Universality for persistence exponents of local times of self-similar processes with stationary increments
aa r X i v : . [ m a t h . P R ] N ov Universality for persistence exponents oflocal times of self-similar processes withstationary increments
Christian MönchDecember 2, 2019
Abstract
We show that P ( ℓ X (0 , T ] ≤
1) = ( c X + o (1)) T − (1 − H ) , where ℓ X is the localtime measure at of any recurrent H -self-similar real-valued process X withstationary increments that admits a sufficiently regular local time and c X issome constant depending only on X . A special case is the Gaussian setting,i.e. when the underlying process is fractional Brownian motion, in whichour result settles a conjecture by Molchan [Commun. Math. Phys. 205, 97-111 (1999)] who obtained the upper bound − H on the decay exponentof P ( ℓ X (0 , T ] ≤ . Our approach establishes a new connection betweenpersistence probabilities and Palm theory for self-similar random measures,thereby providing a general framework which extends far beyond the Gaus-sian case. MSc Classification:
Primary 60G22 Secondary 60G15, 60G18.
Keywords: fractional Brownian motion, local time, marked point process, Palm distribution, persis-tence probability, self-similarity, stationary increments . Introduction: Persistence probabilities for fractionalBrownian motion
We study local times of stochastic processes from the point of view of persistence prob-abilities, i.e. the probabilities that a stochastic process remains inside a relatively smallsubset of its state space for a long time. The problem of calculating persistence prob-abilities is a very active field of mathematical research, see e.g. the recent articles[AB18, AGPP18, LS18, Mol18, PS18, AMZ17], an overview of the developments overthe last decades is given by Aurzada and Simon in [AS15]. The main motivation tostudy the persistence behaviour of stochastic systems is its great significance to certainareas of statistical physics, see e.g. [CD08, CDC +
04, Maj99] and the survey by Bray etal. [BMS13].The starting point of the present investigation are Molchan’s celebrated and by nowclassical results [Mol99] concerning the persistence of linear fractional Brownian mo-tion, which we briefly summarise now. Let B = ( B t ) t ∈ R denote a -dimensional frac-tional Brownian motion (FBM) of Hurst index H ∈ (0 , . B can be characterised asthe unique (up to multiplication by a constant) Gaussian process which is H -self-similar with stationary increments ( H -sssi) . In [Mol99] it is shown that the maximum process ¯ B t = max ≤ s ≤ t B s , t ≥ , of B satisfies P ( ¯ B T ≤
1) = T − (1 − H )+ o (1) . (1)Subsequently, improved bounds on the error estimate implicit in (1) have been derivedby Aurzada [Aur11] and by Aurzada et al. [AGPP18]. Note that, using self-similarity,we may replace the boundary in (1) by any fixed value x > without changing theorder of decay. The probability in (1) is called the persistence probability of B and thecorresponding exponent ¯ κ = 1 − H the persistence exponent of B . In [Mol99], Molchanalso showed that the lower tail probabilities of several other path functionals of FBMare governed by the persistence exponent. In particular, he studied • ℓ (0 , T ] := lim ǫ → (2 ǫ ) − R T { B t ∈ ( − ǫ, ǫ ) } d t , the local time at , • τ + := inf { t ≥ B t = 0 } , the first zero after time , • σ + T := R T { B t > } d t , the time spent in the positive half-axis, • τ max T := arg max { B t , t ∈ [0 , T ] } , the time at which the maximum is achieved;and his results imply that lim T →∞ P ( τ + ≥ T )log T = lim T →∞ P ( σ + T ≤ T = lim T →∞ P ( τ max T ≤ T = H − . (2)These asymptotics can be viewed as a general (and very weak) form of Lévy’s arcsine-laws for Brownian motion. Intuitively, the agreement of exponents can be explained2y observing that the dominating events contributing to each of the probabilities in (1)and (2) are long (negative) excursions of B from B = 0 . This type of event also entailsa small local time at , and one is inclined to believe that the probability of the localtime being small is of the same order. However, the result in [Mol99] for the local timeis only a lower bound, namely that there is a constant b ∈ (0 , ∞ ) such that P ( ℓ (0 , T ] ≤ ≥ T − (1 − H ) b e −√ log T , (3)for sufficiently large T . Hence, the local time persistence exponent κ = − lim T →∞ P ( ℓ (0 , T ] ≤ T of B satisfies κ ≤ − H, and this upper bound with the error estimate given in (3) is still the best known lowertail estimate for the local time of FBM with index H ∈ (0 , \ { / } . The Markoviancase H = / , is of course exceptional – the exact distribution of ℓ (0 , T ] for / -FBM(i.e. Brownian