore follows for any 𝑖 ∈ Θ that R 𝜆 𝑉 𝜆 ( 𝑥, 𝑖 ) = 𝜆 U 𝜆 𝑉 𝜆 ( 𝑥, 𝑖 ) ≤ (cid:0) e 𝜆 𝑥 − e − 𝜆 𝑥 (cid:1) + 𝑏 < 𝑉 𝜆 ( 𝑥, 𝑖 ) + 𝑏, ( 𝑥, 𝑖 ) ∈ ℝ + × Θ , (3.12)which shows that (A.10) holds for 𝛽 = / 𝑏 < ∞ as above. Under the givenassumptions, ( O , J ) is Harris recurrent and there exists an irreducible skeleton chain byProposition 3.8 and Proposition 3.16, hence ( O , J ) is irreducible and aperiodic. Moreover, 𝑉 𝜆 is unbounded off petite sets since 𝑉 𝜆 is increasing and continuous and hence for any 𝑧 >
0, the set {( 𝑥, 𝑖 ) ∈ ℝ + × Θ : 𝑉 𝜆 ( 𝑥, 𝑖 ) ≤ 𝑧 } is compact and hence petite, according toCorollary 3.9. Thus, (A.10) being satisfied for our choice of 𝑉 𝜆 , Theorem 5.2 in [25] impliesthat ( O , J ) is R 𝜆 𝑉 𝜆 -uniformly ergodic.To establish the more explicit rate of convergence for the total variation norm in (3.11),note that (3.12) combined with (A.11) shows that for the petite set 𝐶 ( 𝜀 ) = { 𝑉 𝜆 ≤ 𝑏 / 𝜀 } , 𝜀 ∈ ( , ) and 𝜙 ( 𝑧 ) = ( − 𝜀 ) 𝑧 / R 𝜆 𝑉 𝜆 ( 𝑥, 𝑖 ) ≤ + 𝜀 𝑉 𝜆 ( 𝑥, 𝑖 ) + 𝑏 𝐶 ( 𝜀 ) = 𝑉 𝜆 ( 𝑥, 𝑖 ) − 𝜙 ◦ 𝑉 𝜆 ( 𝑥, 𝑖 ) + 𝑏 𝐶 ( 𝜀 ) , ( 𝑥, 𝑖 ) ∈ ℝ + × Θ , and thus, the claim follows easily from (A.13).(ii) Since e 𝑉 𝜆 ( 𝑥, 𝑖 ) = 𝑉 𝜆 ( 𝑥, 𝑖 ) for 𝑥 ∈ [ , ) , 𝑖 ∈ Θ , it follows from above that ∫ 𝜆 𝑄 𝜆 e 𝑉 𝜆 ( 𝑥, 𝑖 ) Π + 𝑖 ( d 𝑥 ) < ∞ . Moreover, for 𝑥 ≥ 𝜆 𝑄 𝜆 e 𝑉 𝜆 ( 𝑥, 𝑖 ) ≤ 𝑥 𝜆 and thus by our moment assumptions on 𝐻 + , ( 𝑖 ) and Δ + 𝑖,𝑗 ∫ ∞ 𝜆 𝑄 𝜆 e 𝑉 𝜆 ( 𝑥, 𝑖 ) Π + 𝑖 ( d 𝑥 ) < ∞ , 𝔼 [ 𝜆 𝑄 𝜆 e 𝑉 𝜆 ( Δ + 𝑖,𝑗 , 𝑗 )] < ∞ . This shows that e 𝑏 ≔ 𝜆 k 𝚽 + ( 𝜆 ) − k ∞ 𝑛 Õ 𝑖 = © « 𝑑 + 𝑖 + ∫ ∞ 𝑄 𝜆 e 𝑉 𝜆 ( 𝑥, 𝑖 ) Π + 𝑖 ( d 𝑥 ) + Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 𝔼 [ 𝑄 𝜆 e 𝑉 𝜆 ( Δ + 𝑖,𝑗 , 𝑗 )] ª®¬ < ∞ . Observe now that integrating by parts twice yields that for 𝑥 ≥ 𝑖 ∈ Θ , 𝜆 𝑄 𝜆 e 𝑉 𝜆 ( 𝑥, 𝑖 ) ≤ e 𝑉 𝜆 ( 𝑥, 𝑖 ) − 𝑥 𝜆 − + e − 𝜆 𝑥 (cid:16) e 𝜆 + ∫ 𝑥 ( 𝜆 − ) e 𝜆 𝑡 𝑡 𝜆 − d 𝑡 (cid:17) and for 𝑥 ∈ [ , ) , 𝜆 𝑄 𝜆 e 𝑉 𝜆 ( 𝑥, 𝑖 ) ≤ e 𝜆 𝑥 . Thus, for all ( 𝑥, 𝑖 ) ∈ ℝ + × Θ , we have 𝜆 𝑄 𝜆 e 𝑉 𝜆 ( 𝑥, 𝑖 ) ≤ e 𝑉 𝜆 ( 𝑥, 𝑖 ) − ( e 𝑉 𝜆 ( 𝑥, 𝑖 )) 𝜆 − 𝜆 + e − 𝜆 𝑥 (cid:16) e 𝜆 + ∫ 𝑥 ( 𝜆 − ) e 𝜆 𝑡 𝑡 𝜆 − d 𝑡 (cid:17) + e 𝜆 − [ , ] ( 𝑥 ) and hence by the resolvent formula and the definiton of e 𝑏 , R 𝜆 e 𝑉 𝜆 ( 𝑥, 𝑖 ) ≤ e 𝑉 𝜆 ( 𝑥, 𝑖 )−( e 𝑉 𝜆 ( 𝑥, 𝑖 )) 𝜆 − 𝜆 + e − 𝜆 𝑥 (cid:16)e 𝑏 + e 𝜆 + ∫ 𝑥 ( 𝜆 − ) e 𝜆 𝑡 𝑡 𝜆 − d 𝑡 (cid:17) + e 𝜆 − [ , ] ( 𝑥 ) . (3.13)22et 𝑥 ∗ > 𝑥 > 𝑥 ∗ 𝜓 𝜆 ( 𝑥 ) ≔ e − 𝜆 𝑥 (cid:16)e 𝑏 + e 𝜆 + ∫ 𝑥 ( 𝜆 − ) e 𝜆 𝑡 𝑡 𝜆 − d 𝑡 (cid:17) ≤ 𝑥 𝜆 − . By the same arguments as in the previous part, the compact set 𝐶 ≔ [ , 𝑥 ∗ ] × Θ is petiteand it follows from (3.13) that R 𝜆 e 𝑉 𝜆 ≤ e 𝑉 𝜆 − 𝜙 ◦ e 𝑉 𝜆 + e 𝑐 𝐶 , (3.14)where e 𝑐 ≔ e 𝜆 − + max 𝑥 ∈[ ,𝑥 ∗ ] 𝜓 𝜆 ( 𝑥 ) < ∞ and 𝜙 ( 𝑧 ) = 𝑧 − / 𝜆 , 𝑧 ≥
1, is concave, differentiableand increasing. Hence, (A.12) is satisfied. The assertion now follows from (A.13) uponnoting that 𝐻 𝜙 ( 𝑡 ) = ∫ 𝑡 ( / 𝜙 ( 𝑠 )) d 𝑠 = 𝜆 ( 𝑡 / 𝜆 − ) , 𝐻 − 𝜙 ( 𝑡 ) = (cid:16) + 𝑡 𝜆 (cid:17) 𝜆 , and therefore the rate of convergence Ξ ( 𝑡 ) defined in Appendix A is given by Ξ ( 𝑡 ) = /( 𝜙 ◦ 𝐻 − 𝜙 )( 𝑡 ) = (cid:16) + 𝑡 𝜆 (cid:17) − 𝜆 ≤ ( 𝜆 ) 𝜆 − 𝑡 − 𝜆 . (cid:4) Remark . Let us emphasize that the parameter 𝜆 in the exponential 𝜆 -moment assumptionfrom part (i) is only reflected in a multiplicative constant and hence negligible for the speedof convergence. This is particularly nice for statistical considerations, where exact control overthe speed of convergence is decisive for making a minimax approach over entire classes ofMAPs feasible. Moreover, our analysis of the mixing behavior of self-similar Markov processeslater on profits immensly from this exact rate in terms of expliciteness, since the Lamperti–Kiutransform turns the exponential rate into a polynomial one. On the other hand, if the MAPcomponents only have some finite 𝜆 -moment, then the maximal size of 𝜆 is highly relevant forthe polynomial speed of convergence.Finally, we will establish polynomial 𝛽 -mixing of stationary overshoots provided that theprocess converges at polynomial rate and make use of Masuda’s criterion for exponential 𝛽 -mixing given exponential ergodicity of a Markov process stated in (A.16) to establish exponential 𝛽 -mixing of overshoots for any initial distribution with exponential moments. To this end, weneed one more technical result, which is the natural generalization of a result well known forrenewal functions of Lévy subordinators to the MAP situation. Lemma 3.23.
For any 𝑥, 𝑦 > and 𝑖, 𝑗 ∈ Θ it holds 𝑈 + 𝑖,𝑗 ( 𝑥 + 𝑦 ) − 𝑈 + 𝑖,𝑗 ( 𝑥 ) ≤ 𝑈 + 𝑗,𝑗 ( 𝑦 ) . Proof.
Let ( 𝜃 + 𝑡 ) 𝑡 ≥ be the family of transition operators for the Markov process ( 𝐻 + , 𝐽 + ) . Then,with a change of variables and an application of the strong Markov property it follows 𝑈 + 𝑖,𝑗 ( 𝑥 + 𝑦 ) − 𝑈 + 𝑖,𝑗 ( 𝑥 ) = 𝔼 ,𝑖 " ∫ 𝑇 + 𝑥 + 𝑦 𝑇 + 𝑥 { 𝐽 + 𝑡 = 𝑗 } d 𝑡 = 𝔼 ,𝑖 " ∫ 𝑇 + 𝑥 + 𝑦 ◦ 𝜃 + 𝑇 + 𝑥 + 𝑇 + 𝑥 { 𝐽 + 𝑡 + 𝑇 + 𝑥 = 𝑗 } d 𝑡 = 𝔼 ,𝑖 " ∫ 𝑇 + 𝑥 + 𝑦 { 𝐽 + 𝑡 = 𝑗 } d 𝑡 ◦ 𝜃 + 𝑇 + 𝑥 = 𝔼 ,𝑖 " 𝔼 𝐻 + 𝑇 + 𝑥 ,𝐽 + 𝑇 + 𝑥 " ∫ 𝑇 + 𝑥 + 𝑦 { 𝐽 + 𝑡 = 𝑗 } d 𝑡 ≤ 𝔼 ,𝑖 h 𝑈 + 𝐽 + 𝑇 + 𝑥 ,𝑗 ( 𝑦 ) i ≤ 𝑈 + 𝑗,𝑗 ( 𝑦 ) . { 𝑇 + 𝑥 + 𝑦 , ℙ 𝑧,𝑘 } d = { 𝑇 + 𝑥 + 𝑦 − 𝑧 , ℙ ,𝑘 } for any 𝑧 ≥ 𝑧 ↦→ 𝑇 + 𝑧 isincreasing and that 𝐻 + 𝑇 + 𝑥 ≥ 𝑥 by definition. The second inequality follows from 𝑈 + 𝑖,𝑗 ( 𝑦 ) ≤ 𝑈 + 𝑗,𝑗 ( 𝑦 ) thanks to increasing paths of 𝐻 + . (cid:4) Recall the definition of the 𝛽 -mixing coefficient from (A.15). Theorem 3.24.
Suppose that one of the conditions of Proposition 3.16 is satisfied.(i) Suppose that the exponential moment assumption from Theorem 3.21.(i) is satisfied and let 𝜂 bea probability measure on ( ℝ + × Θ , B ( ℝ + × Θ )) such that 𝜂 (· , Θ ) has an exponential 𝜆 -moment.Then, ( O , J ) started in 𝜂 is exponentially 𝛽 -mixing with the 𝛽 -mixing coefficient 𝛽 ( 𝜂 , ·) satisfying 𝛽 ( 𝜂 , 𝑡 ) ≤ 𝜚 ( 𝜂 , 𝜆 , 𝛿 ) e − 𝑡 /( + 𝛿 ) , for 𝜚 ( 𝜂 , 𝜆 ) ≔ ℭ ( 𝛿 ) sup 𝑡 ≥ ∫ ℝ + × Θ R 𝜆 𝑉 𝜆 ( 𝑦, 𝑧 ) ℙ 𝜂 ( O 𝑡 ∈ d 𝑦, J 𝑡 ∈ d 𝑧 ) < ∞ , for some constant ℭ ( 𝛿 ) > and 𝑉 𝜆 ( 𝑥, 𝑖 ) = exp ( 𝜆 𝑥 ) , ( 𝑥, 𝑖 ) ∈ ℝ + × Θ .(ii) Suppose that the 𝜆 -moment assumption from Theorem 3.21.(ii) is satisfied for some 𝜆 > . Then, ( O 𝑡 , J 𝑡 ) 𝑡 ≥ started in its invariant distribution is 𝛽 -mixing with rate 𝛽 ( 𝜌 , 𝑡 ) . 𝑡 − 𝜆 , 𝑡 ≥ . Proof. (i) By Theorem 3.21, for any 𝛿 ∈ ( , ) there exists ℭ ( 𝛿 ) > k ℙ 𝑥,𝑖 ( O 𝑡 ∈ ·) − 𝜌 k TV ≤ ℭ ( 𝛿 ) R 𝜆 𝑉 𝜆 ( 𝑥, 𝑖 ) e − 𝑡 /( + 𝛿 ) , ( 𝑥, 𝑖 ) ∈ ℝ + × Θ . Hence, the assertion will follow from Lemma 3.9 in Masuda [40] if we can establish that 𝜚 ( 𝜂 , 𝜆 , 𝛿 ) < ∞ . To this end, observe that by the explicit form of the 𝜆 -resolvent U 𝜆 of theovershoot process and with the constants 𝑎, 𝑏 appearing in the proof of Theorem 3.21 we have ∫ ℝ + × Θ R 𝜆 𝑉 𝜆 ( 𝑦, 𝑢 ) ℙ 𝜂 ( O 𝑡 ∈ d 𝑦, J 𝑡 ∈ d 𝑢 )≤ 𝑏 + 𝑛 Õ 𝑖,𝑗 = ∫ ℝ + ∫ ℝ + ( e 𝜆 𝑦 − e − 𝜆 𝑦 ) ℙ 𝑥,𝑖 ( O 𝑡 ∈ d 𝑦, J 𝑡 = 𝑗 ) 𝜂 ( d 𝑥, { 𝑖 })≤ 𝑏 + 𝑛 Õ 𝑖,𝑗 = ∫ ℝ + ∫ ∞ e 𝜆 𝑦 ℙ 𝑥,𝑖 ( O 𝑡 ∈ d 𝑦, J 𝑡 = 𝑗 ) 𝜂 ( d 𝑥, { 𝑖 })+ 𝑎 ∫ ℝ + ∫ 𝑦 ℙ 𝑥,𝑖 ( O 𝑡 ∈ d 𝑦, J 𝑡 = 𝑗 ) 𝜂 ( d 𝑥, { 𝑖 }) ! ≤ 𝑏 + 𝑛 Õ 𝑖,𝑗 = ∫ ℝ + ∫ ∞ e 𝜆 𝑦 ℙ 𝑥,𝑖 ( O 𝑡 ∈ d 𝑦, J 𝑡 = 𝑗 ) 𝜂 ( d 𝑥, { 𝑖 }) + 𝑎 ! . Hence, to prove the assertion it suffices to show that for any ( 𝑖, 𝑗 ) ∈ Θ ,sup 𝑡 ≥ ∫ ℝ + ∫ ∞ e 𝜆 𝑦 ℙ 𝑥,𝑖 ( O 𝑡 ∈ d 𝑦, J 𝑡 = 𝑗 ) 𝜂 ( d 𝑥, { 𝑖 }) < ∞ . 