motion) had already been determined by Lévy [Lév48] 50 years prior toMolchan’s paper.Molchan’s proofs rely on the connection of the persistence probability to a certainpath integral functional and this relation is in fact also useful outside the FBM context,see e.g. [AGP15]. Based on the bounds for the persistence probability obtained in thismanner, he then derives the tail bounds for the distribution of the other functionals byexplicitly relating the events in question. However, determining κ this way is harderthan determining ¯ κ – as a functional of the path of B , the local time ℓ is in general an-alytically more involved than the other quantities ¯ B T , σ + T , τ max T and τ + . Thus, relatingdistributional properties of ℓ to the behaviour of B in a path-wise manner is a chal-lenging task. In addition, Molchan’s argument requires some technical tools, namelySlepian’s Lemma and reproducing kernel Hilbert spaces, which are specific to the Gaus-sian setting.The goal of this paper is to show how to circumvent these obstacles and establish theequality κ = 1 − H (4)for FBM directly by studying the local time. In fact, we prove a significantly strongerresult, namely that there is a constant C ∈ (0 , ∞ ) , such that P ( ℓ ((0 , T ] ≤ ∼ CT − (1 − H ) , (5)where here and in what follows we use the notation f ( T ) ∼ g ( T ) to indicate that theratio of the two functions f, g converges to as the argument T approaches ∞ . Our3pproach to show (5) does not use that FBM is a Gaussian process. Consequently, (5)not only holds for FBM, but for any H -sssi process which admits sufficiently regularlocal time measures.A heuristic interpretation of the equality (4) is that it relates the time B spends at to the box-counting dimension of its zero set, which equals − H . Indeed, there is awell known non-rigorous box-counting argument, see e.g. [DY95], which suggests thatthe probability of observing an excursion from of length greater than T is of order T − (1 − H ) . One way of looking at our result is that it makes this connection rigorous;our method indeed enables us to prove (4) using only the invariance properties of theunderlying processes, without recurrence to specific distributional structures such asGaussianity, the Markov or Martingale property, etc.It is immediate from (4) that we have ¯ κ = κ for B and the author believes that thisis also true in a more general context. In particular, one should be able to combine thearguments in this paper with the methods developed by Aurzada et al. in [AGPP18,AM18] to show that both persistence exponents coincide for all H -sssi processes whichare positively associated.The technique for establishing κ = 1 − H proposed below is completely novel in thecontext of persistence probabilities. It combines three principal ingredients: A distribu-tional representation of the local times using Palm theory; a simple bi-variate scalingrelation for an associated point process, which is equivalent to the H -sssi property; anda well-known invariance property of Palm distributions which is the measure theoreticcounterpart to cycle-stationary [Tho95] in the point processes setting. Only the first partrequires a few abstract results from the theory of random measures which are not basedon simple calculations. More precisely, we exploit the fact that the local time of an H -sssi process can be constructed as the Palm distribution associated to certain stationarynon-finite distributions of measures on the real line. This approach was originally de-veloped by Zähle [Zäh88, Zäh90] who applied it to determine the carrying (Hausdorff)dimension of local times and other random measures derived from H -sssi processes[Zäh91].The remainder of this text is organised as follows. In the next section we fix ournotation and present our main result for the local time persistence probabilities in ageneral setting, Theorem 2.1 and in two special cases, namely for FBM and the Rosen-blatt process. The main arguments to prove Theorem 2.1 are given in Section 3, subjectto some auxiliary results which require a more extensive discussion. The subsequentthree sections are devoted to this groundwork. The necessary background for the in- Heuristic scaling arguments often use the box-counting dimension to capture the fractality of a set due toits rather intuitive definition, whereas the Hausdorff dimension is generally preferrable from a math-ematical point of view, see e.g. [Fal04] for a discussion. Both notions of fractal dimension coincidefor many random fractals and in particular for the zero set of FBM. In fact, it was recently shown byMukeru [Muk18] that the level sets of FBM even have Fourier dimension − H .
2. Notation and main results
We assume throughout the remainder of this article that ( X t ) t ∈ R is a real-valued stochas-tic process defined on a complete probability space (Ω , F , P ) , which is continuous inprobability , i.e. lim h → P ( | X t + h − X t | > ǫ ) = 0 , for all t ∈ R . More importantly, X is also taken to be H-sssi, i.e. satisfy the invariance relations ( X t + s − X t ) s ∈ R d = ( X u + s − X u ) s ∈ R , for any t, u ∈ R , (stationarity of increments),and ( X rs ) s ∈ R d = ( r H X s ) s ∈ R , for every r ∈ (0 , , ( H -self-similarity) , where d = denotes equality of finite dimensional distributions. We extend the definitionof H -sssi to processes indexed by [0 , ∞ ) by restricting the stationarity of increments topositive shifts only. Note that, for stationary increment processes, continuity in prob-ability follows from continuity in probability at time . Moreover, self-similarity andcontinuity in probability at imply that P ( X = 0) = 1 [EM09, Lemma 1.1.1] and thuswe may rewrite the stationary increment property as ( X t + s − X t ) s ∈ R d = ( X s ) s ∈ R , for all t ∈ R . Let us now turn our attention to the main object of interest, the local time measure of X at . We use the following notational conventions related to measures: B ( · ) denotesthe Borel- σ -field of the space in brackets. If ν is a measure on B ( R ) , the Borel-sets ofthe real line, and ( a, b ) is an interval, we use the notation ν ( a, b ) instead of ν (( a, b )) andan analogous shorthand for closed and half open intervals. We frequently associate ameasure ν with its additive functional ν t = ( ν (0 , t ] , if t > , − ν ( − t, , if t ≤ , and vice versa. If ν is a random measure, then ( ν t ) t ∈ R is a non-decreasing stochasticprocess. We define the occupation measure of X on B ( R × R ) by setting ψ ( A × B ) = Z A { X r ∈ B } d r, A, B ∈ B ( R ) , (6)5nd recall that (6) yields a well-defined Borel measure as long as the trajectories of X areBorel-functions. We say that X has local times, or shorter X is LT , if for each n = 1 , , . . . , P -a.s., ψ (cid:0) ( − n, n ) × · (cid:1) is absolutely continuous w.r.t. Lebesgue measure . Since X has stationary increments it is in fact sufficient for X to be LT , that the Radon-Nikodym density d ψ ( I, d y ) / d y exists a.s. for an arbitrary open set I . Disintegration yields,for every y outside some Lebesgue-negligible set R , a locally finite measure ℓ y on B ( R ) such that ψ ( A × B ) = Z B ℓ y ( A ) d y, A, B ∈ B ( R ) , (7)and we call ℓ y the local time of X at level y . Moreover, it can be shown, see e.g. [GH74,Lemma (3)], that for every y / ∈ R , we can choose a version of ℓ y (0 , t ] which is right-continuous in the time variable. A similar statement holds for ℓ y ( − t, . Recall that wehave X (0) = 0 a.s., i.e. ℓ = ℓ X (0) , if / ∈ R , but a priori the existence of ℓ cannot be guar-anteed using the above construction of local times. This technical issue is addressed inSection 6, for the time being let us assume that / ∈ R and that ℓ is well-defined.We are chiefly interested in ℓ and therefore just abbreviate ℓ = ℓ and call it local time,without reference to the level . From the construction of ℓ we can straightforwardlyderive a path-wise representation. Let ℓ yǫ ( A ) := 12 ǫ ψ ( A × ( y − ǫ, y + ǫ )) , then we have that for all y / ∈ R lim ǫ → ℓ yǫ ( A ) = ℓ y ( A ) , A ∈ B ( R ) , (8)which shows that our definition agrees with the formula for ℓ given in the introduction.We now introduce two further structural conditions on ℓ which are necessary for ourderivation of the lower tail probabilities of ℓ (0 , T ] . Let supp ( ν ) = { t : ν ( t − ǫ, t + ǫ ) > for all ǫ > } denote the support of a measure ν on B ( R ) and recall that a nowhere dense set of realnumbers is a set whoose (topological) closure does not contain any interval. Assumptions on the local time.
With probability , ℓ has no atoms , AL supp ( ℓ ) is nowhere dense. ND AL is equivalent to demanding that ( ℓ t ) t ∈ R be continuous a.s. Both condi-tions entail a rather erratic behaviour of the trajectories of X , which is not surprising inview of our main example, fractional Brownian motion. The validity of AL and ND areindispensable for the approach to local times taken in this paper. To exclude patholo-gies, we also restrict ourselves to situations where ℓ is a.s. not the zero measure, wethen say ℓ is non-zero . We are now prepared to state our main result. Theorem 2.1 (Persistence of local time for H -sssi processes) . Let X be continuous in prob-ability, H -sssi and LT and denote by ℓ its local time at . If ℓ is non-zero and satisfies AL and ND , then there exists a constant c X ∈ (0 , ∞ ) such that, as T → ∞ , P ( ℓ (0 , T ] ≤ ∼ c X T − (1 − H ) . There are two important observations needed for the proof of Theorem 2.1. The firstone is a result stating that the lengths of the excursions of X from follow, in a cer-tain sense, a hyperbolic distribution. The precise formulation is given in Section 3 asProposition 3.1. The second one is that, under AL and ND , ℓ is entirely encoded in theexcursions of X from , which is manifested in the fact that the right-continuous in-verse of ( ℓ t ) t ∈ R is a.s. a pure jump process. Before we develop the details we devote theremainder of this section to some of the implications of Theorem 2.1.To this end we provide two example processes, for which Theorem 2.1 can be applied.The first one, naturally, is fractional Brownian motion. The local times and level sets ofFBM have been studied by several authors, the pioneering work was done by Kahanein the late 1960’s, see in particular [Kah85, Chapter 18]. Theorem 2.2 (Persistence of local time for FBM) . Let B denote fractional Brownian motionwith Hurst index H ∈ (0 , and denote by ℓ its local time at . Then there is a constant c B ∈ (0 , ∞ ) such that, as T → ∞ , P ( ℓ (0 , T ] ≤ ∼ c B T − (1 − H ) . Proof.