𝑡 ≥ ∫ ℝ + ∫ ∞ e 𝜆 𝑦 ℙ 𝑥,𝑖 ( O 𝑡 ∈ d 𝑦, J 𝑡 = 𝑗 ) 𝜂 ( d 𝑥, { 𝑖 })≤ ∫ 𝑡 ∫ ∞ e 𝜆 𝑦 ℙ ,𝑖 ( O 𝑡 − 𝑥 ∈ d 𝑦, J 𝑡 = 𝑗 ) 𝜂 ( d 𝑥, { 𝑖 }) + ∫ ∞ 𝑡 e 𝜆 ( 𝑥 − 𝑡 ) 𝜂 ( d 𝑥, { 𝑖 })≤ ∫ 𝑡 𝜂 ( d 𝑥, { 𝑖 }) ∫ 𝑡 − 𝑥 𝑈 + 𝑖,𝑗 ( d 𝑦 ) ∫ ∞ Π + 𝑖 ( 𝑡 − 𝑥 − 𝑦 + d 𝑢 ) e 𝜆 𝑢 + Õ 𝑘 ≠ 𝑗 𝑞 + 𝑘,𝑗 ∫ 𝑡 − 𝑥 𝑈 + 𝑖,𝑘 ( d 𝑦 ) ∫ ∞ 𝐹 + 𝑘,𝑗 ( 𝑡 − 𝑥 − 𝑦 + d 𝑢 ) e 𝜆 𝑢 ! + ∫ ∞ e 𝜆 𝑥 𝜂 ( d 𝑥, { 𝑖 }) = ∫ 𝑡 𝜂 ( d 𝑥, { 𝑖 }) ∫ 𝑡 − 𝑥 𝑈 + 𝑖,𝑗 ( d 𝑦 ) ∫ ∞ Π + 𝑖 ( d 𝑢 ) e 𝜆 ( 𝑢 + 𝑦 + 𝑥 − 𝑡 ) + Õ 𝑘 ≠ 𝑗 𝑞 + 𝑘,𝑗 ∫ 𝑡 − 𝑥 𝑈 + 𝑖,𝑘 ( d 𝑦 ) ∫ ∞ 𝐹 + 𝑘,𝑗 ( d 𝑢 ) e 𝜆 ( 𝑢 + 𝑦 + 𝑥 − 𝑡 ) ! + ∫ ∞ e 𝜆 𝑥 𝜂 ( d 𝑥, { 𝑖 }) = ∫ ∞ e 𝜆 𝑢 Π + 𝑖 ( d 𝑢 ) ∫ 𝑡 𝜂 ( d 𝑥, { 𝑖 }) e 𝜆 𝑥 ∫ 𝑡 − 𝑥 e 𝜆 ( 𝑦 − 𝑡 ) 𝑈 + 𝑖,𝑗 ( d 𝑦 )+ Õ 𝑘 ≠ 𝑗 𝑞 + 𝑘,𝑗 ∫ ∞ e 𝜆 𝑢 𝐹 + 𝑘,𝑗 ( d 𝑢 ) ∫ 𝑡 𝜂 ( d 𝑥, { 𝑖 }) e 𝜆 𝑥 ∫ 𝑡 − 𝑥 e 𝜆 ( 𝑦 − 𝑡 ) 𝑈 + 𝑖,𝑘 ( d 𝑦 )+ ∫ ∞ e 𝜆 𝑥 𝜂 ( d 𝑥, { 𝑖 }) From Lemma 3.23 we know that for 𝑡 > 𝑥 and 𝑖, 𝑗 ∈ Θ 𝑈 + 𝑖,𝑗 (( 𝑥, 𝑡 ]) = 𝑈 + 𝑖,𝑗 ( 𝑡 ) − 𝑈 + 𝑖,𝑗 ( 𝑥 ) ≤ 𝑛 Õ 𝑘 = 𝑈 + 𝑘,𝑗 ( 𝑡 − 𝑥 ) and thus ∫ 𝑡 − 𝑥 e 𝜆 ( 𝑦 − 𝑡 ) 𝑈 + 𝑖,𝑗 ( d 𝑦 ) ≤ ∫ 𝑡 e 𝜆 ( 𝑦 − 𝑡 ) 𝑈 + 𝑖,𝑗 ( d 𝑦 ) = 𝜆 ∫ 𝑡 ∫ 𝑦 −∞ e 𝜆 ( 𝑥 − 𝑡 ) d 𝑥 𝑈 + 𝑖,𝑗 ( d 𝑦 ) = 𝜆 ∫ 𝑡 −∞ e 𝜆 ( 𝑥 − 𝑡 ) 𝑈 + 𝑖,𝑗 (( ∨ 𝑥, 𝑡 ]) d 𝑥 = e − 𝜆 𝑡 𝑈 + 𝑖,𝑗 ( 𝑡 ) + 𝜆 ∫ 𝑡 e 𝜆 ( 𝑥 − 𝑡 ) 𝑈 + 𝑖,𝑗 (( 𝑥, 𝑡 ]) d 𝑥 ≤ e − 𝜆 𝑡 𝑈 + 𝑗,𝑗 ( 𝑡 ) + 𝜆 ∫ 𝑡 e 𝜆 ( 𝑥 − 𝑡 ) 𝑈 + 𝑗,𝑗 ( 𝑡 − 𝑥 ) d 𝑥 ≤ e − 𝜆 𝑡 𝑈 + 𝑗,𝑗 ( 𝑡 ) + 𝜆 ∫ ∞ e − 𝜆 𝑧 𝑈 + 𝑗,𝑗 ( 𝑧 ) d 𝑧. Theorem 28 in [21] tells us that 𝑈 + 𝑗,𝑗 ( 𝑧 ) ∼ 𝔼 , 𝝅 + [ 𝐻 + ] − 𝑧 𝜋 + ( 𝑗 ) as 𝑧 → ∞ and since 𝑈 + 𝑗,𝑗 ( 𝑧 ) ismoreover non-negative and increasing, we conclude thatsup 𝑡 ≥ ∫ 𝑡 e 𝜆 ( 𝑦 − 𝑡 ) 𝑈 + 𝑖,𝑗 ( d 𝑦 ) < ∞ . Plugging in now yieldssup 𝑡 ≥ ∫ ℝ + ∫ ∞ e 𝜆 𝑦 ℙ 𝑥,𝑖 ( O 𝑡 ∈ d 𝑦, J 𝑡 = 𝑗 ) 𝜂 ( d 𝑥, { 𝑖 }) ∫ ∞ e 𝜆 𝑢 Π + 𝑖 ( d 𝑢 ) ∫ ∞ e 𝜆 𝑥 𝜂 ( d 𝑥, { 𝑖 }) sup 𝑡 ≥ ∫ 𝑡 e 𝜆 ( 𝑦 − 𝑡 ) 𝑈 + 𝑖,𝑗 ( d 𝑦 )+ Õ 𝑘 ≠ 𝑗 𝑞 + 𝑘,𝑗 ∫ ∞ e 𝜆 𝑢 𝐹 + 𝑘,𝑗 ( d 𝑢 ) ∫ ∞ e 𝜆 𝑥 𝜂 ( d 𝑥, { 𝑖 }) sup 𝑡 ≥ ∫ 𝑡 e 𝜆 ( 𝑦 − 𝑡 ) 𝑈 + 𝑖,𝑘 ( d 𝑦 )+ ∫ ∞ e 𝜆 𝑥 𝜂 ( d 𝑥, Θ ) < ∞ , where finiteness is a consequence of the above discussion and our assumptions that 𝐻 + , ( 𝑖 ) , 𝜂 (· , Θ ) and Δ + 𝑖,𝑗 for 𝑖 ≠ 𝑗 with 𝑞 + 𝑖,𝑗 ≠ 𝜆 -moment. This finishes the proof.(ii) By stationarity, it holds that 𝛽 ( 𝜌 , 𝑡 ) = ∫ ℝ + × Θ k P 𝑡 (( 𝑥, 𝑧 ) , ·) − 𝜌 k TV 𝜌 ( d 𝑥 × d 𝑧 ) = 𝑛 Õ 𝑖 = ∫ ℝ + k P 𝑡 (( 𝑥, 𝑖 ) , ·) − 𝜌 k TV 𝜌 ( d 𝑥, { 𝑖 }) . Since the ( 𝜆 − ) th moments of 𝐻 + , ( 𝑖 ) for all 𝑖 ∈ Θ and Δ + 𝑖,𝑗 for all 𝑖, 𝑗 ∈ Θ such that 𝑞 + 𝑖,𝑗 ≠ 𝛽 ( 𝜌 , 𝑡 ) ≤ e 𝐶𝑡 − 𝜆 𝑛 Õ 𝑖 = ∫ ℝ + R 𝜆 − e 𝑉 𝜆 − ( 𝑥, 𝑖 ) 𝜌 ( d 𝑥, { 𝑖 }) , and hence, to prove the assertion it is enough to show that the integrals on the right-hand sideare finite. From the drift inequality (3.14) established in the proof of Theorem 3.21.(ii) we obtainthat for any 𝑖 ∈ Θ , ∫ ℝ + × Θ R 𝜆 − e 𝑉 𝜆 − ( 𝑥, 𝑖 ) 𝜌 ( d 𝑥, { 𝑖 }) ≤ ∫ e ( 𝜆 − ) 𝑥 𝜌 ( d 𝑥, 𝑖 ) + e 𝑐 𝜌 ( 𝐶 ) + ∫ ∞ 𝑥 𝜆 − 𝜌 ( d 𝑥, { 𝑖 }) . Since by our moment assumptions ∫ ∞ 𝑥 𝜆 − 𝜌 ( d 𝑥, { 𝑖 }) = 𝔼 , 𝝅 + [ 𝐻 + ] ∫ ∞ 𝑥 𝜆 − (cid:16) 𝜋 + ( 𝑖 ) Π + 𝑖 ( 𝑥 ) + Õ 𝑗 ≠ 𝑖 𝜋 + ( 𝑗 ) 𝑞 + 𝑗,𝑖 𝐹 + 𝑗,𝑖 ( 𝑥 ) (cid:17) d 𝑥, ≤ 𝔼 , 𝝅 + [ 𝐻 + ] (cid:16) 𝜋 + ( 𝑖 ) ∫ ∞ 𝑥 𝜆 Π + 𝑖 ( d 𝑥 ) + Õ 𝑗 ≠ 𝑖 𝜋 + ( 𝑗 ) 𝑞 + 𝑗,𝑖 ∫ ∞ 𝑥 𝜆 𝐹 + 𝑗,𝑖 ( d 𝑥 ) (cid:17) < ∞ , (3.15)the assertion follows. (cid:4) Remark . As in (3.15), it is easily established that when the jump measures associated to 𝚷 + have exponential decay, 𝜌 (· , Θ ) possesses an exponential moment and hence for part (i), thestationary overshoot process is exponentially 𝛽 -mixing as well.As a direct corollary we obtain the exponential resp. polynomial 𝛽 -mixing behavior of MAPssampled at first hitting times provided that creeping is possible or the Lévy system has someminor regularity and moreover the respective moment conditions on the MAP are satisfied. Let K 𝑡 ≔ 𝜎 (cid:0) (cid:0) 𝜉 𝑇 𝑠 , 𝐽 𝑇 𝑠 (cid:1) , 𝑠 ≤ 𝑡 (cid:1) , K 𝑡 ≔ 𝜎 (cid:0) (cid:0) 𝜉 𝑇 𝑠 , 𝐽 𝑇 𝑠 (cid:1) , 𝑠 ≥ 𝑡 (cid:1) , 𝑡 ≥ 𝜎 -algebras generated by the MAP sampled at first hitting times up to level 𝑡 and fromlevel 𝑡 onwards, respectively. 26 orollary 3.26. Suppose that the assumptions of Theorem 3.21.(i) are satisfied and let 𝜂 be a probabilitymeasure on ( ℝ + × Θ , B ( ℝ + × Θ )) such that 𝜂 (· , Θ ) has an exponential 𝜆 -moment. Then, for any 𝛿 ∈ ( , ) , sup 𝑡 > 𝛽 ℙ 𝜂 (cid:0) K 𝑡 , K 𝑡 + 𝑠 (cid:1) ≤ 𝜚 ( 𝜂 , 𝜆 , 𝛿 ) e − 𝑠 /( + 𝛿 ) , 𝑠 > , where 𝜚 ( 𝜂 , 𝜆 , 𝛿 ) > is the constant from Theorem 3.24. If instead the assumptions from 3.21.(ii) aresatisfied with 𝜆 > , then sup 𝑡 > 𝛽 ℙ 𝜌 (cid:0) K 𝑡 , K 𝑡 + 𝑠 (cid:1) . 𝑠 − 𝜆 , 𝑠 > .
4. Équations amicales inversés for MAPs
With the help of the spatial Wiener–Hopf factorization for MAPs we can generalize Vigon’séquation amicale inversé for Lévy processes to a characterization of the Lévy system of theascending ladder height MAP in terms of the Lévy system of the parent MAP and the potentialmeasures of the ascending ladder height process of the dual MAP. This is crucial for our resultssince this relation will allow to impose conditions on the parent MAP instead of the ascendingladder height MAP that imply the overshoot convergence results from the previous section. Tothis end, we first need to recall some concepts from distribution theory and introduce morenotation.Let S ( ℝ ) be the Schwartz space of rapidly decreasing smooth functions on ℝ and considerits dual space S ′ ( ℝ ) , the space of tempered distributions. For 𝜇 ∈ S ′ ( ℝ ) the 𝑘 -th derivative 𝜇 ( 𝑘 ) ∈ S ′ ( ℝ ) is defined by (cid:10) 𝜇 ( 𝑘 ) , 𝜙 (cid:11) = (− ) 𝑘 (cid:10) 𝜇 , 𝜙 ( 𝑘 ) (cid:11) , 𝜙 ∈ S ( ℝ ) , 𝑘 ∈ ℕ . If 𝜇 is induced by a function 𝜓 ∈ B ( ℝ ) via h 𝜇 , 𝜙 i = ∫ ℝ 𝜓 ( 𝑥 ) 𝜙 ( 𝑥 ) d 𝑥, 𝜙 ∈ S ( ℝ ) , we just write 𝜇 = 𝜓 , provided that the above integrals are well defined. Similarly, if 𝜇 isa measure on ( ℝ , B ( ℝ )) such that ∫ 𝜙 d 𝜇 is well-defined for any 𝜙 ∈ S ( ℝ ) , we identify thedistribution induced by 𝜙 ↦→ ∫ 𝜙 d 𝜇 with 𝜇 .For a Lévy measure 𝜈 integrating 𝑥 ↦→ | 𝑥 | on [− , ] , let L 𝜈 be the tempered distributiondefined via (cid:10) L 𝜈 , 𝜙 (cid:11) ≔ ∫ ℝ ( 𝜙 ( 𝑥 ) − 𝜙 ( )) 𝜈 ( d 𝑥 ) , 𝜙 ∈ S ( ℝ ) , and for a general Lévy measure 𝜈 let L 𝜈 be the tempered distribution defined via (cid:10) L 𝜈 , 𝜙 (cid:11) ≔ ∫ ℝ ( 𝜙 ( 𝑥 ) − 𝜙 ( ) − 𝜙 ′ ( ) 𝑥 [− , ] ( 𝑥 )) 𝜈 ( d 𝑥 ) , 𝜙 ∈ S ( ℝ ) . Recall that for a tempered distribution 𝜇 ∈ S ′ ( ℝ ) the Fourier transform ℱ 𝜇 is defined by h ℱ 𝜇 , 𝜙 i ≔ h 𝜇 , ℱ 𝜙 i = D 𝜇 , ∫ ℝ e i 𝑥 · 𝜙 ( 𝑥 ) d 𝑥 E , 𝜙 ∈ S ( ℝ ) , and that the Fourier transform is a bijective, continuous mapping on S ′ ( ℝ ) . If 𝛿 is the Diracdelta distribution and letting 𝜓 ( 𝑥 ) = 𝑥 , 𝑥 ∈ ℝ , it is immediate that ℱ 𝛿 = id , ℱ 𝛿 ′ = − i · id , ℱ 𝛿 ′′ = − 𝜓 . 𝜅 , Lévy measure 𝜈 , drift 𝑑 ≥ 𝑞 ≥ (cid:10) ℱ (cid:0) − 𝑞 𝛿 − 𝑑 𝛿 ′ + L 𝜈 (cid:1) , 𝜙 (cid:11) = ∫ ℝ (cid:16) − 𝑞 + i 𝑑 𝜃 + ∫ ℝ (cid:0) e i 𝜃 𝑥 − (cid:1) 𝜈 ( d 𝑥 ) (cid:17) 𝜙 ( 𝜃 ) d 𝜃 = ∫ ℝ 𝜅 ( 𝜃 ) 𝜙 ( 𝜃 ) d 𝜃 , and therefore it holds that ℱ − 𝜅 = − 𝑞 𝛿 − 𝑑 𝛿 ′ + L 𝜈 , i.e. if A denotes the generator of the subordinator, then the inverse Fourier transform of thecharacteristic exponent is equal to − A restricted to the Schwartz space S ( ℝ ) . Similarly, we getfor the characteristic exponent Ψ of a Lévy process with generating triplet ( 𝑎, 𝜎 , 𝜈 ) and killingrate 𝑞 that ℱ − Ψ = − 𝑞 𝛿 − 𝑎 𝛿 ′ + 𝜎 𝛿 ′′ + L 𝜈 . We start with a simple lemma. Let 𝜎 ( 𝐴 ) ≔ sup { Re ( 𝜆 ) : 𝜆 eigenvalue of 𝐴 } , be the spectral bound of a quadratic complex matrix 𝐴 . Lemma 4.1.
For any (non-trivial) MAP with characteristic matrix exponent 𝚿 and 𝜃 ∈ ℝ , it holdsthat 𝜎 ( 𝚿 ( 𝜃 )) ≤ and for any 𝜆 > , 𝜆 𝕀 𝑛 − 𝚿 ( 𝜃 ) is invertible.Proof. Let 𝜆 > e 𝜆 be an independent exponential time with mean 1 / 𝜆 anddefine for 𝑥 ∈ ℝ , 𝑖, 𝑗 ∈ Θ , 𝜆 𝑈 𝑖,𝑗 ( d 𝑥 ) = 𝔼 ,𝑖 h ∫ e 𝜆 { 𝜉 𝑡 ∈ d 𝑥,𝐽 𝑡 = 𝑗 } d 𝑡 i = ∫ ∞ ℙ ,𝑖 ( 𝜉 𝑡 ∈ d 𝑥, 𝐽 𝑡 = 𝑗, 𝑡 < e 𝜆 ) d 𝑡 = ∫ ∞ e − 𝜆 𝑡 ℙ ,𝑖 ( 𝜉 𝑡 ∈ d 𝑥, 𝐽 𝑡 = 𝑗 ) d 𝑡 , i.e. 𝜆 𝑈 𝑖,𝑗 is the (finite) occupation measure of the MAP started in ( , 𝑖 ) , while the modulator 𝐽 is in state 𝑗 , killed at an independent exponential time. Clearly, (cid:8) ℱ 𝜆 𝑈 𝑖,𝑗 (cid:9) ( 𝜃 ) = (cid:16) ∫ ∞ e 𝑡 ( 𝚿 ( 𝜃 )− 𝜆 𝕀 𝑛 ) d 𝑡 (cid:17) 𝑖,𝑗 , where for a matrix valued function 𝑓 : ℝ → ℝ 𝑛 × 𝑛 , such that 𝑓 𝑖,𝑗 is integrable, ∫ ℝ 𝑓 ( 𝑡 ) d 𝑡 ≔ ( ∫ ℝ 𝑓 𝑖,𝑗 ( 𝑡 ) d 𝑡 ) 𝑖,𝑗 = ,...,𝑛 . Hence, if we let 𝜆 𝑈 ≔ ( 𝜆 𝑈 𝑖,𝑗 ) 𝑖,𝑗 ∈ Θ , it follows that (cid:8) ℱ 𝜆 𝑈 (cid:9) ( 𝜃 ) = ∫ ∞ e 𝑡 ( 𝚿 ( 𝜃 )− 𝜆 𝕀 𝑛 ) d 𝑡. Noting that ( 𝜆 𝕀 𝑛 − 𝚿 ( 𝜃 )) ∫ 𝑇 e 𝑡 ( 𝚿 ( 𝜃 )− 𝜆 𝕀 𝑛 ) d 𝑡 = 𝕀 𝑛 − e 𝑇 ( 𝚿 ( 𝜃 )− 𝜆 𝕀 𝑛 ) , (4.1)and that the left-hand side converges to ( 𝜆 𝕀 𝑛 − 𝚿 ( 𝜃 )) · (cid:8) ℱ 𝜆 𝑈 (cid:9) ( 𝜃 ) , as 𝑇 → ∞ , it follows that the matrix exponential e 𝑇 ( 𝚿 ( 𝜃 )− 𝜆 𝕀 𝑛 ) must converge as well as 𝑇 → ∞ .E.g. from Theorem 4.12 of [4], this can only be the case if 𝜎 ( 𝚿 ( 𝜃 ) − 𝜆 𝕀 𝑛 ) ≤
0. But since 𝜆 > 𝜆 >
0, actually 𝜎 ( 𝚿 ( 𝜃 ) − 𝜆 𝕀 𝑛 ) <
0, implying 𝜎 ( 𝚿 ( 𝜃 )) ≤
0. Again by Theorem 4.12 of [4], this implies thatlim 𝑇 →∞ e 𝑇 ( 𝚿 ( 𝜃 )− 𝜆 𝕀 𝑛 ) = 𝑛 × 𝑛 . Thus, (4.1) yields that 𝜆 𝕀 𝑛 − 𝚿 ( 𝜃 ) is invertible with inverse ( 𝜆 𝕀 𝑛 − 𝚿 ( 𝜃 )) − = (cid:8) ℱ 𝜆 𝑈 (cid:9) ( 𝜃 ) . (4.2) (cid:4) Remark . This result generalizes part of Theorem 1 in [30] in the sense that, if we let Υ ( 𝑧 ) = ( 𝔼 ,𝑖 [ exp ( 𝑧 𝜉 ) ; 𝐽 = 𝑗 ]) 𝑖,𝑗 ∈ Θ for 𝑧 ∈ ℂ whenever it is defined, 𝑧 ↦→ det ( Υ ( 𝑧 ) − 𝜆 𝕀 𝑛 ) has no zeroson the imaginary axis, without having to assume anything on the jump structure of ( 𝜉 , 𝐽 ) orirreducibility of 𝐽 .Let us assume for the rest of this section that( 𝒜
3) the modulator 𝐽 of the MAP ( 𝜉 , 𝐽 ) is irreducible, i.e. 𝑸 is an irreducible matrix. Theorem 4.3 (Équations amicales inversés for MAPs) . For an appropriate scaling of local time atthe supremum it holds for any 𝑖, 𝑗 ∈ Θ , 𝑖 ≠ 𝑗 and 𝑥 > that Π + 𝑖 ( d 𝑥 ) = ∫ ∞ Π 𝑖 ( 𝑦 + d 𝑥 ) b 𝑈 + 𝑖,𝑖 ( d 𝑦 ) + Õ 𝑘 ≠ 𝑖 𝜋 ( 𝑘 ) 𝜋 ( 𝑖 ) 𝑞 𝑘,𝑖 ∫ ∞ 𝐹 𝑘,𝑖 ( 𝑦 + d 𝑥 ) b 𝑈 + 𝑘,𝑖 ( d 𝑦 ) , (4.3) 𝑞 + 𝑖,𝑗 𝐹 + 𝑖,𝑗 ( d 𝑥 ) = 𝜋 ( 𝑗 ) 𝜋 ( 𝑖 ) ∫ ∞ Π 𝑗 ( 𝑦 + d 𝑥 ) b 𝑈 + 𝑗,𝑖 ( d 𝑦 ) + Õ 𝑘 ≠ 𝑗 𝜋 ( 𝑘 ) 𝜋 ( 𝑖 ) 𝑞 𝑘,𝑗 ∫ ∞ 𝐹 𝑘,𝑗 ( 𝑦 + d 𝑥 ) b 𝑈 + 𝑘,𝑖 ( d 𝑦 ) . (4.4) and b Π + 𝑖 ( d 𝑥 ) = ∫ ∞ Π 𝑖 (− 𝑦 − d 𝑥 ) 𝑈 + 𝑖,𝑖 ( d 𝑦 ) + Õ 𝑘 ≠ 𝑖 𝑞 𝑖,𝑘 ∫ ∞ 𝐹 𝑖,𝑘 (− 𝑦 − d 𝑥 ) 𝑈 + 𝑘,𝑖 ( d 𝑦 ) , (4.5) b 𝑞 + 𝑖,𝑗 b 𝐹 + 𝑖,𝑗 ( d 𝑥 ) = 𝜋 ( 𝑗 ) 𝜋 ( 𝑖 ) (cid:16) ∫ ∞ Π 𝑗 (− 𝑦 − d 𝑥 ) 𝑈 + 𝑗,𝑖 ( d 𝑦 ) + Õ 𝑘 ≠ 𝑗 𝑞 𝑗,𝑘 ∫ ∞ 𝐹 𝑗,𝑘 (− 𝑦 − d 𝑥 ) 𝑈 + 𝑘,𝑖 ( d 𝑦 ) (cid:17) . (4.6) Remark . If we let 𝚷 ( d 𝑥 ) ≔ ( 𝚷 ( 𝑖, d 𝑥 × { 𝑗 })) 𝑖,𝑗 = ,...,𝑛 , 𝚷 + ( d 𝑥 ) ≔ ( 𝚷 + ( 𝑖, d 𝑥 × { 𝑗 })) 𝑖,𝑗 = ,...,𝑛 and 𝑼 + ( d 𝑥 ) ≔ ( 𝑈 + 𝑖,𝑗 ( d 𝑥 )) 𝑖,𝑗 = ,...,𝑛 (with the analogous definitions for the ascending ladder heightprocess of the dual MAP), then we may compactly express the équations amicales inversés (upto premultiplication of some diagonal matrix corresponding to the scaling of local time at thesupremum) for 𝑥 > 𝚷 + ( d 𝑥 ) = ∫ ∞ 𝚫 − 𝝅 b 𝑼 + ( d 𝑦 ) ⊤ 𝚫 𝝅 𝚷 ( 𝑦 + d 𝑥 ) , b 𝚷 + ( d 𝑥 ) = ∫ ∞ 𝚫 − 𝝅 (cid:0) 𝚷 (− 𝑦 − d 𝑥 ) 𝑼 + ( d 𝑦 ) (cid:1) ⊤ 𝚫 𝝅 , where ∫ ∞ ( 𝑔 𝑖,𝑗 ( 𝑦 ) 𝜈 𝑖,𝑗 ( d 𝑦 )) 𝑖,𝑗 = ,...,𝑛 ≔ ( ∫ ∞ 𝑔 𝑖,𝑗 ( 𝑦 ) 𝜈 𝑖,𝑗 ( d 𝑦 )) 𝑖,𝑗 = ,...,𝑛 for integrable functions 𝑔 𝑖,𝑗 on ( ℝ + , B ( ℝ + ) , 𝜈 𝑖,𝑗 ) . Proof of Theorem
Analogously to Vigon’s [54] idea, we use inverse Fourier transformations ofthe quantities involved in the spatial Wiener–Hopf factorization for MAPs to prove the desiredequalitites. To this end, recall from (2.5) that for an appropriate scaling of local time at thesupremum, it holds that 𝚿 ( 𝜃 ) = − 𝚫 − 𝝅 b 𝚿 + (− 𝜃 ) ⊤ 𝚫 𝝅 𝚿 + ( 𝜃 ) , 𝜃 ∈ ℝ . (4.7)29earranging yields for any 𝜆 > 𝚿 + ( 𝜃 ) = − 𝚫 − 𝝅 (cid:16) (cid:16) b 𝚿 + (− 𝜃 ) − 𝜆 𝕀 𝑛 (cid:17) − (cid:17) ⊤ 𝚫 𝝅 (cid:0) 𝚿 ( 𝜃 ) + 𝜆 𝚿 + ( 𝜃 ) (cid:1) , 𝜃 ∈ ℝ , (4.8)where invertibility of b 𝚿 + (− 𝜃 ) − 𝜆 𝕀 𝑛 is shown in Lemma 4.1. By the form of the characteristicmatrix exponent of a MAP it follows by taking inverse Fourier transformation of the distributioninduced by the left-hand side that ℱ − 𝚿 + 𝑖,𝑗 = { 𝑖 = 𝑗 } (cid:0) ( 𝑞 + 𝑖,𝑖 − † + 𝑖 ) 𝛿 − 𝑑 + 𝑖 𝛿 ′ + L Π + 𝑖 (cid:1) + { 𝑖 ≠ 𝑗 } 𝑞 + 𝑖,𝑗 𝐹 + 𝑖,𝑗 . Note that by (4.2) (cid:16) 𝜆 𝕀 𝑛 − b 𝚿 + (−·) (cid:17) − 𝑖,𝑗 = ℱ 𝜆 e 𝑈 + 𝑖,𝑗 , where for an independent exponentially distributed random variable e 𝜆 with mean 1 / 𝜆 wedefine 𝜆 e 𝑈 + 𝑖,𝑗 ( d 𝑥 ) ≔ b 𝔼 ,𝑖 h ∫ e 𝜆 {− 𝐻 + 𝑡 ∈ d 𝑥,𝐽 + 𝑡 = 𝑗 } d 𝑡 i , 𝑥 ∈ ℝ . With this observation, our previous discussion of inverse Fourier transforms of Lévy character-istic exponents and the property that if we regard two tempered distributions whose Fouriertransforms are induced by some measurable functions, the Fourier transform of the convolu-tion of those distributions becomes the tempered distribution induced by the product of thefunctions, it follows that the inverse Fourier transformation of the distribution induced by theright-hand side of (4.8) may be written as − ℱ − (cid:16) 𝚫 − 𝝅 (cid:16) (cid:16) b 𝚿 + (−·) − 𝜆 𝕀 𝑛 (cid:17) − (cid:17) ⊤ 𝚫 𝝅 (cid:0) 𝚿 + 𝜆 𝚿 + (cid:1) (cid:17) 𝑖,𝑗 = − 𝑛 Õ 𝑘 = 𝜋 ( 𝑘 ) 𝜋 ( 𝑖 ) ℱ − (cid:16) (cid:16) 𝚿 + 𝜆 𝚿 + (cid:17) 𝑘,𝑗 (cid:16) b 𝚿 + (−·) − 𝜆 𝕀 𝑛 (cid:17) − 𝑘,𝑖 (cid:17) = 𝜋 ( 𝑗 ) 𝜋 ( 𝑖 ) (cid:16) L Π 𝑗 + 𝜆 L Π + 𝑗 − ( 𝑎 𝑗 + 𝜆 𝑑 + 𝑗 ) 𝛿 ′ + 𝜎 𝛿 ′′ + ( 𝑞 𝑗,𝑗 + 𝜆 ( 𝑞 + 𝑗,𝑗 − † + 𝑗 )) 𝛿 (cid:17) ∗ 𝜆 e 𝑈 + 𝑗,𝑖 + Õ 𝑘 ≠ 𝑗 𝜋 ( 𝑘 ) 𝜋 ( 𝑖 ) (cid:0) 𝑞 𝑘,𝑗 𝐹 𝑘,𝑗 + 𝜆 𝑞 + 𝑘,𝑗 𝐹 + 𝑘,𝑗 (cid:1) ∗ 𝜆 e 𝑈 + 𝑘,𝑖 . Observe that the restriction of L Π + 𝑗 and L Π 𝑗 to the space D + of smooth functions on ℝ withcompact support in ( , ∞) is equal to the distributions induced by Π + 𝑗 and Π 𝑗 on this space,see also Propriété 3.9 in [54]. Restricting to ( , ∞) and equating both sides therefore yields theequality of distributions on D ′+ , { 𝑖 = 𝑗 } Π + 𝑖 + { 𝑖 ≠ 𝑗 } 𝑞 + 𝑖,𝑗 𝐹 + 𝑖,𝑗 = 𝜋 ( 𝑗 ) 𝜋 ( 𝑖 ) (cid:0) Π 𝑗 + 𝜆 Π + 𝑗 (cid:1) ∗ 𝜆 e 𝑈 + 𝑗,𝑖 + Õ 𝑘 ≠ 𝑗 𝜋 ( 𝑘 ) 𝜋 ( 𝑖 ) (cid:0) 𝑞 𝑘,𝑗 𝐹 𝑘 𝑗 + 𝜆 𝑞 + 𝑘,𝑗 𝐹 + 𝑘,𝑗 (cid:1) ∗ 𝜆 e 𝑈 + 𝑘,𝑖 . (4.9)Here, we used that for a measure 𝜇 on ℝ such that the distribution 𝜇 ∗ 𝜆 e 𝑈 + 𝑘,𝑖 , is well-defined itholds that (cid:0) 𝜇 ∗ 𝜆 e 𝑈 + 𝑘,𝑖 (cid:1)(cid:12)(cid:12) ( , ∞) = (cid:0) 𝜇 (cid:12)(cid:12) ( , ∞) ∗ 𝜆 e 𝑈 + 𝑘,𝑖 (cid:1)(cid:12)(cid:12) ( , ∞) , since 𝜆 e 𝑈 + 𝑘,𝑖 has support ℝ − , see also Propriété 3.8 in [54]. Denote e 𝑈 + 𝑘,𝑖 ( d 𝑥 ) ≔ b 𝔼 ,𝑘 h ∫ ∞ {− 𝐻 + 𝑡 ∈ d 𝑥,𝐽 + 𝑡 = 𝑖 } d 𝑡 i , 𝑥 ∈ ℝ , 𝜙 ∈ D + be non-negative with support supp ( 𝜙 ) ⊂ ( 𝑎, 𝑏 ) , where 0 < 𝑎 < 𝑏 < ∞ . Utilizingthe strong Markov property and spatial homogeneity of the dual ascending ladder heightprocess we can calculate as follows: (cid:10) Π + 𝑗 ∗ e 𝑈 + 𝑗,𝑖 , 𝜙 (cid:11) = ∫ −∞ ∫ ( 𝑎,𝑏 ) 𝜙 ( 𝑧 ) Π + 𝑗 ( d 𝑧 − 𝑦 ) e 𝑈 + 𝑗,𝑖 ( d 𝑦 )≤ b 𝔼 ,𝑗 h ∫ ∞ ∫ ( 𝑎 + 𝐻 + 𝑡 ,𝑏 + 𝐻 + 𝑡 ) 𝜙 ( 𝑧 − 𝐻 + 𝑡 ) Π + 𝑗 ( d 𝑧 ) { 𝐽 + 𝑡 = 𝑖 } d 𝑡 i ≤ k 𝜙 k ∞ ∫ ( 𝑎, ∞) b 𝔼 ,𝑗 h ∫ ( 𝑇 + 𝑧 − 𝑏 ,𝑇 + 𝑧 − 𝑎 ) { 𝐽 + 𝑡 = 𝑖 } d 𝑡 i Π + 𝑗 ( d 𝑧 ) = k 𝜙 k ∞ ∫ ( 𝑎, ∞) b 𝔼 ,𝑗 hb 𝔼 𝐻 + 𝑇 + 𝑧 − 𝑏 ,𝐽 + 𝑇 + 𝑧 − 𝑏 h ∫ ( ,𝑇 + 𝑧 − 𝑎 ) { 𝐽 + 𝑡 = 𝑖 } d 𝑡 i i Π + 𝑗 ( d 𝑧 )≤ k 𝜙 k ∞ ∫ ( 𝑎, ∞) b 𝔼 ,𝑗 hb 𝔼 ,𝐽 + 𝑇 + 𝑧 − 𝑏 h ∫ ( ,𝑇 + 𝑏 − 𝑎 ) { 𝐽 + 𝑡 = 𝑖 } d 𝑡 i i Π + 𝑗 ( d 𝑧 )≤ k 𝜙 k ∞ Π + 𝑗 (( 𝑎, ∞)) 𝑛 Õ 𝑘 = b 𝑈 + 𝑘,𝑖 ( 𝑏 − 𝑎 ) < ∞ . Since by monotone convergence, h Π + 𝑗 ∗ 𝜆 e 𝑈 + 𝑗,𝑖 , 𝜙 i → h Π + 𝑗 ∗ e 𝑈 + 𝑗,𝑖 , 𝜙 i as 𝜆 ↓
0, it follows that h 𝜆 Π + 𝑗 ∗ 𝜆 e 𝑈 + 𝑗,𝑖 , 𝜙 i → 𝜆 Π + 𝑗 ∗ 𝜆 e 𝑈 + 𝑗,𝑖 → D ′+ as 𝜆 ↓
0. Similarly, we obtain 𝜆 𝐹 + 𝑘,𝑗 ∗ 𝜆 e 𝑈 + 𝑘,𝑖 → 𝜆 ↓
0. Thus, letting 𝜆 ↓ D + we have { 𝑖 = 𝑗 } Π + 𝑖 + { 𝑖 ≠ 𝑗 } 𝑞 + 𝑖,𝑗 𝐹 + 𝑖,𝑗 = 𝜋 ( 𝑗 ) 𝜋 ( 𝑖 ) Π 𝑗 ∗ e 𝑈 + 𝑗,𝑖 + Õ 𝑘 ≠ 𝑗 𝜋 ( 𝑘 ) 𝜋 ( 𝑖 ) 𝑞 𝑘,𝑗 𝐹 𝑘,𝑗 ∗ e 𝑈 + 𝑘,𝑖 . The relations (4.3) and (4.4) follow upon noting that by a monotone class argument measureswith support on ( , ∞) are uniquely characterized by their action on D + and observing that b 𝑈 + 𝑖,𝑗 ( d 𝑦 ) = e 𝑈 + 𝑖,𝑗 (− d 𝑦 ) for 𝑦 ≥
0. Relations (4.5) and (4.6) are proved similarly by taking inverseFourier transforms on both sides of of b 𝚿 + = − 𝚫 − 𝝅 (cid:16) (cid:0) 𝚿 + (−·) − 𝜆 𝕀 𝑛 (cid:1) − (cid:17) ⊤ (cid:0) 𝚿 (−·) ⊤ + 𝜆 𝚫 𝝅 b 𝚿 + 𝚫 − 𝝅 (cid:1) 𝚫 𝝅 , 𝜆 > , which is a rearranged version of (4.7). (cid:4) Without loss of generality, for the remainder of this section we fix a scaling of local time at thesupremum such that (2.5) is satisfied and hence the formulas given in Theorem 4.3 hold withoutfurther multiplicative constants. As a first consequence of the équations amicales inversés, weobtain a characterization of 𝑸 + in terms of the transitional jumps of ( 𝜉 , 𝐽 ) , which we made useof in Proposition 3.5. Lemma 4.5.
Suppose that for 𝑖, 𝑗 ∈ Θ with 𝑖 ≠ 𝑗 , we have supp ( 𝑞 𝑖,𝑗 𝐹 𝑖,𝑗 ) ∩ ( , ∞) ≠ ∅ . Then, 𝑞 + 𝑖,𝑗 > . Proof.
By assumption, there exists 𝜀 > 𝑞 𝑖,𝑗 𝐹 𝑖,𝑗 ( 𝑧 ) > 𝑧 ∈ ( , 𝜀 ) . Note also that b 𝑈 + 𝑖,𝑖 ([ , 𝜀 )) > ( 𝐻 + , 𝐽 + ) under b ℙ ,𝑖 . Plugging ( , ∞) into (4.4) therefore yields 𝑞 + 𝑖,𝑗 ≥ ∫ ∞ 𝑞 𝑖,𝑗 𝐹 𝑖,𝑗 ( 𝑧 ) b 𝑈 + 𝑖,𝑖 ( d 𝑧 ) ≥ ∫ 𝜀 𝑞 𝑖,𝑗 𝐹 𝑖,𝑗 ( 𝑧 ) b 𝑈 + 𝑖,𝑖 ( d 𝑧 ) > . (cid:4) Lemma 4.6.
If for some 𝑗 ∈ Θ , 𝜉 ( 𝑗 ) has infinite jump activity on ℝ + , i.e. Π 𝑗 ( ℝ + ) = ∞ , then b 𝑈 + 𝑗,𝑖 doesnot have an atom at for all 𝑖 ≠ 𝑗 . Proof.