We only need to verify conditions AL and ND . The continuity in time of fractionalBrownian local time is well known, e.g. by applying the criterion proposed by Geman[Gem76] for Gaussian processes. Let Z B denote the set of zeroes of the FBM trajectory.Kahane [Kah85] showed that the Hausdorff dimension dim Z B of the zero set equals − H < a.s. Since B is a.s. continuous, it follows that Z B is closed. Together with itsnon-integer dimension this implies that Z B is nowhere dense and therefore supp ( ℓ ) isnowhere dense, since supp ( ℓ ) ⊂ Z B .To illustrate the power of our approach, we now discuss a non-Gaussian example,namely the Rosenblatt process R = ( R t ) t ∈ R . This process was introduced by Taqqu However, the author strongly believes that the conditions listed are not minimal an that in particularcondition ND is a consequence of condition AL for any H -sssi process which is continuous in probability,but is not aware of any proof of this implication. R which are relevant to verifythe local time persistence result. A comprehensive source for all stated facts is Taqqu’ssurvey article [Taq11]. Unlike FBM, the Rosenblatt process can only be defined for H ∈ ( / , . For any such H , R is uniquely defined (up to multiplication by a constant)and satisfies • R has Hölder continuous paths a.s. for any Hölder exponent γ < H , • R is H -sssi. Theorem 2.3 (Persistence of local time for the Rosenblatt process) . Let R denote the H -sssi Rosenblatt process, H ∈ ( / , and denote by ℓ its local time at . Then there is a constant c R ∈ (0 , ∞ ) such that, as T → ∞ , P ( ℓ (0 , T ] ≤ ∼ c R T − (1 − H ) Proof.
In principle, we can apply the same arguments as for FBM, but the correspondingpreliminary results for the Rosenblatt process needed to verify conditions AL and ND are less well known. We thus give a slightly more explicit version of the argument.Existence of square integrable (in space) local times for R has been shown in [She11].Let us show continuity of the cumulative local time process ( ℓ t ) t ≥ . Geman’s sufficientcriterion [Gem76, Therorem B (I)] for the continuity of the local time can be restated asfollows for two-sided stationary increment processes: Z − sup ǫ> ǫ P ( | R s | < ǫ ) d s < ∞ . (9)To show that (9) holds, we use results of Veillette and Taqqu [VT13], who studied thedistribution of R extensively. In particular, they show that R has a smooth densityand the same holds, by self-similarity, for R s , s ∈ R \ { } . Let f s denote the density of R s . Then (9) is satisfied, if g ( s ) := lim sup ǫ ↓ ǫ Z ǫ − ǫ f s ( u ) d u is integrable around the origin. But by smoothness of f s , we have g ( s ) = f s (0) < ∞ and thus g ( s ) = s − H g (1) for any s > . Consequently, g is integrable around and (9)is satisfied.Turning to ND , we argue using the well known fact, see e.g [Kah85], about Höldercontinuity and Hausdorff dimension of the level sets of a real function f : If f is γ -Hölder continuous with γ ∈ (0 , then the Hausdorff dimension of its level sets isat most − γ. This is sufficient to complete the argument in the same fashion as forFBM. 8 emark 2.4.
The following general recipe may be used to verify the conditions of Theorem 2.1:If an H -sssi LT process has sufficiently high moments, then Hölder-continuity of the paths canalways be inferred from the Kolmogorov-Chentsov Continuity Theorem [Che56] and ND is thenalways satisfied. Additionally, if the transition density at exists, then AL is always satisfied.
3. Proof of Theorem 2.1
Let X be as in Theorem 2.1and let ℓ be its local time at . Recall that the correspondingadditive functional ( ℓ t ) t ∈ R , is given by ℓ t = ( ℓ (0 , t ] , if t > − ℓ ( t, , if t ≤ , and denote its right-continuous inverse by L = ( L x ) x ∈ R . It is straightforward from(8) that ( ℓ t ) t ∈ R is (1 − H ) -self-similar, hence L is / − H self-similar. By AL , ℓ has noatoms and hence L is strictly increasing. By ND , supp ( ℓ ) is nowhere dense. L is thena monotone pure jump process. Consequently, it induces a purely atomic randommeasure ˆ ℓ ∈ M . We say a random measure is β -scale-invariant, if its additive func-tional is a β -self-similar process. It thus follows from self-similarity of L that ˆ ℓ is / − H -scale-invariant. Because ˆ ℓ is purely atomic, we may identify it with a point process on R × (0 , ∞ ) , see Lemma 5.2. This point process is denoted by ˆ N and its intensity mea-sure by ˆΛ . The key observation of our argument is that ˆΛ is entirely determined (up toa multiplicative constant) by the invariance properties of ˆ ℓ . Proposition 3.1.