Suppose that there exists 𝑖 ≠ 𝑗 s.t. b 𝑈 + 𝑗,𝑖 ({ }) = 𝛼 >
0. Then, again plugging ( , ∞) into(4.4), implies 𝑞 + 𝑖,𝑗 ≥ 𝜋 ( 𝑗 ) 𝜋 ( 𝑖 ) 𝛼 Π 𝑗 ( ℝ + ) = ∞ , which is impossible. (cid:4) We can also use the équations amicales inversés to express our assumptions from Section 3 onthe ascending ladder height process ( 𝐻 + , 𝐽 + ) needed for ergodicity. That is, we can verify theconditions on the smoothness of the Lévy system required in Proposition 3.16 and the momentassumptions on the underlying Lévy processes and the transitional jumps required in Theorem3.21 for exponential or polynomial ergodicity of overshoots, in terms of related conditions onthe parent MAP ( 𝜉 , 𝐽 ) . Lemma 4.7. (i) If there exists 𝑖 ∈ Θ and ≤ 𝑎 < 𝑏 ≤ ∞ such that Leb | ( 𝑎,𝑏 ) ≪ Π 𝑖 | ( 𝑎,𝑏 ) , then also Leb | ( 𝑎,𝑏 ) ≪ Π + 𝑖 | ( 𝑎,𝑏 ) .(ii) If there exists 𝑖, 𝑗 ∈ Θ with 𝑖 ≠ 𝑗 and ≤ 𝑎 < 𝑏 ≤ ∞ such that Leb | ( 𝑎,𝑏 ) ≪ 𝑞 𝑖,𝑗 𝐹 𝑖,𝑗 | ( 𝑎,𝑏 ) , then also Leb | ( 𝑎,𝑏 ) ≪ 𝑞 + 𝑖,𝑗 𝐹 + 𝑖,𝑗 | ( 𝑎,𝑏 ) .(iii) For fixed 𝑖 ∈ Θ , 𝔼 [ exp ( 𝜆 𝐻 + , ( 𝑖 ) )] < ∞ if ∫ ∞ e 𝜆 𝑥 Π 𝑖 ( d 𝑥 ) + Õ 𝑘 ≠ 𝑖 𝑞 𝑘,𝑖 ∫ ∞ e 𝜆 𝑥 𝐹 𝑘,𝑖 ( d 𝑥 ) < ∞ . (4.10) (iv) For fixed 𝑖, 𝑗 ∈ Θ such that 𝑞 + 𝑗,𝑖 ≠ and 𝜆 > , 𝔼 [ exp ( 𝜆 Δ + 𝑗,𝑖 )] < ∞ if (4.10) holds.(v) If lim 𝑡 →∞ 𝜉 𝑡 = ∞ a.s., then for 𝜆 > and 𝑖 ∈ Θ , 𝔼 [( 𝐻 + , ( 𝑖 ) ) 𝜆 ] < ∞ if ∫ ∞ 𝑥 𝜆 Π 𝑖 ( d 𝑥 ) + Õ 𝑘 ≠ 𝑖 𝑞 𝑘,𝑖 ∫ ∞ 𝑥 𝜆 𝐹 𝑘,𝑖 ( d 𝑥 ) < ∞ , and for 𝑖, 𝑗 ∈ Θ such that 𝑞 + 𝑖,𝑗 ≠ , 𝔼 [( Δ + 𝑖,𝑗 ) 𝜆 ] < ∞ if ∫ ∞ 𝑥 𝜆 Π 𝑗 ( d 𝑥 ) + Õ 𝑘 ≠ 𝑖 𝑞 𝑘,𝑗 ∫ ∞ 𝑥 𝜆 𝐹 𝑘,𝑗 ( d 𝑥 ) < ∞ . Proof. (i) Let 𝐵 ⊂ ( 𝑎, 𝑏 ) be a Borel set s.t. Leb ( 𝐵 ) >
0. We may assume that sup 𝐵 < 𝑏 and hence 𝐵 + 𝑧 ⊂ ( 𝑎, 𝑏 ) for all 𝑧 ∈ ( , 𝑏 − sup 𝐵 ) . By translation invariance of the Lebesgue measure,we have Leb ( 𝐵 + 𝑧 ) > Π 𝑖 ( 𝐵 + 𝑧 ) > 𝑧 ∈ ( , 𝑏 − sup 𝐵 ) .From (4.3) it follows Π + 𝑖 ( 𝐵 ) ≥ ∫ ∞ b 𝑈 + 𝑖,𝑖 ( d 𝑧 ) Π 𝑖 ( 𝐵 + 𝑧 ) ≥ ∫ 𝑏 − sup 𝐵 b 𝑈 + 𝑖,𝑖 ( d 𝑧 ) Π 𝑖 ( 𝐵 + 𝑧 ) and since b 𝑈 + 𝑖,𝑖 ([ , 𝑏 − sup 𝐵 )) > ( 𝜉 , 𝐽 ) , it follows Π + 𝑖 ( 𝐵 ) >
0, implying Leb | ( 𝑎,𝑏 ) ≪ Π + 𝑖 | ( 𝑎,𝑏 ) . 32ii) This is immediate from (4.4) in Theorem 4.3 and the same arguments as in part (i).(iii) Since 𝐽 is irreducible, it follows from the proof of the Wiener–Hopf factorization in Theo-rem 26 of [21] that b 𝚽 + is invertible and hence, for any 𝑖, 𝑗 ∈ Θ we have ∫ ∞ e − 𝜆 𝑦 b 𝑈 + 𝑖,𝑗 ( d 𝑦 ) = (cid:0) b 𝚽 + ( 𝜆 ) − (cid:1) 𝑖,𝑗 . Thus, with Fubini and (4.3) ∫ ∞ e 𝜆 𝑥 Π + 𝑖 ( d 𝑥 ) = ∫ ∞ ∫ ∞ e 𝜆 𝑥 Π 𝑖 ( 𝑦 + d 𝑥 ) b 𝑈 + 𝑖,𝑖 ( d 𝑦 ) + Õ 𝑘 ≠ 𝑖 𝜋 ( 𝑘 ) 𝜋 ( 𝑖 ) 𝑞 𝑘,𝑖 ∫ ∞ ∫ ∞ e 𝜆 𝑥 𝐹 𝑘,𝑖 ( 𝑦 + d 𝑥 ) b 𝑈 + 𝑘,𝑖 ( d 𝑦 ) = ∫ ∞ ∫ ∞ + 𝑦 e 𝜆 𝑥 Π 𝑖 ( d 𝑥 ) e − 𝜆 𝑦 b 𝑈 + 𝑖,𝑖 ( d 𝑦 ) + Õ 𝑘 ≠ 𝑖 𝜋 ( 𝑘 ) 𝜋 ( 𝑖 ) 𝑞 𝑘,𝑖 ∫ ∞ ∫ ∞ + 𝑦 e 𝜆 𝑥 𝐹 𝑘,𝑖 ( d 𝑥 ) e − 𝜆 𝑦 b 𝑈 + 𝑘,𝑖 ( d 𝑦 ) , ≤ ∫ ∞ e 𝜆 𝑥 Π 𝑖 ( d 𝑥 ) ∫ ∞ e − 𝜆 𝑦 b 𝑈 + 𝑖,𝑖 ( d 𝑦 ) + Õ 𝑘 ≠ 𝑖 𝜋 ( 𝑘 ) 𝜋 ( 𝑖 ) 𝑞 𝑘,𝑖 ∫ ∞ e 𝜆 𝑥 𝐹 𝑘,𝑖 ( d 𝑥 ) ∫ ∞ e − 𝜆 𝑦 b 𝑈 + 𝑘,𝑖 ( d 𝑦 ) = ∫ ∞ e 𝜆 𝑥 Π 𝑖 ( d 𝑥 ) (cid:0) b 𝚽 + ( 𝜆 ) − (cid:1) 𝑖,𝑖 + Õ 𝑘 ≠ 𝑖 𝜋 ( 𝑘 ) 𝜋 ( 𝑖 ) 𝑞 𝑘,𝑖 ∫ ∞ e 𝜆 𝑥 𝐹 𝑘,𝑖 ( d 𝑥 ) (cid:0) b 𝚽 + ( 𝜆 ) − (cid:1) 𝑘,𝑖 , which is finite given the assumption.(iv) Analogously to (iii).(v) Under the assumption lim 𝑡 →∞ 𝜉 𝑡 = ∞ a.s., the ascending ladder height process of the dualof ( 𝜉 , 𝐽 ) is killed a.s. and hence for any 𝑖, 𝑗 ∈ Θ , b 𝑈 + 𝑖,𝑗 is a finite measure. Thus, again by(4.3), (4.4) and a change of variables, ∫ ∞ 𝑥 𝜆 Π + 𝑖 ( d 𝑥 ) ≤ b 𝑈 + 𝑖,𝑖 ( ℝ + ) ∫ ∞ 𝑥 𝜆 Π 𝑖 ( d 𝑥 ) + Õ 𝑘 ≠ 𝑖 𝜋 ( 𝑘 ) 𝜋 ( 𝑖 ) 𝑞 𝑘,𝑖 b 𝑈 + 𝑘,𝑖 ( ℝ + ) ∫ ∞ 𝑥 𝜆 𝐹 𝑘,𝑖 ( d 𝑥 ) < ∞ and 𝑞 + 𝑖,𝑗 ∫ ∞ 𝑥 𝜆 𝐹 + 𝑖,𝑗 ( d 𝑥 ) ≤ b 𝑈 + 𝑗,𝑖 ( ℝ + ) 𝜋 ( 𝑗 ) 𝜋 ( 𝑖 ) ∫ ∞ 𝑥 𝜆 Π 𝑗 ( d 𝑥 ) + Õ 𝑘 ≠ 𝑗 𝜋 ( 𝑘 ) 𝜋 ( 𝑖 ) 𝑞 𝑘,𝑗 b 𝑈 + 𝑘,𝑖 ( ℝ + ) ∫ ∞ 𝑥 𝜆 𝐹 𝑘,𝑗 ( d 𝑥 ) < ∞ . (cid:4) Remark . (i) Conditions (4.10) are sufficient but not necessary conditions for exponentialmoments of the components of the Lévy system 𝚷 + , since b 𝑈 + 𝑘,𝑖 is trivial for some 𝑘 ≠ 𝑖 whenever 𝐽 + is not irreducible under ( b ℙ ,𝑖 ) 𝑖 ∈ Θ . It is however true that if 𝔼 [ exp ( 𝜆 𝐻 + , ( 𝑖 ) )] < ∞ , we must necessarily have ∫ ∞ e 𝜆 𝑥 Π 𝑖 ( d 𝑥 ) < ∞ and if 𝔼 [ exp ( 𝜆 Δ + 𝑖,𝑗 )] < ∞ , it must hold ∫ ∞ e 𝜆 𝑥 𝐹 𝑖,𝑗 ( d 𝑥 ) < ∞ , since the on-diagonal potential measures b 𝑈 + 𝑖,𝑖 are non-trivial.(ii) We restrict to the case lim 𝑡 →∞ 𝜉 𝑡 = ∞ a.s. in (v). The oscillatory case lim sup 𝑡 →∞ 𝜉 𝑡 = − lim inf 𝑡 →∞ 𝜉 𝑡 = ∞ a.s. is more difficult to handle since in this case ( 𝐻 + , 𝐽 + ) is unkilledunder the dual measures b ℙ ,𝑖 and we have no control over b 𝑼 + solely in terms of thecharacteristics of ( 𝜉 , 𝐽 ) . In [21] the authors establish the necessary and sufficient integralcriterion given in (3.8) for finiteness of the first moment of 𝐻 + in the oscillatory regime bytaking a detour via random walk theory, building on the strategy for the related problemfor Lévy processes in [23]. Taking into account Theorem 1 of [18], such an ansatz, eventhough out of scope of this paper, is a possible strategy to tackle the problem at hand inour case as well. 33 . Application to real self-similar Markov processes In this section we show how to apply our results on the exponential mixing behavior of Markovadditive processes sampled at first hitting times to the class of 𝛼 -self-similar Markov processesand in particular strictly 𝛼 -stable Lévy processes. Even in the case of 𝛼 -stable processes theapplication is non-trivial since such Lévy processes do not satisfy the fundamental assumptionof finite mean of the associated ascending ladder height Lévy process, since in fact the ascendingladder height process is an 𝛼 -stable subordinator with 𝛼 ∈ ( , ) and thus does not have a finitefirst moment. Because of non-ergodicity of the associated overshoots, we can therefore notexpect a strong mixing behavior of the stable process sampled at first hitting times. However,making use of the Lamperti–Kiu transform for real self-similar Markov processes, we can givebounds on the 𝛽 -mixing coefficient of the 𝜎 -algebras generated by the past and the future of 𝛼 -self-similar process sampled at first hitting times given appropriate properties of the associatedMAP. By considering the Lamperti-stable MAP and its explicit characterization found in [17],we are thus able to bound the 𝛽 -mixing coefficient of the above 𝜎 -algebras for transient 𝛼 -stableprocesses. To this end, let us first recall the precise definitions of real 𝛼 -self-similar Markovprocesses and 𝛼 -stable Lévy processes and give a brief overview on the Lamperti–Kiu transformand its implications.We say that a real-valued Feller process ( Ω , G , ( G 𝑡 ) 𝑡 ≥ , ( 𝑍 𝑡 ) 𝑡 ≥ , ( P 𝑥 ) 𝑥 ∈ ℝ ) is an 𝛼 -self-similarMarkov process, if it satisfies the scaling property that for any 𝑐 > { 𝑍, P 𝑐𝑥 } d = (cid:8) (cid:0) 𝑐𝑍 𝑐 − 𝛼 𝑡 (cid:1) 𝑡 ≥ , P 𝑥 (cid:1) , 𝑥 ∈ ℝ . (5.1)An (unkilled) Lévy process 𝑋 = ( 𝑋 𝑡 ) 𝑡 ≥ with associated family of probability measures ( P 𝑥 ) 𝑥 ∈ ℝ is a strictly 𝛼 -stable Lévy process (or simply stable Lévy process for short if there is no roomfor confusion) for 𝛼 ∈ ( , ] if it satisfies (5.1). The case 𝛼 = 𝛼 -stable Lévy processes aretherefore particular representatives of 𝛼 -self-similar Markov processes.Taking the perspective commonly encountered in the literature to parametrize the stableprocess through its index of self-similarity 𝛼 and the positivity parameter 𝜌 ≔ P ( 𝑋 𝑡 ≥ ) , theLévy measure Π of 𝑋 is absolutely continuous with density 𝜋 satisfying 𝜋 ( 𝑥 ) = 𝑐 + 𝑥 −( 𝛼 + ) ( , ∞) ( 𝑥 ) + 𝑐 − | 𝑥 | −( 𝛼 + ) (−∞ , ) ( 𝑥 ) , 𝑥 ∈ ℝ , where 𝑐 + = Γ ( 𝛼 + ) sin ( 𝜋𝛼𝜌 ) 𝜋 , and 𝑐 − = Γ ( 𝛼 + ) sin ( 𝜋𝛼 ( − 𝜌 )) 𝜋 . The Lévy–Khintchine exponent Ψ is given by Ψ ( 𝜃 ) = 𝑐 | 𝜃 | 𝛼 (cid:0) − i 𝛽 tan 𝜋𝛼 sgn ( 𝜃 ) (cid:1) , 𝜃 ∈ ℝ , where 𝛽 = ( 𝑐 + − 𝑐 − )/( 𝑐 + + 𝑐 − ) and our specific parametrization forces 𝑐 = cos ( 𝜋𝛼 ( 𝜌 − / )) . Forall of the above statements we refer to Kyprianou [37].We now come to the one-to-one correspondence between self-similar Markov processes on ℝ and Markov additive processes on ℝ × {− , } expressed through the Lamperti–Kiu transform,which is investigated in [34] and [17] for the real valued setting, and, more generally for arbitrarystate spaces, in [1]. If we let 𝑍 be an 𝛼 -self-similar Markov process on ℝ absorbed at 0 withlifetime 𝜏 = inf { 𝑡 > 𝑋 𝑡 = } and define ℙ 𝑥,𝑖 = P 𝑖 e 𝑥 for ( 𝑥, 𝑖 ) ∈ ℝ × {− , } and ℙ −∞ , 𝜛 = P ,then the process ( 𝜉 , 𝐽 ) defined by ( 𝜉 𝑡 = log | 𝑍 𝜏 ( 𝑡 ) | and 𝐽 𝑡 = sgn ( 𝑍 𝜏 ( 𝑡 ) ) , if 𝑡 < ∫ 𝜏 | 𝑍 𝑠 | − 𝛼 d 𝑠 , ( 𝜉 𝑡 , 𝐽 𝑡 ) = 𝜗 ≕ (−∞ , 𝜛 ) , if 𝑡 ≥ ∫ 𝜏 | 𝑍 𝑠 | − 𝛼 d 𝑠 , 𝑡 ↦→ 𝜏 ( 𝑡 ) is the time change given by the right-continuous inverse 𝜏 ( 𝑡 ) ≔ inf { 𝑠 ≥ ∫ 𝑠 | 𝑍 𝑢 | − 𝛼 d 𝑢 > 𝑡 } , of the continuous additive functional ( 𝐴 𝑡 ) 𝑡 ≥ of 𝑍 , given by 𝐴 𝑡 ≔ ∫ 𝑡 ∧ 𝜏 | 𝑍 𝑠 | − 𝛼 d 𝑠 , 𝑡 ≥ , and 𝜛 is some isolated state, then (( 𝜉 , 𝐽 ) , ( ℙ 𝑥 ) 𝑥 ∈( ℝ ×{− , }) 𝜗 ) is a MAP on ℝ × {− , } with lifetime 𝜁 = ∫ 𝜏 | 𝑍 𝑠 | − 𝛼 d 𝑠 and underlying filtration ( F 𝑡 = G 𝜏 ( 𝑡 ) ) 𝑡 ≥ . Moreover, we have the followingtrichotomy characterizing the long-time behavior of the MAPs ordinator in terms of the hittingproperties of 𝑍 at 0 (one can indeed verify that self-similarity of 𝑍 guarantees that these are theonly possible cases):(a) if P 𝑥 ( 𝜏 < ∞) = 𝑥 ≠
0, then lim 𝑡 →∞ 𝜉 𝑡 = ∞ almost surely;(b) if P 𝑥 ( 𝜏 < ∞ , 𝑍 𝜏 − = ) = 𝑥 ≠
0, then lim 𝑡 →∞ 𝜉 𝑡 = −∞ almost surely;(c) if P 𝑥 ( 𝜏 < ∞ , 𝑍 𝜏 − ≠ ) = 𝑥 ≠
0, then the MAP is almost surely killed and itslifetime 𝜁 is exponentially distributed with a rate not depending on its initial distribution.Conversely, for a given MAP ( 𝜉 , 𝐽 ) with lifetime 𝜁 , 𝑍 𝑡 = 𝐽 𝜎 ( 𝑡 ) e 𝜉 𝜎 ( 𝑡 ) (cid:8) 𝑡 < ∫ 𝜁 e 𝛼𝜉 𝑠 d 𝑠 (cid:9) , 𝑡 ≥ , where 𝜎 ( 𝑡 ) = inf { 𝑠 ≥ ∫ 𝑠 e 𝛼𝜉 𝑢 d 𝑢 > 𝑡 } , 𝑡 ≥ , defines an 𝛼 -self-similar Markov process absorbed in 0 with lifetime 𝜏 = ∫ 𝜁 e 𝛼𝜉 𝑠 d 𝑠. This ishowever not the direction we are interested in and we refer the reader to the relevant literaturecited above for details. Note also that in case of 𝑍 being strictly positive almost surely upto its lifetime, the Lamperti–Kiu transform boils down to the Lamperti transform for positiveself-similar Markov processes and the associated MAP can be projected onto a killed Lévyprocess.With the help of the Lamperti–Kiu transform we obtain the following result on the 𝛽 -mixingcoefficient of the 𝜎 -algebras generated by 𝛼 -self-similar Markov processes sampled at past andfuture hitting times. While the Lamperti-stable MAP is exponentially 𝛽 -mixing under the givenassumptions, the 𝛽 -mixing coefficient for the 𝛼 -self similar Markov process sampled at firsthitting times shows non-homogeneous decay with almost square root rate as a result of thelogarithm present in the Lamperti–Kiu transform. Proposition 5.1.
Suppose that 𝑍 is 𝛼 -self-similar such that P 𝑥 ( 𝜏 < ∞) = for all 𝑥 ≠ andmoreover its associated Lamperti–Kiu MAP satisfies the assumptions from Theorem 3.21.(i). If 𝜂 is somedistribution on ( ℝ , B ( ℝ )) without atom at such that ∫ ℝ | 𝑥 | 𝜆 𝜂 ( d 𝑥 ) < ∞ , for some 𝜆 > , then for any 𝛿 ∈ ( , ) there exists a constant 𝐶 ( 𝜆 , 𝜂 , 𝛿 ) > such that for any 𝑡 ≥ wehave 𝛽 P 𝜂 ( N 𝑡 , N 𝑡 + 𝑠 ) ≤ 𝐶 ( 𝜆 , 𝜂 ) (cid:16) 𝑡 + 𝑠𝑡 (cid:17) − /( + 𝛿 ) , 𝑠 > , where we denoted N 𝑡 = 𝜎 (cid:0) 𝑍 𝑇 𝑍𝑠 , 𝑠 ≤ 𝑡 (cid:1) , N 𝑡 = 𝜎 (cid:0) 𝑍 𝑇 𝑍𝑠 , 𝑠 ≥ 𝑡 (cid:1) . roof. First, observe that 𝑍 not hitting 0 almost surely when started away from the originimplies that the time change ( 𝜏 ( 𝑡 )) 𝑡 ≥ is strictly increasing and continuous almost surely. Thus,the overshoot process of ( log | 𝑍 𝑡 | , sgn ( 𝑍 𝑡 )) 𝑡 ≥ is indistinguishable from the overshoot process ofthe associated Lamperti-MAP ( 𝜉 , 𝐽 ) . Moreover, the mapping 𝜙 : ℝ × {− , } → ℝ \ { } , ( 𝑥, 𝑖 ) ↦→ 𝑖 e 𝑥 , is a homeomorphism and 𝑍 𝑡 = 𝜙 ( log | 𝑍 𝑡 | , sgn ( 𝑍 𝑡 )) for all 𝑡 ≥ Λ = { 𝜔 ∈ Ω : 𝑍 𝑡 ( 𝜔 ) ≠ 𝑡 ≥ } , which is of P 𝜇 -measure 1 for any distribution 𝜇 on ( ℝ , B ( ℝ )) not having anatom at 0. It follows for any 𝑡 ≥ P 𝜇 -nullset 𝑁 𝜇 𝑡 such that N 𝑡 ∨ 𝑁 𝜇 𝑡 = (cid:0) (cid:0) 𝜉 𝑇 𝑠 , 𝐽 𝑇 𝑠 (cid:1) , 𝑠 ≤ log 𝑡 (cid:1) ∨ 𝑁 𝜇 𝑡 = K log ( 𝑡 ) ∨ 𝑁 𝜇 𝑡 , and N 𝑡 ∨ 𝑁 𝜇 𝑡 = (cid:0) (cid:0) 𝜉 𝑇 𝑠 , 𝐽 𝑇 𝑠 (cid:1) , 𝑠 ≥ log 𝑡 (cid:1) ∨ 𝑁 𝜇 𝑡 = K log ( 𝑡 ) ∨ 𝑁 𝜇 𝑡 , where for two 𝜎 -algebras A , B we write A ∨ B = 𝜎 ( A ∪ B ) . Here we used the definition of theLamperti–Kiu transform and the fact that for any 𝑡 ≥ 𝑇 | 𝑍 | 𝑡 = 𝑇 log 𝑡 . Since moreover P 𝜂 = ℙ 𝜂 ◦ 𝜙 and by assumption ∫ ℝ ×{− , } e 𝜆 𝑥 𝜂 ◦ 𝜙 ( d 𝑥, d 𝑖 ) = ∫ ℝ \{ } e 𝜆 log | 𝑧 | 𝜂 ( d 𝑧 ) = ∫ ℝ \{ } | 𝑧 | 𝜆 𝜂 ( d 𝑧 ) < ∞ , it follows from Corollary 3.26 and the assumptions on the Lamperti-MAP ( 𝜉 , 𝐽 ) that for any 𝛿 ∈ ( , ) there exists 𝐶 ( 𝜆 , 𝜂 , 𝛿 ) > 𝑡 ≥ 𝑠 > 𝛽 P 𝜂 ( N 𝑡 , N 𝑡 + 𝑠 ) = 𝛽 ℙ 𝜂 ◦ 𝜙 ( K log 𝑡 , K log ( 𝑡 + 𝑠 ) ) ≤ 𝐶 ( 𝜆 , 𝜂 , 𝛿 ) e −( log ( 𝑡 + 𝑠 )− log 𝑡 )/( + 𝛿 ) = 𝐶 ( 𝜆 , 𝜂 , 𝛿 ) (cid:16) 𝑡 + 𝑠𝑡 (cid:17) − /( + 𝛿 ) , as claimed. Note here that the nullsets 𝑁 𝜂 𝑡 and 𝑁 𝜂 𝑡 + 𝑠 from above have no influence on the 𝛽 -mixing coefficient by its definition. (cid:4) Consider now a scalar 𝛼 -stable process ( 𝑋 𝑡 ) 𝑡 ≥ absorbed upon hitting of the origin, i.e. for 𝜏 = inf { 𝑠 ≥ 𝑋 𝑠 = } , 𝑋 𝑡 = 𝑋 𝑡 [ , 𝜏 ) ( 𝑡 ) , 𝑡 ≥ . We show that 𝑋 satisfies the assumptions from Proposition 5.1 that yield 𝛽 -mixing of overshootsof the corresponding MAP ( 𝜉 , 𝐽 ) obtained through the Lamperti–Kiu transform, which wehenceforth will refer to as the Lamperti-stable MAP .Since the assumptions are couched in form of the ascending ladder height process ( 𝐻 + , 𝐽 + ) ,one direct approach would be to make use of the deep factorization of 𝑋 given in [37], where theMAP exponents 𝚽 + and b 𝚽 + of the ascending ladder height processes of ( 𝜉 , 𝐽 ) and its dual wereexplicitly calculated. However, for the sake of exposition, we go another route by making useof the results based on Vigon’s équations amicales inversés from Section 4 to infer the neededproperties of ( 𝐻 + , 𝐽 + ) from those of ( 𝜉 , 𝐽 ) . The characteristics of the latter were calculated inTheorem 10 and Corollary 11 of Chaumont et al. [17], giving 𝜎 ± =
0, i.e. the underlying Lévyprocesses have no Gaussian component, Π ± ( d 𝑥 ) = e 𝑥 𝜋 (±( e 𝑥 − )) d 𝑥, 𝑥 ∈ ℝ ,𝐹 ± , ∓ ( d 𝑥 ) = 𝛼 e 𝑥 ( + e 𝑥 ) 𝛼 + d 𝑥, 𝑥 ∈ ℝ , and 𝑞 ± , ∓ = 𝑐 ∓ 𝛼 .
36f we assume that 𝑋 does not have one-sided jumps, then 𝑐 ± > 𝐽 is irreducible.Since Π ± has a strictly positive Lebesgue density on ( , ∞) it follows by Lemma 4.7 thatLeb | ( , ∞) ≪ Π +± | ( , ∞) as well. Further, we have for 𝜆 > ∫ ∞ e 𝜆 𝑥 Π ( d 𝑥 ) = 𝑐 + ∫ ∞ e ( 𝜆 + ) 𝑥 ( e 𝑥 − ) −( 𝛼 + ) d 𝑥, and hence ∫ ∞ e 𝜆 𝑥 Π ( d 𝑥 ) < ∞ ⇐⇒ 𝜆 ∈ ( , 𝛼 ) . Similarly, we obtain ∫ ∞ e 𝜆 𝑥 Π − ( d 𝑥 ) < ∞ ⇐⇒ 𝜆 ∈ ( , 𝛼 ) , and hence 𝔼 [ exp ( 𝜆𝜉 (± ) )] < ∞ iff 𝜆 ∈ ( , 𝛼 ) . Moreover, ∫ ℝ e 𝜆 𝑥 𝐹 ± ( d 𝑥 ) = 𝛼 ∫ ℝ e ( 𝜆 + ) 𝑥 ( + e 𝑥 ) −( 𝛼 + ) d 𝑥 < ∞ ⇐⇒ 𝜆 ∈ ( , 𝛼 ) . Again by Lemma 4.7 we conclude that 𝐻 + , (± ) and Δ +± , ∓ all possess an exponential 𝜆 -momentwhenever 𝜆 ∈ ( , 𝛼 ) .Recall now that 𝑋 does not hit 0 if and only if 𝛼 ∈ ( , ) and hence the ordinator 𝜉 ofthe Lamperti-stable MAP satisfies lim sup 𝑡 →∞ 𝜉 𝑡 = ∞ almost surely if and only if 𝛼 ∈ ( , ) .Since our asymptotic approach on overshoots of MAPs requires this property, we will restrictto this case and can therefore identify 𝑋 = 𝑋 almost surely. All that remains to show forexponential 𝛽 -mixing of the Lamperti-stable MAP is now upward regularity and irreducibilityof 𝐽 + . Irreducibility of 𝐽 + is a direct consequence of Proposition 3.5 since 𝐽 is irreducible andthe support of Π ± is unbounded. To verify upward regularity, we observe that by Theorem1 in Kuznetsov and Pardo [36], 𝜉 ( ) killed at an independent exponential time with rate 𝑐 − / 𝛼 belongs to the class of hypergeometric Lévy processes with parameters ( − 𝛼 ( − 𝜌 ) , 𝛼𝜌 , ( − 𝛼 )( − 𝜌 ) , 𝛼 ( − 𝜌 )) . The ascending ladder height process 𝐻 of such a hypergeometric Lévy process isa 𝛽 -subordinator with parameters ( 𝛼 ( − 𝜌 ) , 𝛼 ( − 𝜌 ) , − 𝛼𝜌 ) , whose Lévy measure is given by Π 𝐻 ( d 𝑥 ) = − 𝛼𝜌 Γ ( 𝛼𝜌 ) ( − e − 𝑥 ) 𝛼𝜌 − e −( + 𝛼 ( − 𝜌 )) 𝑥 d 𝑥, 𝑥 > . Clearly, Π 𝐻 (( , )) = ∞ and hence 𝐻 is not compound Poisson, which shows that the associatedhypergeometric Lévy process is upward regular. Since killing has no influence on upwardregularity, this now shows that 𝜉 ( ) is indeed upward regular. Upward regularity of 𝜉 (− ) can beargued analogously once we observe that 𝜉 (− ) killed at rate 𝑐 + / 𝛼 is the hypergeometric processobtained from killing the dual process b 𝑋 of 𝑋 upon entering (−∞ , ] . Hence, with the ergodicanalysis of overshoots from Section 3 and Proposition 5.1, we have proved the following. Proposition 5.2.
Let 𝛼 ∈ ( , ) and 𝑋 be strictly 𝛼 -stable. Then the overshoot process of the Lamperti-stable MAP associated to 𝑋 is R 𝜆 𝑉 𝜆 -uniformly ergodic and for any starting distribution 𝜇 such that 𝜇 (· , {− , }) has an exponential 𝜆 -moment for some 𝜆 ∈ ( , 𝛼 ) , the overshoot process is exponentially 𝛽 -mixing. Moreover, for any distribution 𝜂 on ( ℝ , B ( ℝ )) without atom at such that for some 𝜆 ∈ ( , 𝛼 ) , ∫ ℝ | 𝑥 | 𝜆 𝜂 ( d 𝑥 ) < ∞ , there exists a constant 𝐶 ( 𝜆 , 𝜂 , 𝛿 ) > for any 𝛿 ∈ ( , ) such that for any 𝑡 ≥ we have 𝛽 P 𝜂 ( N 𝑡 , N 𝑡 + 𝑠 ) ≤ 𝐶 ( 𝜆 , 𝜂 , 𝛿 ) (cid:16) 𝑡 + 𝑠𝑡 (cid:17) − /( + 𝛿 ) , 𝑠 > , where we denoted N 𝑡 = 𝜎 (cid:0) 𝑋 𝑇 𝑋𝑠 , 𝑠 ≤ 𝑡 (cid:1) , N 𝑡 = 𝜎 (cid:0) 𝑋 𝑇 𝑋𝑠 , 𝑠 ≥ 𝑡 (cid:1) . orollary 5.3. Let 𝛼 ∈ ( , ) and 𝑋 be strictly 𝛼 -stable. Then, for any 𝑥 > and 𝛿 ∈ ( , ) there existsa constant 𝐶 ( 𝑥, 𝛿 ) sich that for any 𝑡 ≥ , 𝛽 P ( N 𝑡 , N 𝑡 + 𝑠 ) ≤ 𝐶 ( 𝑥, 𝛿 ) (cid:16) 𝑡 + 𝑥 + 𝑠𝑡 + 𝑥 (cid:17) − /( + 𝛿 ) , 𝑠 > . Proof.
Fix 𝑥 > 𝜆 ∈ ( , 𝛼 ) . By spatial homogeneity of 𝑋 we have n (cid:16) 𝑋 𝑇 𝑋𝑡 + 𝑥 , 𝑡 ≥ (cid:17) , P 𝑥 o d = n (cid:16) 𝑋 𝑇 𝑋𝑡 + 𝑥, 𝑡 ≥ (cid:17) , P o and therefore, using Proposition 5.2 𝛽 P ( N 𝑡 , N 𝑡 + 𝑠 ) = 𝛽 P 𝑥 ( N 𝑡 + 𝑥 , N 𝑡 + 𝑥 + 𝑠 ) ≤ 𝐶 ( 𝑥, 𝛿 ) (cid:16) 𝑡 + 𝑥 + 𝑠𝑡 + 𝑥 (cid:17) − /( + 𝛿 ) , where 𝐶 ( 𝑥 ) ≔ 𝐶 ( 𝜆 , 𝛿 𝑥 , 𝛿 ) . (cid:4) A. Stability of Markov processes
The theory of stability of time-continuous Markov processes is the essential tool for our analysisof MAPs sampled at first hitting times through studying ergodic properties of its overshoots.Stability of discrete-time Markov chains has a long history dating back at least to the 1930s,whereas the systematic treatment of stability of continuous-time Markov processes is compar-atively young. The Meyn and Tweedie approach developed in the 1990s, which we will follow,infers recurrence and ergodic properties of continuous-time processes through techniques de-veloped for the discrete-time case by means of sampling the process on a countable grid of(random) times. In this way accessible criteria are established in terms of the transition semi-group, the generator or the resolvent of the Markov process, at least one of which is availablefor the specific processes under consideration. Let us therefore introduce the most importantterminology and results that we will need in the following.Let X = ( 𝑋 𝑡 ) 𝑡 ≥ be a continuous-time Borel right Markov process on a locally compact,separable space ( X , B ( X )) with cemetery state 𝜗 , lifetime 𝜁 and underlying family of probabilitymeasures ( ℙ 𝑥 ) 𝑥 ∈ X 𝜗 , where X 𝜗 denotes the Alexandrov one-point compactification of X . Denoteits sub-Markov transition semigroup by ( 𝑃 𝑡 ) 𝑡 ≥ , which is induced by ( ℙ 𝑥 ) 𝑥 ∈ X 𝜗 via 𝑃 𝑡 ( 𝑥, 𝐵 ) = ℙ 𝑥 ( 𝑋 𝑡 ∈ 𝐵, 𝑡 < 𝜁 ) for ( 𝑥, 𝐵 ) ∈ X 𝜗 × B ( X 𝜗 ) . Note that by our convention to extend functions 𝑓 ∈ B ( X ) to B ( X 𝜗 ) by setting 𝑓 ( 𝜗 ) = ( 𝑃 𝑡 ) 𝑡 ≥ restricted to B 𝑏 ( X ) coincides with the Markovtransition semigroup of X . Borel right Markov processes are the most general class of strongMarkov processes widely used in the literature and are the cornerstone of the théorie générale of Markov processes developed mainly in the 1960s to 1980s based on fundamental works ofDynkin, Feller and others. Their precise potential theoretically motivated definition can befound in Sharpe [51, Definition 8.1], but for our purposes it will be enough to know that Fellerprocesses as defined below are Borel right, which follows from Theorem II.2.12 in [12].We understand Feller processes as càdlàg Markov processes with right-continuous and com-plete filtration, whose sub-Markov transition semigroup ( 𝑃 𝑡 ) 𝑡 ≥ is (i) strongly continuous, i.e. k 𝑃 𝑡 𝑓 − 𝑓 k ∞ → 𝑡 → 𝑓 ∈ C ( X ) , and (ii) 𝑃 𝑡 C ( X ) ⊂ C ( X ) for all 𝑡 ≥
0, where C ( X ) isthe space of continuous functions on X vanishing at infinity. Note that in presence of the Fellerproperty (ii), strong continuity (i) is actually satisfied whenever we have pointwise convergence 𝑃 𝑡 𝑓 ( 𝑥 ) → 𝑓 ( 𝑥 ) as 𝑡 → 𝑥 ∈ X , see e.g. Kallenberg [32, Theorem 19.6].The resolvent ( 𝑈 𝜆 ) 𝜆> of X is the operator defined by 𝑈 𝜆 𝑓 ( 𝑥 ) = 𝔼 𝑥 h ∫ ∞ e − 𝜆 𝑡 𝑓 ( 𝑋 𝑡 ) d 𝑡 i = ∫ ∞ e − 𝜆 𝑡 𝔼 𝑥 [ 𝑓 ( 𝑋 𝑡 )] d 𝑡 , 𝜆 > , 𝑥 ∈ X 𝜗 , 𝑓 ∈ B 𝑏 ( X 𝜗 ) ∪ B + ( X 𝜗 ) , 𝑥 ∈ X and 𝑓 ∈ B 𝑏 ( X ) can be written as 𝑈 𝜆 𝑓 ( 𝑥 ) = ∫ ∞ e − 𝜆 𝑡 𝑃 𝑡 𝑓 ( 𝑥 ) d 𝑡. The 𝜆 -resolvent 𝑈 𝜆 can be interpreted as the potential 𝑈 of the Markov process X killed at anindependent exponential time with rate 1 / 𝜆 , where for 𝑥 ∈ X 𝜗 , the potential 𝑈 ( 𝑥, ·) defined by 𝑈 ( 𝑥, 𝐵 ) ≔ ∫ ∞ ℙ 𝑥 ( 𝑋 𝑡 ∈ 𝐵 ) d 𝑡 , 𝐵 ∈ B ( X 𝜗 ) , is the expected sojourn time of X in 𝐵 when started in X .Suppose from here on that 𝜁 is almost surely infinite and thus X is an unkilled Borel rightMarkov process, which is the setting in which Meyn and Tweedie’s stability theory is embeddedin. We say that a 𝜎 -finite measure 𝜒 on ( X , B ( X )) is an invariant measure for X , if ∀ 𝐵 ∈ B ( X ) : ℙ 𝜒 ( 𝐵 ) ≔ ∫ ∞ ℙ 𝑥 ( 𝑋 𝑡 ∈ 𝐵 ) 𝜒 ( d 𝑥 ) = 𝜒 ( 𝐵 ) . Note that an invariant measure is never unique, since any scaling of the measure is againinvariant. We therefore say that an invariant measure 𝜒 is essentially unique if it is unique up toconstant multiples. If 𝜒 ( X ) =
1, we call 𝜒 an invariant distribution (which is unique under Harrisrecurrence, which we define below). The following proposition gives a criterion in terms of theresolvent, which is helpful for detecting an invariant measure provided the resolvent can bedetermined analytically. The statement has a very classical flavor, but we were not able to findit in the literature. We will use it in combination with an analytical treatment of the resolventof the overshoot process to determine an essentially unique measure for this process. Proposition A.1.
Suppose that H ⊂ B 𝑏 ( X ) ∩ B + ( X ) such that 𝑃 𝑡 H ⊂ H for any 𝑡 ≥ and there isa non-trivial measure 𝜒 on ( X , B ( X )) and a family ( 𝛼 𝜆 ) 𝜆> of finite measures on ( X , B ( X )) satisfying lim 𝜆 ↓ 𝛼 𝜆 ( X ) = such that for any 𝑓 ∈ H lim 𝜆 ↓ ∫ X 𝑈 𝜆 𝑓 ( 𝑥 ) 𝛼 𝜆 ( d 𝑥 ) = 𝜒 ( 𝑓 ) ≔ ∫ X 𝑓 ( 𝑦 ) 𝜒 ( d 𝑦 ) . (A.1) Then, for any 𝑡 ≥ and 𝑓 ∈ H , ∫ X 𝑃 𝑡 𝑓 ( 𝑦 ) 𝜒 ( d 𝑦 ) = 𝜒 ( 𝑓 ) . In particular, if H = B 𝑏 ( X ) ∩ B + ( X ) (i.e. 𝑈 𝛼 𝜆 𝜆 ≔ ∫ X 𝑈 𝜆 ( 𝑥, d 𝑦 ) 𝛼 𝜆 ( d 𝑥 ) converges strongly to 𝜒 as 𝜆 ↓ ), then 𝜒 is an invariant measure of X . Proof.