Let ˆ N denote the point process representation of the inverse local time mea-sure ˆ ℓ . Then the corresponding intensity measure ˆΛ is given by ˆΛ( d x × d m ) = cm − − (1 − H ) d x d m, (10) for some finite constant c > . We postpone the proof of Proposition 3.1 to the end of Section 6, but note that subjectto the validity of Proposition 3.1, all that remains to establish Theorem 2.1 is to relatethe tail behaviour of ˆΛ to the tail behaviour of ℓ . Proof of Theorem 2.1.
We observe that P ( ℓ (0 , T ] ≤
1) = P ( ℓ (0 , T ] <
1) = P ( L > T ) , T > , i.e. we obtain lower tail bounds for ℓ t from upper tail bounds for L . Let ˆ N denotethe point process representation of ˆ ℓ and note that L = R R ∞ m ˆ N ( d x × d m ) . Fix any r > . Since ˆ N is a simple point process and ˆ ℓ is purely atomic, we have by standardresults from random measure theory, e.g. [DVJ07, Prop. 9.1.III(v)], ˆ N ([0 , × ( r, ∞ )) = lim n →∞ n X k =1 (cid:26) ˆ ℓ (cid:18) k − n , kn (cid:21) > r (cid:27) , a.s . (11)9et P k,n := P (cid:18) ˆ ℓ (cid:18) k − n , kn (cid:21) > r (cid:19) , ≤ k ≤ n, n = 1 , , . . . , taking expectations in (11), we obtain lim sup n →∞ n X k =1 P k,n ≤ E ˆ N ([0 , × ( r, ∞ )) ≤ lim inf n →∞ n X k =1 P k,n , having applied Fatou’s Lemma and the inverse Fatou’s Lemma, i.e. n X k =1 P k,n n →∞ −→ E ˆ N ([0 , × ( r, ∞ )) . For any δ ∈ (0 , we may thus fix ≤ N δ < ∞ such that (1 − δ ) Z ∞ r cm − − (1 − H ) d m ≤ N δ X k =1 P k,N δ ≤ (1 + δ ) Z ∞ r cm − − (1 − H ) d m, (12)where we have used that E ˆ N ([0 , × ( r, ∞ )) = ˆΛ([0 , × ( r, ∞ )) = Z ∞ r cm − − (1 − H ) d m, according to Proposition 3.1. Using that P k,N δ = P ( L / Nδ > r ) by the stationarity of theincrements of L , we can rewrite (12) as (1 − δ ) c − H r − (1 − H ) ≤ N δ P ( L / Nδ > r ) ≤ (1 + δ ) c − H r − (1 − H ) , and applying the / − H -self-similarity of L and rearranging terms yields (1 − δ ) c − H (cid:16) rN / − H δ (cid:17) − (1 − H ) ≤ P (cid:16) L > rN / − H δ (cid:17) ≤ (1 + δ ) c − H (cid:16) rN / − H δ (cid:17) − (1 − H ) , i.e. P ( L > T ) = c − H (1 + o (1)) T − (1 − H ) , as T → ∞ , and Theorem 2.1 is proved, subject to Proposition 3.1.
4. Palm distributions and duality
We now provide the background needed to complete the proof of Theorem 2.1, startingwith some basics of random measure theory, namely we introduce Palm distributionsand discuss some of their key properties.Here and in the following two sections, we take a general point of view on ℓ and itsdistributional properties as a random measure, in particular we can forget about the10rocess X from which it is derived. We instead consider some complete measurablespace (Ω , F ) , equipped with a σ -finite measure Q . We denote by E Q ( · ) integration withrespect to Q . A measurable map ξ from Ω into the space ( M , B ( M )) of locally finitemeasures on R , equipped with its Borel- σ -field is called a random measure , even thoughwe stress that Q need not be a probability distribution. We call Q a quasi-distribution , todistinguish it from the measures which are elements of M and set Q ξ = Q ◦ ξ − , i.e. Q ξ ( G ) = Q ( ξ ∈ G ) , G ∈ B ( M ) . The intensity measure of ξ under Q (or Q ξ ) is given by Λ ξ ( A ) := E Q ξ ( A ) = Z ν ( A ) Q ξ ( d ν ) , A ∈ B ( R ) . In what follows, we frequently consider Q directly as a quasi-distribution on B ( M ) without explicit reference to a (canonical) random measure ξ with distribution Q , con-sequently we denote the associated intensity measure by Λ Q . Whenever we discussa probability measure, then we indicate this by using blackboard-face symbols, e.g. P , P ξ , E ξ , etc. Futhermore, to formalise our discussion of stationarity properties, weuse the shift group ( θ t ) t ∈ R on R . Note that the θ t , t ∈ R , act measurably on ( M , B ( M )) , and in particular we have that θ − t ν ( A ) = ν ( A + t ) = ν ◦ θ t ( A ) , A ∈ B ( R ) , where A + t := { a + t, a ∈ A } . A quasi-distribution Q on B ( M ) is invariant under theshifts ( θ t ) t ∈ R , i.e. stationary , if Q ( G ) = Q ( { θ t ν, ν ∈ G } ) , t ∈ R . If Q is stationary and satisfies λ Q := Λ Q ((0 , ∈ (0 , ∞ ) then it follows immediately that Λ Q ( d s ) = λ Q d s , i.e. Λ Q is a constant multiple ofLebesgue