Let 𝑓 ∈ H such that (A.1) holds and 𝑡 ≥
0. We have for any 𝜆 > ( 𝑃 𝑡 ) 𝑡 ≥ 𝑈 𝛼 𝜆 𝜆 𝑃 𝑡 𝑓 = ∫ X ∫ ∞ e − 𝜆 𝑠 𝑃 𝑠 𝑃 𝑡 𝑓 ( 𝑥 ) d 𝑠 𝛼 𝜆 ( d 𝑥 ) = ∫ X ∫ ∞ e − 𝜆 𝑠 𝑃 𝑠 + 𝑡 𝑓 ( 𝑥 ) d 𝑠 𝛼 𝜆 ( d 𝑥 ) = e 𝜆 𝑡 ∫ X ∫ ∞ 𝑡 e − 𝜆 𝑠 𝑃 𝑠 𝑓 ( 𝑥 ) d 𝑠 𝛼 𝜆 ( d 𝑥 ) = e 𝜆 𝑡 (cid:18) 𝑈 𝛼 𝜆 𝜆 𝑓 − ∫ X ∫ 𝑡 e − 𝜆 𝑠 𝑃 𝑠 𝑓 ( 𝑥 ) d 𝑠 𝛼 𝜆 ( d 𝑥 ) (cid:19) . | ∫ X ∫ 𝑡 e − 𝜆 𝑠 𝑃 𝑠 𝑓 ( 𝑥 ) d 𝑠 𝛼 𝜆 ( d 𝑥 )| ≤ 𝑡 k 𝑓 k ∞ 𝛼 𝜆 ( X ) it therefore follows by our assumption that 𝛼 𝜆 ( X ) → 𝑈 𝛼 𝜆 𝜆 𝑓 → 𝜒 ( 𝑓 ) as 𝜆 ↓ 𝜆 ↓ 𝑈 𝛼 𝜆 𝜆 𝑃 𝑡 𝑓 = 𝜒 ( 𝑓 ) . On the other hand, our assumptions and 𝑃 𝑡 𝑓 ∈ H yield thatlim 𝜆 ↓ 𝑈 𝛼 𝜆 𝜆 𝑃 𝑡 𝑓 = ∫ X 𝑃 𝑡 𝑓 ( 𝑦 ) 𝜒 ( d 𝑦 ) and hence ∫ X 𝑃 𝑡 𝑓 ( 𝑦 ) 𝜒 ( d 𝑦 ) = 𝜒 ( 𝑓 ) follows. If H = B 𝑏 ( X ) ∩ B + ( X ) , then for any 𝐵 ∈ B ( X ) the choice 𝑓 = 𝐵 shows that 𝑃 𝑡 ( 𝜒 , 𝐵 ) = 𝜒 ( 𝐵 ) , ∀ 𝑡 ≥ , i.e. 𝜒 is an invariant measure. (cid:4) A 𝜎 -finite measure 𝜓 is called irreducibility measure of X , if for any Borel set 𝐵 , 𝜓 ( 𝐵 ) > 𝑈 ( 𝑥, 𝐵 ) > 𝑥 ∈ X . Whenever such a measure exists, we say that X is 𝜓 - irreducible orsimply irreducible when the specific measure does not matter. If X is irreducible, there existsa maximal irreducibility measure 𝜓 in the sense that for any irreducibility measure 𝜈 of X itholds that 𝜈 ≪ 𝜓 , see Tweedie [53, Theorem 2.1]. We define B + ( X ) ≔ { 𝐵 ∈ B ( X ) : 𝜓 ( 𝐵 ) > } and call sets in B + ( X ) accessible. Note that maximal irreducibility measures are clearly non-unique. Moreover, if X is 𝜓 -irreducible and admits an invariant measure 𝜒 , then 𝜒 is a maximalirreducibility measure. To see this, let 𝜓 ( 𝐵 ) >
0, then 𝑡 𝜒 ( 𝐵 ) = ∫ 𝑡 (cid:16) ∫ X ℙ 𝑥 ( 𝑋 𝑠 ∈ 𝐵 ) 𝜒 ( d 𝑥 ) (cid:17) d 𝑠 = ∫ X (cid:16) ∫ 𝑡 ℙ 𝑥 ( 𝑋 𝑠 ∈ 𝐵 ) d 𝑠 (cid:17) 𝜒 ( d 𝑥 ) and by monotone convergence the right hand side converges to 𝑈 ( 𝜒 , 𝐵 ) ≔ ∫ X 𝑈 ( 𝑥, 𝐵 ) 𝜒 ( d 𝑥 ) > 𝑈 ( 𝑥, 𝐵 ) > 𝑥 ∈ X by our choice of 𝐵 . Hence, 𝜓 ≪ 𝜒 . The next important concept, Harris recurrence , is an even stronger property than irreducibility. We say that X is 𝜇 - Harrisrecurrent if there exists a 𝜎 -finite measure 𝜇 on the state space s.t. ∀ 𝐵 ∈ B ( X ) : 𝜇 ( 𝐵 ) > = ⇒ ℙ 𝑥 (cid:16) ∫ ∞ 𝐵 ( 𝑋 𝑡 ) d 𝑡 = ∞ (cid:17) = , ∀ 𝑥 ∈ X , (A.2)i.e. if 𝜇 ( 𝐵 ) >
0, the process almost surely spends infinitely much time in the set 𝐵 . A powerfulimplication of Harris recurrence is that an invariant measure of a Markov process having thisproperty (we call such processes positive Harris recurrent ) is essentially unique, see [3, Théorème2.5]. Moreover, by the remark succeding this theorem in [3], an invariant measure 𝜒 of a Harrisrecurrent process is a Harris measure. Thus, it is maximal Harris in the sense that it dominatesany other Harris measure, since any Harris measure is in particular an irreducibility measureand 𝜒 is a maximal irreducibility measure, as discussed above. The defining condition for Harrisrecurrence is often hard to check directly, however, Kaspi and Mandelbaum [33, Theorem 1]provide us with a simpler equivalent criterion for Borel right Markov processes: suppose thatthere exists a 𝜎 -finite measure 𝜈 such that for any Borel set 𝐵 we have the implication 𝜈 ( 𝐵 ) > = ⇒ ℙ 𝑥 ( 𝑇 𝐵 < ∞) = , ∀ 𝑥 ∈ X , (A.3)40here 𝑇 𝐵 ≔ inf { 𝑡 ≥ 𝑋 𝑡 ∈ 𝐵 } is the first hitting time of 𝐵 . Then, X is Harris recurrent and aHarris recurrence measure 𝜇 is given by 𝜇 ( 𝐵 ) = 𝔼 𝜈 h ∫ ∞ e − 𝑡 𝐵 ( 𝑋 𝑡 ) d 𝑡 i = 𝑈 ( 𝜈 , 𝐵 ) , 𝐵 ∈ B ( X ) . (A.4)Let us now recall the notion of petite and small sets, with the former concept being a general-ization of the latter. We say that a non-empty set 𝐶 ∈ B ( X ) is petite, if there exists a samplingdistribution 𝑎 on (( , ∞) , B ( , ∞)) and a non-trivial measure 𝜈 𝑎 on the state space such that forthe sampled kernel 𝐾 𝑎 ( 𝑥, d 𝑦 ) ≔ ∫ ∞ + 𝑃 𝑡 ( 𝑥, d 𝑦 ) 𝑎 ( d 𝑡 ) , 𝑥, 𝑦 ∈ X , it holds that 𝐾 𝑎 ( 𝑥, ·) ≥ 𝜈 𝑎 (·) , 𝑥 ∈ 𝐶. The sampled kernel corresponds to the transition kernel of the discrete-time Markov chain ob-tained from X by sampling at renewal times of an independent renewal process with incrementdistribution 𝑎 . An important special case is the 𝜆 -resolvent kernel 𝑅 𝜆 ( 𝑥, d 𝑦 ) ≔ ∫ ∞ + 𝜆 e − 𝜆 𝑡 𝑃 𝑡 ( 𝑥, d 𝑦 ) d 𝑡 = 𝜆 𝑈 𝜆 ( 𝑥, d 𝑦 ) , 𝑥, 𝑦 ∈ X , obtained for the sampling distribution 𝑎 = Exp ( 𝜆 ) , 𝜆 >
0. If 𝑎 = 𝛿 Δ for some Δ >
0, then 𝐶 is called a small set and we refer to the sampled chain X Δ ≔ ( 𝑋 𝑛 Δ ) 𝑛 ∈ ℕ as the Δ -skeleton of X .The importance of petite sets comes from the fact, that petite sets are small for the sampledchain and small sets in discrete time Markov chain theory allow to construct a related Markovchain possessing an atom via the technique of Nummelin splitting, which then makes reasoningwell-known for Markov chains on countable state spaces transferrable to the general state spacesituation. We refer to Meyn and Tweedie [42] for a comprehensive account. Let us also remarkthat petite sets are by no means rare. E.g. consider the case that X is a 𝑇 -process, that is thereexists a non-trivial continuous component 𝑇 for the sampled kernel 𝐾 𝑎 for some samplingdistribution 𝑎 , meaning that(a) 𝑥 ↦→ 𝑇 ( 𝑥, 𝐵 ) is lower semicontinuous for all 𝐵 ∈ B ( X ) ;(b) 𝐾 𝑎 ( 𝑥, 𝐵 ) ≥ 𝑇 ( 𝑥, 𝐵 ) for all 𝑥 ∈ X and 𝐵 ∈ B ( X ) . Then every compact subset of X is petite, provided that X is irreducible, see Theorem 5.1 in [53].The final concept that we need is aperiodicity . We say that X is aperiodic, if there exists a petiteset 𝐶 ∈ B + ( X ) (i.e. 𝐶 must be accessible) and some 𝑇 ≥ ∀ 𝑡 ≥ 𝑇 , 𝑥 ∈ 𝐶 : 𝑃 𝑡 ( 𝑥, 𝐶 ) > . Alternatively, X is called aperiodic in [42] if there exists some Δ > Δ -skeleton X Δ is irreducible, i.e. there exists a 𝜎 -finite measure 𝜇 on ( X , B ( X )) such that 𝜇 ( 𝐵 ) > = ⇒ ∀ 𝑥 ∈ X : ∞ Õ 𝑛 = ℙ 𝑥 ( 𝑋 𝑛 Δ ∈ 𝐵 ) > . It seems to be well-known in the literature that the existence of an irreducible skeleton chain fora Harris recurrent Markov process implies aperiodicity, but there is no concrete statement tobe found. Proposition 6.1 in [44], which [24] refers to, does not quite state that irreducibility ofskeletons implies aperiodicity, but indeed provides the right tool to prove it. For completenesswe give the short proof and make the additional simple observation that if the petite set 𝐶 inthe definition of aperiodicity is a singleton set, then aperiodicity also implies the existence ofan irreducible skeleton chain, which will be useful later on.41 emma A.2. Suppose that the 𝜓 -irreducible Markov process X is positive Harris recurrent, Borel rightand its state space is locally compact and separable. Then, if there exists some irreducible skeleton chain, X is aperiodic. Conversely, if X is aperiodic and the defining set 𝐶 is a singleton set, then any Δ -skeletonis irreducible. Proof.
Suppose first that there exists some irreducible Δ -skeleton. Then, the assumptions onthe process allow to use Proposition 6.1 from [43], which states that for any petite set 𝐶 thereexists some non-trivial measure 𝜇 and and a 𝑇 > 𝑡 ≥ 𝑇 we have ℙ 𝑥 ( 𝑋 𝑡 ∈ ·) ≥ 𝜇 (·) , ∀ 𝑡 ≥ 𝑇 , 𝑥 ∈ 𝐶 , (A.5)which implies in particular that 𝐶 is even a small set. By the Markov property it thus followsfor 𝑠 ≥ ℙ 𝑥 ( 𝑋 𝑡 + 𝑠 ∈ ·) = ∫ X ℙ 𝑥 ( 𝑋 𝑡 ∈ d 𝑦 ) ℙ 𝑦 ( 𝑋 𝑠 ∈ ·) ≥ ∫ X 𝜇 ( d 𝑦 ) ℙ 𝑦 ( 𝑋 𝑠 ∈ ·) = ℙ 𝜇 ( 𝑋 𝑠 ∈ ·) , ∀ 𝑡 ≥ 𝑇 , 𝑥 ∈ 𝐶. (A.6)By Proposition 3.4 of Meyn and Tweedie [41] the state space X can be covered by countably manypetite sets ( = small sets in our case), hence we may assume that 𝜓 ( 𝐶 ) >
0, i.e. 𝐶 ∈ B + ( X ) . Notethat 𝑈 ( 𝑥, 𝐶 ) > 𝑥 ∈ X and non-triviality of 𝜇 then yield that 𝑈 ( 𝜇 , 𝐶 ) = ∫ X 𝑈 ( 𝑥, 𝐶 ) 𝜇 ( d 𝑥 ) > 𝑈 ( 𝜇 , 𝐶 ) = ∫ ∞ ℙ 𝜇 ( 𝑋 𝑡 ∈ 𝐶 ) d 𝑡 it follows that there exists 𝑠 > ℙ 𝜇 ( 𝑋 𝑠 ∈ 𝐶 ) >
0. From (A.6) it thus follows that for such 𝑠 and all 𝑡 ≥ 𝑇 + 𝑠 and 𝑥 ∈ 𝐶 it holdsthat ℙ 𝑥 ( 𝑋 𝑡 ∈ 𝐶 ) ≥ ℙ 𝜇 ( 𝑋 𝑠 ∈ 𝐶 ) > , which proves aperiodicity of X .Suppose now that X is aperiodic with defining small singleton set 𝐶 = { 𝑐 } ∈ B + ( X ) for some 𝑐 ∈ X . Then, there exists 𝑇 > ℙ 𝑐 ( 𝑋 𝑡 = 𝑐 ) > , ∀ 𝑡 ≥ 𝑇 , and 𝛿 𝑐 is an irreducibility measure. Then, for given 𝑥 ∈ X , there exist 𝑡 𝑥 such that ℙ 𝑥 ( 𝑋 𝑡 𝑥 = 𝑐 ) > 𝑡 ≥ 𝑇 ℙ 𝑥 ( 𝑋 𝑡 𝑥 + 𝑡 = 𝑐 ) ≥ ℙ 𝑥 ( 𝑋 𝑡 𝑥 + 𝑡 = 𝑐, 𝑋 𝑡 𝑥 = 𝑐 ) = ℙ 𝑥 ( 𝑋 𝑡 𝑥 = 𝑐 ) ℙ 𝑐 ( 𝑋 𝑡 = 𝑐 ) > . Hence, for given Δ >
0, if we choose 𝑛 ∈ ℕ such that 𝑛 Δ ≥ 𝑡 𝑥 + 𝑇 , it follows that ℙ 𝑥 ( 𝑋 𝑛 Δ = 𝑐 ) > X Δ is 𝛿 𝑐 -irreducible. (cid:4) We are now well-suited to discuss ergodicity of a Markov process. Let k·k TV denote the totalvariation norm on the space of signed finite measures M 𝑠𝑏 ( X , B ( X )) on ( X , B ( X )) , defined by k 𝜈 k TV ≔ sup | 𝑔 |≤ | 𝜈 ( 𝑔 )| , 𝜈 ∈ M 𝑠𝑏 ( X , B ( X )) . We say that X having a stationary distribution 𝜌 is ergodic if ∀ 𝑥 ∈ X : lim 𝑡 →∞ k ℙ 𝑥 ( 𝑋 𝑡 ∈ ·) − 𝜌 k TV = . Clearly, ergodicity implies weak convergence of the marginal distributions of X to its invariantdistribution. If X is positive Harris recurrent, Theorem 6.1 in Meyn and Tweedie [44] providesus with a necessary and sufficient criterion for ergodicity in terms of skeletons of the process: X is ergodic ⇐⇒ ∃ Δ > X Δ is irreducible . (A.7)Once we know that X is ergodic, a natural question is the rate of convergence of the marginals tothe invariant distribution. To this end, [25] investigate convergence in the so called 𝑓 -norm. For42 strictly positive, measurable function 𝑓 ∈ B ( X ) satisfying 𝑓 ≥
1, the 𝑓 -norm on M 𝑠𝑏 ( X , B ( X )) on ( X , B ( X )) is given by k 𝜈 k 𝑓 ≔ sup | 𝑔 |≤ 𝑓 | 𝜈 ( 𝑔 )| , 𝜈 ∈ M ( X , B ( X )) , where the supremum is taken over all measurable functions 𝑔 bounded by 𝑓 . Note that for 𝑓 ≡
1, the 𝑓 -norm reduces to the total variation norm. We say that the Markov process X withstationary distribution 𝜌 is 𝑓 - uniformly ergodic if there exist constants 𝐷, 𝜅 > k 𝑃 𝑡 ( 𝑥, ·) − 𝜇 k 𝑓 ≤ 𝐷 𝑓 ( 𝑥 ) e − 𝜅 𝑡 , 𝑥 ∈ X , (A.8)which in particular implies that the marginal distributions of X converge to the stationarydistribution at an exponential rate in total variation. For the latter, we also refer to the processas being exponentially or geometrically ergodic .[25] give conditions in terms of drift criteria for the generator, semigroup and resolventkernel for 𝑓 -uniform ergodicity. For our treatment of overshoots based on their resolvent, wewill choose the resolvent drift criterion for determining the convergence speed of overshoots.More precisely, if X is irreducible and aperiodic and for some 𝜆 > 𝑏 ∈ ℝ + , 𝛽 ∈ ( , ) , a petite set 𝐶 and a measurable function 𝑉 𝜆 ≥ 𝑅 𝜆 𝑉 𝜆 ≤ 𝛽 𝑉 𝜆 ( 𝑥 ) + 𝑏 𝐶 , (A.9)Theorem 5.2 in [25] tells us that X is 𝑅 𝜆 𝑉 𝜆 -uniformly ergodic. If 𝑉 𝜆 is unbounded off petite sets ,that is { 𝑥 ∈ X : 𝑉 𝜆 ( 𝑥 ) ≤ 𝑧 } is petite for any 𝑧 >
0, (A.9) is equivalent to demanding that thereexists 𝛽 ∈ ( , ) such that 𝑅 𝜆 𝑉 𝜆 ≤ 𝛽 𝑉 𝜆 ( 𝑥 ) + 𝑏. (A.10)To see this, for 𝛼 > 𝐶 ( 𝛼 ) ≔ { 𝑥 ∈ X : 𝑉 𝜆 ( 𝑥 ) ≤ 𝛼 𝑏 /( − 𝛽 )} , then 𝛽 𝑉 𝜆 + 𝑏 ≤ 𝛽 𝑉 𝜆 + 𝑏 𝐶 + 𝛼 ( − 𝛽 ) 𝑉 𝜆 𝐶 c ≤ 𝛼 ( + ( 𝛼 − ) 𝛽 ) 𝑉 𝜆 + 𝑏 𝐶 , (A.11)showing that for any choice of 𝛼 >
1, (A.10) implies (A.9) with 𝐶 = 𝐶 ( 𝛼 ) and 𝛽 = ( + ( 𝛼 − ) 𝛽 )/ 𝛼 ∈ ( , ) . The converse relation is obvious.General drift criteria for the speed of convergence to the invariant distribution were extendedin [24] to the case of subgeometric rates. The combined conclusions of Theorem 3.2 and Theorem4.9 in [24] read that if X is ergodic and for some 𝜆 > • a closed, petite set 𝐶 and a constant 𝑏 < ∞ , • a function e 𝑉 𝜆 : X → [ , ∞) , • an increasing, differentiable and concave function 𝜙 : [ , ∞) → ( , ∞) ,such that 𝑅 𝜆 e 𝑉 𝜆 ≤ e 𝑉 𝜆 − 𝜙 ◦ e 𝑉 𝜆 + 𝑏 𝐶 , (A.12)then, provided 𝑅 𝜆 e 𝑉 𝜆 is continuous, there exists some constant 𝑐 > k 𝑃 𝑡 ( 𝑥, ·) − 𝜇 k TV ≤ 𝑐 R 𝜆 e 𝑉 𝜆 ( 𝑥 ) Ξ ( 𝑡 ) , 𝑡 ≥ , 𝑥 ∈ X , (A.13)where Ξ ( 𝑡 ) = /( 𝜙 ◦ 𝐻 − 𝜙 )( 𝑡 ) for 𝐻 𝜙 ( 𝑡 ) = ∫ 𝑡 ( / 𝜙 ( 𝑠 )) d 𝑠. Note that (A.9) can be recovered for linear 𝜙 , in which case Ξ ( 𝑡 ) = e − 𝜅 𝑡 for some 𝜅 >
0, and hence exponential ergodicity can be regardedas a special case of this general result. 43tudying exponential and subgeometric convergence is not only interesting in its own right,but does have direct implications on the mixing behavior of the Markov process, which weare ultimately going after in this article. For two 𝜎 -algebras G and H and a given probabilitymeasure P , introduce the 𝛽 -mixing coefficient 𝛽 P ( G , H ) ≔ sup 𝐶 ∈ G ⊗ H (cid:12)(cid:12) P | G ⊗ H ( 𝐶 ) − P | G ⊗ P | H ( 𝐶 ) (cid:12)(cid:12) , (A.14)where P | G ⊗ H is the restriction to ( Ω × Ω , G ⊗ H ) of the image measure of P under the canonicalinjection 𝜄 ( 𝜔 ) = ( 𝜔 , 𝜔 ) . Noting that for 𝐴 × 𝐵 ∈ G ⊗ H , it holds that P | G ⊗ H ( 𝐴 × 𝐵 ) = P ( 𝐴 ∩ 𝐵 ) ,it is clear that the 𝛽 -mixing coefficient should be interpreted as a measure of independence ofthe 𝜎 -algebras. For the Markov process X with natural filtration 𝔽 = ( F 𝑡 ) 𝑡 ≥ and a given initialdistribution 𝜂 let us now define 𝛽 ( 𝜂 , 𝑡 ) = sup 𝑠 ≥ 𝛽 ℙ 𝜂 ( F 𝑠 , F 𝑠 + 𝑡 ) , 𝑡 > , (A.15)where we denoted by F 𝑡 = 𝜎 ( 𝑋 𝑠 , 𝑠 ≥ 𝑡 ) the 𝜎 -algebra of the future after time 𝑡 . We then say that X is 𝛽 -mixing when started in 𝜂 , if lim 𝑡 →∞ 𝛽 ( 𝜂 , 𝑡 ) = X is 𝛽 -mixing we can roughlystate that there is an asymptotic independence between the past and the future of the Markovprocess. If there even exist constants 𝐶 , 𝜅 > 𝛽 ( 𝜂 , 𝑡 ) ≤ 𝐶 e − 𝜅 𝑡 , we call X exponentially 𝛽 -mixing.[55, Lemma 1.4] gives 𝛽 ℙ 𝜂 ( F 𝑠 , F 𝑡 + 𝑠 ) = 𝔼 𝜂 h sup 𝐵 ∈ F 𝑡 + 𝑠 | ℙ 𝜂 ( 𝐵 | F 𝑠 ) − ℙ 𝜂 ( 𝐵 )| i . Proposition 1 in [20] therefore demonstrates that 𝛽 ( 𝜂 , 𝑡 ) = sup 𝑠 ≥ ∫ X k ℙ 𝑥 ( 𝑋 𝑠 ∈ ·) − ℙ 𝜂 ( 𝑋 𝑡 + 𝑠 ∈ ·)k TV ℙ 𝜂 ( 𝑋 𝑡 ∈ d 𝑥 ) , 𝑡 > . Masuda [40, Lemma 3.9] uses this characterization to establish that if we have (sub)geometricdecay as in (A.13) for X and moreover 𝜚 ( 𝜂 ) ≔ sup 𝑡 ≥ 𝑐 𝔼 𝜂 [ 𝑅 𝜆 e 𝑉 𝜆 ( 𝑋 𝑡 )] < ∞ , (A.16)then X started in 𝜂 is 𝛽 -mixing at rate Ξ ( 𝑡 ) with 𝛽 ( 𝜂 , 𝑡 ) ≤ 𝜚 ( 𝜂 ) Ξ ( 𝑡 ) , 𝑡 > . B. Proof of the resolvent formula
Proof of Theorem 3.6.