measure. We call λ Q ∈ [0 , ∞ ] the intensity of Q . Note that the quasi-distribution Q is a place holder for a stationarised version of the distribution of the local time mea-sure ℓ . The corresponding construction is given in Section 6. At the moment it is morebeneficial to stay in the general setting. However, the following assumptions on thesupport of Q S Q := supp ( Q ) = M \ [ N ∈ B ( M ): Q ( N )=0 N are justified in view of our applications. Let o denote the -measure, then o / ∈ S Q , i.e. Q is non-zero ,11nd { ν t , t ∈ R } = R , for all ν ∈ S Q . R We have λ Q > , if Q is non-zero and stationary. Note that R is in congruence withcondition AL , the details are given in Lemma A.1.We now turn to the subject of Palm distributions of a stationary, non-zero quasi-distribution Q . Fix any A ∈ B ( R ) with finite and positive Lebesgue measure, thenthe quasi-distribution defined by P Q ( G ) = 1 R A d s Z Z A { θ − t ν ∈ G } ν ( d t ) Q ( d ν ) , G ∈ B ( M ) , is independent of the choice of A , see Lemma A.2. It is referred to as the Palm measure of Q . If Q has finite intensity λ Q then a probability distribution is defined by P Q ( G ) = P Q ( G ) λ Q , G ∈ B ( M ) , and is called the Palm distribution of Q . Conversely, we say that a probability distri-bution P (on B ( M ) ) is Palm-distributed , if it is the Palm distribution of some stationaryquasi-distribution Q . Similarly, a random measure is Palm distributed if its distributionis the Palm distribution of some Q . It is well known, see e.g. [Zäh88, Lemma 3.3], thatthe almost sure properties of Q and P Q agree (up to shifts). We may thus assume thatany ν ∈ S ( P Q ) has the property indicated in R . The latter entails that the right contin-uous inverse ( ν − x ) x ∈ R of the additive functional ( ν t ) t ∈ R of ν is strictly increasing andthat we have ν ν − x = x, for all x ∈ R . This allows us to define the measurable group of random time shifts ˆ θ x := ˆ θ x ( ν ) := θ ν − x , x ∈ R . Note that we may pick A = [0 , in the definition of P Q and change variables accordingto the random time change to obtain the alternative representation P Q ( G ) = 1 λ Q Z Z ν { ˆ θ − x ν ∈ G } d xQ ( d ν ) , G ∈ B ( M ) . (13)The following basic lemma is crucial for our argument. It is a generalisation of [MN94,Lemma 2.3], see also [MNS00, Theorem 3.1]. Lemma 4.1 (Duality lemma) . Let Q be a stationary non-zero measure on B ( M ) with λ Q < ∞ , then its Palm distribution P Q is stationary with respect to the random shifts (ˆ θ x ) x ∈ R . Before we give the proof, we briefly discuss the intuition behind Palm distributionsin general and Lemma 4.1 in particular. Palm distributions originated in queuing the-ory [Pal43]. The concept is easiest understood for simple stationary point processes,12hich corresponds to Q being the distribution of a random counting measures in oursetting. In this case, the Palm distribution may be interpreted as a description of thedistribution of the point process seen from a ‘typical point’, an intuition which can bemade precise using ergodic theory, see e.g. the discussion in [Tho00, Chapter 8]. At theheart of Palm theory lies a duality principle [Tho95], which can be paraphrased as“A point process is stationary, if and only if its Palm version is stationary under point-shifts.”For the purpose of this paper, the backward implication in this statement is not needed.We only rely on the observation, that a Palm distributed random measure is stationaryw.r.t. to intrinsic shifts, i.e. shifts by mass points of its realisation, which is exactly whatis expressed in Lemma 4.1. Proof of Lemma 4.1.
Let G ∈ B ( M ) and fix any r > . Then the stationarity of Q impliesthat, for any y > , Z Z ν (0 ,r ]+ yν (0 ,r ] G ◦ ˆ θ x d xQ ( d ν ) = Z Z ν (0 ,r ]+ yν (0 ,r ] G ◦ ˆ θ x d xQ ( d ν ◦ θ − r )= Z Z − ν ( r, y − ν ( r, G ◦ ˆ θ x − ν ( − r, d xQ ( d ν )= Z Z y G ◦ ˆ θ x d xQ ( d ν ) , and an analogous calculation can be made for r < . Thus we have that, using (13), P Q (ˆ θ − x G ) = 1 λ Q Z Z ν G ◦ ˆ θ z + x d zQ ( d ν )= 1 λ Q (cid:18)Z Z ν G ◦ ˆ θ z d zQ ( d ν ) − Z Z x G ◦ ˆ θ z d zQ ( d ν )+ Z Z ν + xν G ◦ ˆ θ z d zQ ( d ν ) (cid:19) = 1 λ Q Z Z ν G ◦ ˆ θ z d zQ ( d ν ) = P Q ( G ) .
5. Bi-scale-invariance
So far, we have focussed our discussion of random measures on invariance with re-spect to time shifts only. Now we additionally consider scale-invariance of measures,which is the counterpart to self-similarity of processes. The essential observation of13his section is that stationarity combined with scale-invariance of a quasi-distributionor Palm distribution determines the corresponding intensity measures entirely up to amultiplicative constant.We illustrate this by means of marked point processes on the real line. We consider (Ω , F , P ) , i.e. we work under a probability measure. An extended marked point processwith positive marks (EMPP) is a point process on R × (0 , ∞ ) which is a.s. finite on all setsof the form A × M for bounded A ∈ B ( R ) and Borel sets M ⊂ (cid:18) ǫ, ǫ (cid:19) , for some ǫ > . Fix β > and r ∈ (0 , . We define a rescaled point process by S βr N ( A × M ) := N ( rA × r β M ) , where cM := { cm, m ∈ M } for any c ∈ R \ { } , hence a contraction by factor r in thetime domain is combined with a contraction by r β in the mark space into the operator S βr . An EMPP N on R × (0 , ∞ ) is called β -bi-scale-invariant if, for any r ∈ (0 , , P ◦ ( S βr N ) − = P ◦ N − . Similarly, the EMPP is stationary, if its distribution is invariant under the shifts ( θ t ) t ∈ R applied to the time domain only. We now recall a well known result about point pro-cesses: if the intensity measure of the EMPP is finite, then it follows from bi-scale in-variance together with stationarity that the intensity measure must be a product of amultiple of Lebesgue measure in time and a hyperbolic law on the marks. That thisis the case can be seen by noting that stationarity implies homogeneity in time of theintensity measure, i.e. it must be a multiple of Lebesgue measure. Additionally, bi-scale-invariance is transferred into stationarity on the mark space when mapped to logarith-mic coordinates, hence the intensity on the marks is a multiple of Lebesgue mesasurein logarithmic coordinates which is translated into a hyperbolic law when reversing thecoordinate transform. Proposition 5.1.
Let N be a bi-scale invariant, stationary, extended marked point process on R × (0 , ∞ ) with positive and locally finite intensity measure Λ N . Then Λ N necessarily is of theform Λ N ( d t × d m ) = cm − − / β d t d m. (14) Proof.
We only give an outline of the precise argument. A similar derivation in moredetail can be found in [DVJ07, Chapter 12], for the case of an extended marked Poissonprocess. Let Λ N denote the intensity measure of N . We assume that Λ N ( { } × (0 , ∞ )) = 0 , Instead of using the notion of an extended point process, one can also consider a classical locally finitepoint process if one equips the closure of the mark space with a metric which places infinitely faraway from any positive mark. N has almost surely no points at and that Λ N is absolutely continuouswith respect to -dimensional Lebesgue measure. We use a logarithmic change of coor-dinates, which makes it necessary to decompose N into a marked point process N + on (0 , ∞ ) and a marked point process N − on ( −∞ , with associated intensity measures Λ ± .Let us first consider Λ + only. By the logarithmic change of coordinates ( t, m ) (log t, β log t − log m ) , t ∈ R , m ∈ (0 , ∞ ) , bi-scale-invariance is turned into shift-invariance and consequently under the new co-ordinates, Λ + must be a product of Lebesgue measure and some absolutely continuousmeasure ρ + on the (coordinate transformed) mark space, whenever the intensity in timeis finite. In particular, reversing the coordinate transform, we obtain Λ + ( d t × d m ) = φ + ( t β /m ) tm d t d m for some locally integrable density φ + of ρ + on (0 , ∞ ) , see [DVJ07, p. 258] . A similarrepresentation holds for Λ − with a density φ − . The additional assumption of stationar-ity in the time domain now implies that we must have φ + ( m ) = φ − ( m ) = cm − / β , for some c > and thus (14) must be satisfied.To apply this representation to random measures we recall that atomic random mea-sures can be bijectively mapped to EMPPs. Let N be the space of locally finite extendedmarked point processes with positive marks N on ( R × (0 , ∞ )) satisfying Z ( m ∧ N ( A × d m ) < ∞ (15)for any bounded Borel set A , and let M a ⊂ M denote the locally finite, purely atomicrandom measures on B ( R ) . Lemma 5.2 ([DVJ07, Lemma 9.1.VII]) . There is a bijection mapping N onto M a . In principle, the bijection of Lemma 5.2 just consists of interpreting, for given ξ ∈ M a ,a point x ∈ supp ( ξ ) with ξ ( { x } ) = m > as a pair ( x, m ) ∈ R × (0 , ∞ ) and the collectionof all such points forms a marked