Note first that by assumed irreducibility of 𝐽 + , it follows as a consequenceof the Perron–Frobenius theorem that 𝚽 + ( 𝜆 ) is invertible for any 𝜆 >
0, see Corollary 2.4 inStephenson [52] or Remark 2.2 in Ivanovs et al. [30], and hence the statement of the theoremmakes formally sense. Fix ( 𝑥, 𝑖 ) ∈ ℝ + × Θ . Let 𝜏 ≔ inf { 𝑡 ≥ O 𝑡 = } , which is clearly finiteand a stopping time for ( O 𝑡 , J 𝑡 ) since the process is Feller by Proposition 3.3. By the sawtoothstructure of ( O , J ) , see also Figure 3.1 for an illustration, we have 𝜏 = 𝑥 and ( O 𝑡 , J 𝑡 ) = ( 𝑥 − 𝑡 , 𝑖 ) for 𝑡 ∈ [ , 𝑥 ] , ℙ 𝑥,𝑖 -a.s.. Together with the strong Markov property of ( O , J ) , we therefore obtainfor 𝑓 ∈ B 𝑏 ( ℝ + × Θ ) U 𝜆 𝑓 ( 𝑥, 𝑖 ) = 𝑄 𝜆 𝑓 ( 𝑥, 𝑖 ) + e − 𝜆 𝑥 U 𝜆 𝑓 ( , 𝑖 ) . Hence, we only need to calculate U 𝜆 𝑓 ( , 𝑖 ) . 44e start with the case that the Lévy measures Π + 𝑖 , 𝑖 ∈ Θ are finite and then proceed by anapproximation argument to the general case. Our assumption of upward regularity of ( 𝜉 , 𝐽 ) then forces 𝑑 + 𝑖 > 𝑖 ∈ Θ , that is the processes 𝐻 + 𝑖 are compound Poisson processeswith drift. Denote for 𝑖 ∈ Θ by 𝑌 ( 𝑖 ) random variables independent of ( 𝜉 , 𝐽 ) correspondingto the jumps of 𝐻 + , ( 𝑖 ) , whose distribution is given by Π + 𝑖 ( d 𝑥 )/ Π + 𝑖 ( ℝ + ) . Moreover, denote by 𝜎 ≔ inf { 𝑡 ≥ 𝐽 + 𝑡 ≠ 𝐽 + } the first jump time of 𝐽 + and by 𝜏 = inf { 𝑡 ≥ Δ 𝐻 + , ,𝐽 + 𝑡 > } thefirst jump time of the Lévy process driving the ascending ladder height process before the firstphase transition. Then, from Proposition 2.1 and indistinguishability of ( O + , J + ) and ( O , J ) wecan infer that under ℙ ,𝑖 , it holds that 𝑇 ≔ inf { 𝑡 ≥ Δ ( O 𝑡 , J 𝑡 ) ≠ } = 𝐻 + , ,𝑖 ( 𝜎 ∧ 𝜏 )− = 𝑑 + 𝑖 ( 𝜎 ∧ 𝜏 ) almost surely (consult again Figure 3.1 for an illustration). U 𝜆 𝑓 ( , 𝑖 ) = 𝔼 ,𝑖 h ∫ 𝑇 + ∫ 𝑇 + 𝜏 ◦ 𝜃 𝑇 𝑇 + ∫ ∞ 𝑇 + 𝜏 ◦ 𝜃 𝑇 e − 𝜆 𝑡 𝑓 ( O 𝑡 , J 𝑡 ) d 𝑡 i ≕ 𝐼 + 𝐼 + 𝐼 , (B.1)where ( 𝜃 𝑡 ) 𝑡 ≥ denotes the transition operator of ( O , J ) . Since under ℙ ,𝑖 , 𝜏 d = Exp ( Π + 𝑖 ( ℝ + )) isindependent of 𝜎 d = Exp (− 𝑞 + 𝑖,𝑖 ) by Proposition 2.1, it follows that 𝑇 d = Exp (( Π + 𝑖 ( ℝ + ) − 𝑞 + 𝑖,𝑖 )/ 𝑑 + 𝑖 ) and hence 𝐼 = 𝔼 ,𝑖 h ∫ 𝑇 e − 𝜆 𝑡 𝑓 ( , 𝑖 ) d 𝑡 i = 𝑓 ( , 𝑖 ) 𝜆 (cid:0) − 𝔼 ,𝑖 [ e − 𝜆 𝑇 ] (cid:1) = 𝑓 ( , 𝑖 ) 𝜆 (cid:16) − Π + 𝑖 ( ℝ + ) − 𝑞 + 𝑖,𝑖 𝑑 + 𝑖 𝜆 + Π + 𝑖 ( ℝ + ) − 𝑞 + 𝑖,𝑖 (cid:17) = 𝑓 ( , 𝑖 ) 𝑑 + 𝑖 𝑑 + 𝑖 𝜆 + Π + 𝑖 ( ℝ + ) − 𝑞 + 𝑖,𝑖 . For the second integral, we use that ℙ ,𝑖 ( 𝐽 + 𝜎 = 𝑗 ) = − 𝑞 + 𝑖,𝑗 / 𝑞 + 𝑖,𝑖 , independence of 𝜎 , 𝐽 + 𝜎 and 𝑌 ( 𝑖 ) incombination with Proposition 2.1 and the strong Markov property to obtain 𝐼 = 𝔼 ,𝑖 h e − 𝜆 𝑇 𝔼 ,𝑖 h ∫ 𝜏 e − 𝜆 𝑡 𝑓 ( O 𝑡 , J 𝑡 ) d 𝑡 ◦ 𝜃 𝑇 (cid:12)(cid:12)(cid:12) G 𝑇 i i = 𝔼 ,𝑖 h e − 𝜆 𝑇 𝔼 O 𝑇 , J 𝑇 h ∫ 𝜏 e − 𝜆 𝑡 𝑓 ( O 𝑡 , J 𝑡 ) d 𝑡 i i = 𝔼 ,𝑖 (cid:2) e − 𝜆 𝑑 + 𝑖 𝜏 𝑄 𝜆 ( 𝑌 ( 𝑖 ) , 𝑖 ) ; 𝜏 < 𝜎 (cid:3) + 𝔼 ,𝑖 (cid:2) e − 𝜆 𝑑 + 𝑖 𝜎 𝑄 𝜆 (cid:0) Δ + , 𝑖,𝐽 + 𝜎 , 𝐽 + 𝜎 (cid:1) ; 𝜎 < 𝜏 (cid:3) = 𝔼 ,𝑖 [ e − 𝜆 𝑑 + 𝑖 𝜏 ; 𝜏 < 𝜎 ] 𝔼 ,𝑖 [ 𝑄 𝜆 ( 𝑌 ( 𝑖 ) , 𝑖 )] + 𝔼 ,𝑖 (cid:2) e − 𝜆 𝑑 + 𝑖 𝜎 ; 𝜎 < 𝜏 ] 𝔼 ,𝑖 (cid:2) 𝑄 𝜆 (cid:0) Δ + , 𝑖,𝐽 + 𝜎 , 𝐽 + 𝜎 (cid:1) (cid:3) = Π + 𝑖 ( ℝ + ) 𝜆 𝑑 + 𝑖 + Π + 𝑖 ( ℝ + ) − 𝑞 + 𝑖,𝑖 ∫ ∞ 𝑄 𝜆 𝑓 ( 𝑦, 𝑖 ) Π + 𝑖 ( d 𝑦 )/ Π + 𝑖 ( ℝ + )+ − 𝑞 + 𝑖,𝑖 𝜆 𝑑 + 𝑖 + Π + 𝑖 ( ℝ + ) − 𝑞 + 𝑖,𝑖 Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 − 𝑞 + 𝑖,𝑖 ∫ ∞ 𝑄 𝜆 𝑓 ( 𝑦, 𝑗 ) 𝐹 + 𝑖,𝑗 ( d 𝑦 ) = 𝜆 𝑑 + 𝑖 + Π + 𝑖 ( ℝ + ) − 𝑞 + 𝑖,𝑖 (cid:16) ∫ ∞ 𝑄 𝜆 𝑓 ( 𝑦, 𝑖 ) Π + 𝑖 ( d 𝑦 ) + Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 ∫ ∞ 𝑄 𝜆 𝑓 ( 𝑦, 𝑗 ) 𝐹 + 𝑖,𝑗 ( d 𝑦 ) (cid:17) With the same arguments as above we also obtain 𝐼 = 𝔼 ,𝑖 h e − 𝜆 𝑇 𝔼 ,𝑖 h ∫ ∞ 𝜏 e − 𝜆 𝑡 𝑓 ( O 𝑡 , J 𝑡 ) d 𝑡 (cid:12)(cid:12)(cid:12) G 𝑇 i i = 𝔼 ,𝑖 h e − 𝜆 𝑇 𝔼 O 𝑇 , J 𝑇 h ∫ ∞ 𝜏 e − 𝜆 𝑡 𝑓 ( O 𝑡 , J 𝑡 ) d 𝑡 i i = 𝔼 ,𝑖 [ e − 𝜆 𝑑 + 𝑖 𝜏 ; 𝜏 < 𝜎 ] 𝔼 ,𝑖 h 𝔼 𝑦,𝑖 h e − 𝜆𝜏 𝔼 𝑦,𝑖 h ∫ ∞ e − 𝜆 𝑡 𝑓 ( O 𝑡 , J 𝑡 ) d 𝑡 ◦ 𝜃 𝜏 (cid:12)(cid:12)(cid:12) G 𝜏 i i (cid:12)(cid:12)(cid:12) 𝑦 = 𝑌 ( 𝑖 ) i 𝔼 ,𝑖 [ e − 𝜆 𝑑 + 𝑖 𝜎 ; 𝜎 < 𝜏 ] Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 − 𝑞 + 𝑖,𝑖 𝔼 ,𝑖 h 𝔼 𝑦,𝑗 h e − 𝜆𝜏 𝔼 𝑦,𝑗 h ∫ ∞ e − 𝜆 𝑡 𝑓 ( O 𝑡 , J 𝑡 ) d 𝑡 ◦ 𝜃 𝜏 (cid:12)(cid:12)(cid:12) G 𝜏 i i(cid:12)(cid:12)(cid:12) 𝑦 =Δ + , 𝑖,𝑗 i = Π + 𝑖 ( ℝ + ) 𝜆 𝑑 + 𝑖 + Π + 𝑖 ( ℝ + ) − 𝑞 + 𝑖,𝑖 U 𝜆 𝑓 ( , 𝑖 ) 𝔼 ,𝑖 (cid:2) e − 𝜆 𝑌 ( 𝑖 ) (cid:3) + 𝜆 𝑑 + 𝑖 + Π + 𝑖 ( ℝ + ) − 𝑞 + 𝑖,𝑖 Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 U 𝜆 𝑓 ( , 𝑗 ) 𝔼 ,𝑖 (cid:2) e − 𝜆 Δ + , 𝑖,𝑗 (cid:3) = 𝜆 𝑑 + 𝑖 + Π + 𝑖 ( ℝ + ) − 𝑞 + 𝑖,𝑖 (cid:16) U 𝜆 𝑓 ( , 𝑖 ) ∫ ∞ e − 𝜆 𝑦 Π + 𝑖 ( d 𝑦 ) + Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 U 𝜆 𝑓 ( , 𝑗 ) ∫ ∞ e − 𝜆 𝑦 𝐹 + 𝑖,𝑗 ( d 𝑦 ) (cid:17) . Plugging into (B.1), using 𝐺 + 𝑖𝑗 ( 𝜆 ) = ∫ ∞ exp (− 𝜆 𝑦 ) 𝐹 + 𝑖𝑗 ( d 𝑦 ) and rearranging now yields U 𝜆 𝑓 ( , 𝑖 ) (cid:16) 𝑑 + 𝑖 𝜆 + ∫ ∞ ( − e − 𝜆 𝑦 ) Π + 𝑖 ( d 𝑦 ) − 𝑞 + 𝑖,𝑖 (cid:17) − Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 𝐺 + 𝑖𝑗 ( 𝜆 ) U 𝜆 𝑓 ( , 𝑗 ) = 𝑑 + 𝑖 𝑓 ( , 𝑖 ) + ∫ ∞ 𝑄 𝜆 𝑓 ( 𝑦, 𝑖 ) Π + 𝑖 ( d 𝑦 ) + Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 ∫ ∞ 𝑄 𝜆 𝑓 ( 𝑦, 𝑖 ) 𝐹 + 𝑖,𝑗 ( d 𝑦 ) . By (2.4) the left hand side is equal to U 𝜆 𝑓 ( , 𝑖 )( Φ + 𝑖 ( 𝜆 ) − 𝑞 + 𝑖,𝑖 ) − Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 𝐺 + 𝑖𝑗 ( 𝜆 ) U 𝜆 𝑓 ( , 𝑗 ) = (cid:0) 𝚽 + ( 𝜆 ) · ( U 𝜆 𝑓 ( , 𝑗 )) ⊤ 𝑗 = ,...,𝑛 (cid:1) 𝑖 and hence we conclude that ( U 𝜆 𝑓 ( , 𝑖 )) ⊤ 𝑖 = ,...,𝑛 = 𝚽 + ( 𝜆 ) − · (cid:16) 𝑑 + 𝑖 𝑓 ( , 𝑖 ) + ∫ ∞ 𝑄 𝜆 𝑓 ( 𝑥, 𝑖 ) Π + 𝑖 ( d 𝑥 ) + Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 𝔼 [ 𝑄 𝜆 𝑓 ( Δ + 𝑖,𝑗 , 𝑗 )] (cid:17) ⊤ 𝑖 = ,...,𝑛 , (B.2)which proves the assertion in case that ( 𝐻 + , 𝐽 + ) is a compound Poisson Markov additive subor-dinator. For the general case, suppose that ( 𝜉 , 𝐽 ) is an upward regular MAP and let for 𝜀 > ( 𝜀 𝐻 + , 𝐽 + ) be the ascending ladder process corresponding to the ordinator constructed from theLévy subordinators 𝜀 𝐻 + , ( 𝑖 ) defined by 𝜀 𝐻 + , ( 𝑖 ) 𝑡 ≔ ( 𝑑 + 𝑖 + 𝜀 ) 𝑡 + Õ 𝑠 ≤ 𝑡 Δ 𝐻 + , ( 𝑖 ) 𝑠 ( 𝜀 , ∞) ( Δ 𝐻 + , ( 𝑖 ) 𝑠 ) , 𝑡 ≥ , i.e. 𝜀 𝐻 + , ( 𝑖 ) is obtained from 𝐻 + by deleting jumps smaller than 𝜀 and adding an additional drift 𝜀 . This ensures that 𝜀 𝐻 + , ( 𝑖 ) is a compound Poisson subordinator with drift 𝑑 + 𝑖 + 𝜀 and Lévymeasure Π + , 𝜀 𝑖 = Π + 𝑖 (· ∩ ( 𝜀 , ∞)) and hence we may apply (B.2) for the 𝜆 resolvent of the overshootprocess ( 𝜀 O + 𝑡 , 𝜀 J + 𝑡 ) 𝑡 ≥ ≔ ( 𝜀 𝐻 + 𝑇 + , 𝜀 𝑡 − 𝑡 , 𝐽 + 𝑇 + , 𝜀 𝑡 ) 𝑡 ≥ , where 𝑇 + , 𝜀 𝑡 ≔ inf { 𝑠 ≥ 𝜀 𝐻 + 𝑠 > 𝑡 } , 𝑡 ≥ . We first observe that for any 𝑡 > 𝑠 ≤ 𝑡 | 𝜀 𝐻 + 𝑠 − 𝐻 + 𝑠 | ≤ 𝜀 𝑡 + Õ 𝑠 ≤ 𝑡 Δ 𝐻 + 𝑠 { Δ 𝐻 + 𝑠 <𝜀 } , and since Í 𝑠 ≤ 𝑡 Δ 𝐻 + 𝑠 converges we obtain by dominated convergence that almost surelysup 𝑠 ≤ 𝑡 | 𝜀 𝐻 + 𝑠 − 𝐻 + 𝑠 | → , as 𝜀 ↓ 𝜀 𝐻 + converges to 𝐻 + uniformly on compact sets almost surely as 𝜀 ↓
0. Let Ξ be the set of ℙ -measure 1 on which 𝜀 𝐻 + and ( 𝐻 + , 𝐽 + ) have càdlàg paths and on which the above convergence46olds. Let 𝜔 ∈ Ξ . Then 𝜀 𝐻 +· ( 𝜔 ) , 𝐻 +· ( 𝜔 ) ∈ D ( ℝ + ) , the space of càdlàg functions mapping from ℝ + to ℝ + , which we endow with Skorokhods 𝐽 -topology. Since 𝜀 𝐻 +· ( 𝜔 ) converges uniformlyon compact time sets to 𝐻 +· ( 𝜔 ) , Proposition VI.1.17 in [31] tells us that 𝜀 𝐻 +· ( 𝜔 ) also convergeswith respect to the metric inducing the Skorokhod topology to 𝐻 +· ( 𝜔 ) . For 𝑡 ≥ 𝑆 𝑡 : D ( ℝ + ) → [ , ∞] , 𝛼 ↦→ inf { 𝑠 ≥ | 𝛼 ( 𝑠 )| ≥ 𝑡 or | 𝛼 ( 𝑠 −)| ≥ 𝑡 } . Since 𝜀 𝐻 +· ( 𝜔 ) and 𝐻 +· ( 𝜔 ) are strictly increasing it follows that 𝑇 + , 𝜀 𝑡 ( 𝜔 ) = 𝑆 𝑡 ( 𝜀 𝐻 +· ( 𝜔 )) and 𝑇 + 𝑡 ( 𝜔 ) = 𝑆 𝑡 ( 𝐻 +· ( 𝜔 )) . Moreover, the set { 𝑡 > 𝑆 𝑡 ( 𝐻 +· ( 𝜔 )) ≠ 𝑆 𝑡 + ( 𝐻 +· ( 𝜔 ))} is empty by strictly increasingpaths of 𝐻 +· ( 𝜔 ) . Hence, we obtain from Proposition 2.11 and the proof of part c) of PropositionVI.2.12 in [31] that 𝑇 + , 𝜀 𝑡 ( 𝜔 ) = 𝑆 𝑡 ( 𝜀 𝐻 +· ( 𝜔 )) → 𝑆 𝑡 ( 𝐻 +· ( 𝜔 )) = 𝑇 + 𝑡 ( 𝜔 ) , as 𝜀 ↓ , (B.3)and that for 𝑡 ∉ Λ ( 𝜔 ) = { 𝑡 > Δ 𝐻 + 𝑇 + 𝑡 ( 𝜔 ) > 𝐻 + 𝑇 + 𝑡 − ( 𝜔 ) = 𝑡 } we have 𝜀 𝐻 + 𝑇 + , 𝜀 𝑡 ( 𝜔 ) → 𝐻 + 𝑇 + 𝑡 ( 𝜔 ) , as 𝜀 ↓ . (B.4)But from the sawtooth structure of the paths of O it is easy to see that Λ ( 𝜔 ) = { 𝑡 > Δ O + 𝑡 ( 𝜔 ) > } , which is countable (alternatively, see Lemma VI.2.10.(d) in [31] for the same conclusion),hence non-convergence of 𝜀 O + 𝑡 ( 𝜔 ) to O + 𝑡 ( 𝜔 ) only takes place on a set of Lebesgue measure 0.Furthermore, from (B.3) it follows that 𝜀 J + 𝑡 ( 𝜔 ) converges to J + 𝑡 ( 𝜔 ) as 𝜀 ↓ Λ ′ ( 𝜔 ) ≔ { 𝑡 > 𝐽 + 𝑇 + 𝑡 ( 𝜔 ) ≠ 𝐽 + 𝑇 + 𝑡 − ( 𝜔 )} = { 𝑡 > Δ 𝐽 + 𝑇 + 𝑡 ( 𝜔 ) ≠ , Δ 𝐻 + 𝑇 + 𝑡 ( 𝜔 ) > } ∪ { 𝑡 > Δ 𝐽 + 𝑇 + 𝑡 ( 𝜔 ) ≠ , Δ 𝐻 + 𝑇 + 𝑡 ( 𝜔 ) = } ≕ Λ ′ ( 𝜔 ) ∪ Λ ′ ( 𝜔 ) . For 𝑡 ∈ Λ ′ ( 𝜔 ) we have that in case 𝐻 + 𝑇 + 𝑡 − ( 𝜔 ) < 𝑡 ≤ 𝐻 + 𝑇 + 𝑡 ( 𝜔 ) it holds that 𝑇 + 𝑠 ( 𝜔 ) = 𝑇 + 𝑡 ( 𝜔 ) for 𝑠 ∈ [ 𝐻 𝑇 + 𝑡 − ( 𝜔 ) , 𝑡 ] . Right-continuity of 𝑠 ↦→ 𝑇 + 𝑠 ( 𝜔 ) and 𝑠 ↦→ 𝐽 + 𝑠 ( 𝜔 ) therefore imply that forsuch 𝑡 we also have 𝜀 J + 𝑡 ( 𝜔 ) → J + 𝑡 ( 𝜔 ) as 𝜀 ↓
0. Further, since 𝑡 ↦→ 𝐻 + 𝑡 ( 𝜔 ) is continuous in 𝑇 + 𝑡 ( 𝜔 ) if Δ 𝐻 + 𝑇 + 𝑡 ( 𝜔 ) =
0, it follows from strictly increasing paths that for 𝑠 , 𝑡 ∈ Λ ′ ( 𝜔 ) we have 𝑇 + 𝑠 ( 𝜔 ) ≠ 𝑇 + 𝑡 ( 𝜔 ) . Hence, 𝑡 ↦→ 𝑇 + 𝑡 ( 𝜔 ) is injective on Λ ′ ( 𝜔 ) . Since 𝑇 +· ( 𝜔 )( Λ ′ ( 𝜔 )) = { 𝑡 > Δ 𝐽 + 𝑡 ( 𝜔 ) ≠ , Δ 𝐻 + 𝑡 ( 𝜔 ) = } ⊂ { 𝑡 > Δ 𝐽 + 𝑡 ( 𝜔 ) ≠ } , and the set on the right-hand side is countable thanks to 𝐽 +· ( 𝜔 ) being càdlàg, it follows that Λ ′ ( 𝜔 ) is countable as well. The above discussion therefore yields that the set of times 𝑡 > 𝐽 + 𝑇 + , 𝜀 𝑡 ( 𝜔 ) does not converge to J + 𝑡 ( 𝜔 ) is given by Λ ′′ ( 𝜔 ) ≔ { 𝑡 > Δ 𝐽 + 𝑇 + 𝑡 ( 𝜔 ) ≠ , 𝐻 + 𝑇 + 𝑡 − ( 𝜔 ) = 𝑡 < 𝐻 + 𝑇 + 𝑡 ( 𝜔 )} ∪ Λ ′ ( 𝜔 ) ⊂ { 𝑡 > Δ O + 𝑡 ( 𝜔 ) > } ∪ Λ ′ ( 𝜔 ) is countable and therefore has Lebesgue measure 0 as well. It follows that for any 𝜔 ∈ Ξ wehave for 𝑓 ∈ C 𝑏 ( ℝ + × Θ ) by dominated convergencelim 𝜀 ↓ ∫ ∞ 𝑓 ( 𝜀 O + 𝑡 ( 𝜔 ) , 𝜀 J + 𝑡 ( 𝜔 )) d 𝑡 = ∫ ( Λ ( 𝜔 )∪ Λ ′′ ( 𝜔 )) c lim 𝜀 ↓ 𝑓 ( 𝜀 O + 𝑡 ( 𝜔 ) , 𝜀 J + 𝑡 ( 𝜔 )) d 𝑡 = ∫ ( Λ ( 𝜔 )∪ Λ ′′ ( 𝜔 )) c 𝑓 ( O + 𝑡 ( 𝜔 ) , J + 𝑡 ( 𝜔 )) d 𝑡 = ∫ ∞ 𝑓 ( O + 𝑡 ( 𝜔 ) , J + 𝑡 ( 𝜔 )) d 𝑡. 𝑈 𝜀𝜆 the 𝜆 -resolvent for ( 𝜀 O + , 𝜀 J + ) , the set Ξ having ℙ -measure 1implies that for any 𝑓 ∈ C 𝑏 ( ℝ + × Θ )( U 𝜆 𝑓 ( , 𝑖 )) 𝑖 = ,...,𝑛 = (cid:16) ∫ Ξ lim 𝜀 ↓ ∫ ∞ 𝑓 ( 𝜀 O + 𝑡 ( 𝜔 ) , 𝜀 J + 𝑡 ( 𝜔 )) d 𝑡 ℙ ,𝑖 ( d 𝜔 ) (cid:17) 𝑖 = ,...,𝑛 = lim 𝜀 ↓ (cid:16) ∫ Ξ ∫ ∞ 𝑓 ( 𝜀 O + 𝑡 ( 𝜔 ) , 𝜀 J + 𝑡 ( 𝜔 )) d 𝑡 ℙ ,𝑖 ( d 𝜔 ) (cid:17) 𝑖 = ,...,𝑛 = lim 𝜀 ↓ ( 𝑈 𝜀𝜆 𝑓 ( , 𝑖 )) 𝑖 = ,...,𝑛 = lim 𝜀 ↓ 𝜀 𝚽 + ( 𝜆 ) − · (cid:16) ( 𝑑 + 𝑖 + 𝜀 ) 𝑓 ( , 𝑖 ) + ∫ ∞ 𝜀 𝑄 𝜆 𝑓 ( 𝑥, 𝑖 ) Π + 𝑖 ( d 𝑥 ) + Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 𝔼 [ 𝑄 𝜆 𝑓 ( Δ + 𝑖,𝑗 , 𝑗 )] (cid:17) ⊤ 𝑖 = ,...,𝑛 = 𝚽 + ( 𝜆 ) − · (cid:16) 𝑑 + 𝑖 𝑓 ( , 𝑖 ) + ∫ ∞ 𝑄 𝜆 𝑓 ( 𝑥, 𝑖 ) Π + 𝑖 ( d 𝑥 ) + Õ 𝑗 ≠ 𝑖 𝑞 + 𝑖,𝑗 𝔼 [ 𝑄 𝜆 𝑓 ( Δ + 𝑖,𝑗 , 𝑗 )] (cid:17) ⊤ 𝑖 = ,...,𝑛 ≕ 𝚼 ( 𝜆 ) , where we used dominated convergence for the second and (B.2) for the fourth equality. Itremains to extend this result to any 𝑓 ∈ B + ( ℝ + × Θ ) ∪ B 𝑏 ( ℝ + × Θ ) . To this end, let M ≔ (cid:8) 𝑓 ∈ B 𝑏 ( ℝ + × Θ ) : ( U 𝜆 𝑓 ( , 𝑖 )) 𝑖 = ,...,𝑛 = 𝚼 ( 𝜆 ) (cid:9) . Clearly, M is a vector space over ℝ + by linearity of the Lebesgue integral and since C 𝑏 ( ℝ + × Θ ) ⊂ M , the constant function ℝ + × Θ is contained in M . Moreover, dominated convergence showsthat M is closed under convergence of an increasing family of functions 𝑓 𝑛 converging to some 𝑓 ∈ B 𝑏 ( ℝ + × Θ ) . Hence, M is a monotone vector space and since C 𝑏 ( ℝ + ) is closed undermultiplication and contained in M , the functional Monotone Class Theorem A.0.6 from [51]implies that 𝜎 ( C 𝑏 ( ℝ + × Θ )) ⊂ M . But since ℝ + × Θ is a locally compact metric space withcountable base, C 𝑏 ( ℝ + × Θ ) is dense in B 𝑏 ( ℝ + × Θ ) and hence M = B 𝑏 ( ℝ + × Θ ) follows. Forgeneral 𝑓 ∈ B + ( ℝ + × Θ ) let 𝑓 𝑛 ≔ 𝑓 { 𝑓 ∈[ ,𝑛 ]} ∈ B 𝑏 ( ℝ + × Θ ) and apply monotone convergence todeduce that (B.2) also holds for 𝑓 ∈ B + ( ℝ + × Θ ) . This finishes the proof. (cid:4)
References [1] L. Alili, L. Chaumont, P. Graczyk, and T. Żak. “Inversion, duality and Doob ℎ -transformsfor self-similar Markov processes”. In: Electron. J. Probab.
22 (2017), Paper No. 20, 18. doi : .[2] S. Asmussen. Applied probability and queues . 2nd ed. Vol. 51. Applications of Mathematics(New York). Stochastic Modelling and Applied Probability. Springer-Verlag, New York,2003, pp. xii+438. isbn : 0-387-00211-1.[3] J. Azéma, M. Duflo, and D. Revuz. “Mesure invariante des processus de Markov récur-rents”. In:
Séminaire de Probabilités, III (Univ. Strasbourg, 1967/68) . Lecture Notes in Mathe-matics, Vol. 88. Springer, Berlin, 1969, pp. 24–33.[4] A. Bátkai, M. Kramar Fijavž, and A. Rhandi.
Positive operator semigroups . Vol. 257. OperatorTheory: Advances and Applications. Birkhäuser/Springer, Cham, 2017, pp. xvii+364. isbn :978-3-319-42811-6; 978-3-319-42813-0. doi : .[5] D. Belomestny, F. Comte, V. Genon-Catalot, H. Masuda, and M. Reiß. Lévy matters IV -Estimation for discretely observed Lévy processes . Lecture Notes in Mathematics, Vol. 2128.Springer, Cham, 2015. 486] J. Bertoin, K. van Harn, and F. W. Steutel. “Renewal theory and level passage by subordina-tors”. In:
Statist. Probab. Lett. issn : 0167-7152. doi : .[7] J. Bertoin. Lévy processes . Vol. 121. Cambridge Tracts in Mathematics. Cambridge Univer-sity Press, Cambridge, 1996, pp. x+265. isbn : 0-521-56243-0.[8] J. Bertoin.
Subordinators, Lévy processes with no negative jumps and branching processes . Ma-PhySto Lecture Notes Series No. 8. Aarhus, 2000.[9] J. Bertoin, T. Budd, N. Curien, and I. Kortchemski. “Martingales in self-similar growth-fragmentations and their connections with random planar maps”. In:
Probab. Theory Re-lated Fields issn : 0178-8051. doi : .[10] J. Bertoin and M. Savov. “Some applications of duality for Lévy processes in a half-line”.In: Bull. Lond. Math. Soc. issn : 0024-6093. doi : .[11] R. M. Blumenthal and R. K. Getoor. “Local times for Markov processes”. In: Z. Wahrschein-lichkeitstheorie und Verw. Gebiete doi : .[12] R. M. Blumenthal and R. K. Getoor. Markov processes and potential theory . Pure and AppliedMathematics, Vol. 29. Academic Press, New York-London, 1968, pp. x+313.[13] R. M. Blumenthal.
Excursions of Markov processes . Probability and its Applications. BirkhäuserBoston Inc., Boston, MA, 1992, pp. xii+275. isbn : 0-8176-3575-0. doi : .[14] E. Çinlar. “Lévy systems of Markov additive processes”. In: Z. Wahrscheinlichkeitstheorieund Verw. Gebiete
31 (1974/75), pp. 175–185. doi : .[15] E. Çinlar. “Markov additive processes. I, II”. In: Z. Wahrscheinlichkeitstheorie und Verw.Gebiete
24 (1972), 85–93; ibid. 24 (1972), 95–121. doi : .[16] L. Chaumont, A. Kyprianou, J. C. Pardo, and V. Rivero. “Fluctuation theory and exitsystems for positive self-similar Markov processes”. In: Ann. Probab. issn : 0091-1798. doi : .[17] L. Chaumont, H. Pantí, and V. Rivero. “The Lamperti representation of real-valued self-similar Markov processes”. In: Bernoulli issn : 1350-7265. doi : .[18] Y. S. Chow. “On moments of ladder height variables”. In: Adv. in Appl. Math. issn : 0196-8858. doi : .[19] S. Christensen and T. Sohr. “A solution technique for Lévy driven long term average im-pulse control problems”. In: Stochastic Process. Appl. (2020). doi : https://doi.org/10.1016/j.spa.2020.07.016 .[20] J. A. Davydov. “Mixing conditions for Markov chains”. In: Teor. Verojatnost. i Primenen. issn : 0040-361x.[21] S. Dereich, L. Döring, and A. E. Kyprianou. “Real self-similar processes started from theorigin”. In:
Ann. Probab. issn : 0091-1798. doi : .[22] N. Dexheimer, C. Strauch, and L. Trottner. Mixing it up: A general framework for Markovianstatistics beyond reversibility and the minimax paradigm . 2020. arXiv: .[23] R. A. Doney and R. A. Maller. “Stability of the overshoot for Lévy processes”. In:
Ann.Probab. issn : 0091-1798. doi : .[24] R. Douc, G. Fort, and A. Guillin. “Subgeometric rates of convergence of 𝑓 -ergodic strongMarkov processes”. In: Stochastic Process. Appl. issn : 0304-4149. doi : .[25] D. Down, S. P. Meyn, and R. L. Tweedie. “Exponential and uniform ergodicity of Markovprocesses”. In: Ann. Probab. issn : 0091-1798.[26] R. K. Getoor. “Excursions of a Markov process”. In:
Ann. Probab. issn : 0091-1798. 4927] P. S. Griffin. “Sample path behavior of a Lévy insurance risk process approaching ruin, un-der the Cramér-Lundberg and convolution equivalent conditions”. In:
Ann. Appl. Probab. issn : 1050-5164. doi : .[28] K. Itô. Poisson point processes and their application to Markov processes . SpringerBriefs inProbability and Mathematical Statistics. Springer, Singapore, 2015, pp. xi+43. isbn : 978-981-10-0271-7; 978-981-10-0272-4. doi : .[29] J. Ivanovs. “One-sided Markov Additive Processes and Related Exit Problems”. PhDthesis. University of Amsterdam, 2007.[30] J. Ivanovs, O. Boxma, and M. Mandjes. “Singularities of the matrix exponent of a Markovadditive process with one-sided jumps”. In: Stochastic Process. Appl. issn : 0304-4149. doi : .[31] J. Jacod and A. N. Shiryaev. Limit theorems for stochastic processes . 2nd ed. Vol. 288.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Math-ematical Sciences]. Springer-Verlag, Berlin, 2003, pp. xx+661. isbn : 3-540-43932-3. doi : .[32] O. Kallenberg. Foundations of modern probability . 2nd ed. Probability and its Applica-tions (New York). Springer-Verlag, New York, 2002, pp. xx+638. isbn : 0-387-95313-2. doi : .[33] H. Kaspi and A. Mandelbaum. “On Harris recurrence in continuous time”. In: Math. Oper.Res. issn : 0364-765X. doi : .[34] S. W. Kiu. “Semistable Markov processes in R 𝑛 ”. In: Stochastic Process. Appl. issn : 0304-4149. doi : .[35] C. Klüppelberg, A. E. Kyprianou, and R. A. Maller. “Ruin probabilities and overshoots forgeneral Lévy insurance risk processes”. In: Ann. Appl. Probab. issn : 1050-5164. doi : .[36] A. Kuznetsov and J. C. Pardo. “Fluctuations of stable processes and exponential function-als of hypergeometric Lévy processes”. In: Acta Appl. Math.
123 (2013), pp. 113–139. issn :0167-8019. doi : .[37] A. E. Kyprianou. “Deep factorisation of the stable process”. In: Electron. J. Probab.
21 (2016),Paper No. 23, 28. doi : .[38] A. E. Kyprianou. Fluctuations of Lévy processes with applications . 2nd ed. Universitext. In-troductory lectures. Springer, Heidelberg, 2014, pp. xviii+455. isbn : 978-3-642-37631-3;978-3-642-37632-0. doi : .[39] A. E. Kyprianou, V. Rivero, B. Şengül, and T. Yang. “Entrance laws at the origin of self-similar Markov processes in high dimensions”. In: Trans. Amer. Math. Soc. issn : 0002-9947. doi : .[40] H. Masuda. “Ergodicity and exponential 𝛽 -mixing bounds for multidimensional diffu-sions with jumps”. In: Stochastic Processes and their Applications issn : 0304-4149. doi : https://doi.org/10.1016/j.spa.2006.04.010 .[41] S. P. Meyn and R. L. Tweedie. “Generalized resolvents and Harris recurrence of Markovprocesses”. In: Doeblin and modern probability (Blaubeuren, 1991) . Vol. 149. Contemp. Math.Amer. Math. Soc., Providence, RI, 1993, pp. 227–250. doi : .[42] S. Meyn and R. L. Tweedie. Markov chains and stochastic stability . Second. With a prologueby Peter W. Glynn. Cambridge University Press, Cambridge, 2009, pp. xxviii+594. isbn :978-0-521-73182-9. doi : .5043] S. P. Meyn and R. L. Tweedie. “Stability of Markovian processes. I. Criteria for discrete-time chains”. In: Adv. in Appl. Probab. issn : 0001-8678. doi : .[44] S. P. Meyn and R. L. Tweedie. “Stability of Markovian processes. II. Continuous-timeprocesses and sampled chains”. In: Adv. in Appl. Probab. issn :0001-8678. doi : .[45] S. P. Meyn and R. L. Tweedie. “Stability of Markovian processes. III. Foster-Lyapunovcriteria for continuous-time processes”. In: Adv. in Appl. Probab. issn : 0001-8678. doi : .[46] E. Nummelin. General irreducible Markov chains and nonnegative operators . Vol. 83. Cam-bridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1984, pp. xi+156. isbn : 0-521-25005-6. doi : .[47] H. S. Park and R. Maller. “Moment and MGF convergence of overshoots and undershootsfor Lévy insurance risk processes”. In: Adv. in Appl. Probab. issn :0001-8678. doi : .[48] M. Rosenbaum and P. Tankov. “Asymptotic results for time-changed Lévy processessampled at hitting times”. In: Stochastic Process. Appl. issn :0304-4149. doi : .[49] N. Sandrić. “A note on the Birkhoff ergodic theorem”. In: Results Math. issn : 1422-6383. doi : .[50] K.-i. Sato. Lévy processes and infinitely divisible distributions . Cambridge Studies in AdvancedMathematics, Vol. 68. Translated from the 1990 Japanese original, Revised by the author.Cambridge University Press, Cambridge, 1999.[51] M. Sharpe.
General theory of Markov processes . Vol. 133. Pure and Applied Mathematics.Academic Press Inc., Boston, MA, 1988, pp. xii+419. isbn : 0-12-639060-6.[52] R. Stephenson. “On the exponential functional of Markov additive processes, and appli-cations to multi-type self-similar fragmentation processes and trees”. In:
ALEA Lat. Am.J. Probab. Math. Stat. doi : .[53] R. L. Tweedie. “Topological conditions enabling use of Harris methods in discrete andcontinuous time”. In: Acta Appl. Math. issn : 0167-8019. doi : .[54] V. Vigon. “Votre Lévy rampe-t-il?” In: J. London Math. Soc. (2) issn : 0024-6107. doi : .[55] V. A. Volkonski˘ı and J. A. Rozanov. “Some limit theorems for random functions. II”. In: Teor. Verojatnost. i Primenen. issnissn