point process N ξ . To deal with accumulation pointsof supp ( ξ ) , one needs to consider extended MPPs. Note that if the intensity measure Λ ξ of some random measure ξ under this bijection assigns infinite mass to a bounded open As noted above, the result there is for a Poisson process, however the statement transfers immediatelyto the general case. This is also discussed in [DVJ07] on pp. 260-262, however not in as much detail asthe Poisson case. A , then by local finiteness of ξ this must be the consequence of infinitely manysmaller and smaller atoms. Thus N ξ puts finite mass on any open set A × ( δ, ∞ ) for δ > and is thus locally finite on (0 , ∞ ) in the extended sense. Since we wish to applyLemma 5.2 to measures without fixed atoms, we observe that the bijection is preservedwhen restricted to the subspace of measures without fixed points and the subspace ofwithout fixed points, respectively.The version of β -bi-scale-invariance for quasi-distributions is just called β -scale-invariance,as defined Section 3 for probability distributions. Let us quickly recall this definition inthe notation of this section. Let P be a probability distribution on B ( M ) . P is called β -scale-invariant, if for any r ∈ (0 , , B ∈ B ( R ) we have P ( G ) = P (cid:16) { R βr ν, ν ∈ G } (cid:17) , G ∈ B ( M ) where (cid:16) R βr ν (cid:17) ( A ) := r − β ν ( rA ) , or short P ◦ ( R βr ) − = P . When considering a non-finite quasi-distribution, one needs toadd an additional factor rescaling the total mass: A stationary non-finite quasi-distribution Q with finite intensity λ Q is called β -scale-invariant if Q ◦ (cid:0) R βr (cid:1) − = r β − Q. That this is the correct notion of scale-invariance for quasi-distributions can be seen bylooking at their Palm distributions:
Lemma 5.3 ([Zäh88, Statement 2.3]) . A stationary, non-zero, non-finite quasi distribution Q is β -scale-invariant if and only if its Palm distribution P Q is β -scale-invariant.
6. Local time as a Palm distribution
Zähle observed in [Zäh88] that scale-invariance of random measures can be based on anotion of scaling around a typical point of mass of the measure, i.e. by scale-invarianceof Palm distributions. Clearly, the local time of a centered self-similar process is alwaysdistributionally scale-invariant in the usual sense, i.e. when scaling is performed w.r.t.to the origin. To fit local time and other fractal measures derived from a H -sssi pro-cess X into the framework developed in [Zäh88, Zäh90] one has to express them asPalm distributions. This is done in [Zäh91]. We do not require the main results of thiswork which are concerned with the carrying Hausdorff dimension of the realisationsof random measures derived from X , but remark in passing, that they can be usefulfor verifying assumption ND , cf. the proofs of Theorems 2.2 and 2.3. In view of theprevious two sections, we only need to know that the local time of an H -sssi processcan be viewed as a Palm distribution. The precise result, in the notation introduced inSection 5 is as follows: 16 roposition 6.1 ([Zäh91, Proposition 6.9.]) . If X is H -sssi and LT , then ℓ is an (1 − H ) -scale-invariant, Palm-distributed random measure. We may rephrase the statement of Proposition 6.1 as a statement about P ℓ , the distri-bution of the local time as a random measure: There exists some quasi-distribution Q ℓ ,such that P ℓ is the Palm-distribution of Q ℓ . Remark 6.2.
As mentioned in Section 2, it cannot be a priori excluded that ∈ R and thus ℓ = ℓ = ℓ X (0) is not defined. Note that in [Zäh91], this is avoided by working with theaveraged version ˜ P ℓ ( · ) = Z P ℓ,t ( · ) d t, where P ℓ,t ( · ) denotes the distribution of θ − t ℓ X t , t ∈ R . This trick can always be used in thestationary increment case, but it has the drawback that this version of the local time does nothave the usual path-wise interpretation. It is, however, easily seen that ˜ P ℓ and P ℓ have the samedistribution as random measures, if the latter is defined, see the discussion in [Zäh91, p. 132].We remark that the use of ˜ P ℓ is obsolete, if X a.s. has continuous paths. Aside from the technical issue of Remark 6.2, we can give a path-wise interpretationto the quasi-distribution Q ℓ . Let us set Q ℓ ( G ) = E Z { ℓ y ( · ) ∈ G } d y, G ∈ B ( M ) , o / ∈ G, recalling that o is the trivial measure. We can think of Q ℓ ( G ) as the ‘local time at theorigin’ of a trajectory in the flow of X , i.e. the quasi-distribution Q X on trajectoriesobtained via Q X ( H ) = Z { X + y ∈ H } d y, H ∈ Z ( C ) , where C is a suitable path space equipped with the σ -field Z ( · ) of cylinder sets. Q X canbe thought of as mixture of the law of X w.r.t. Lebesgue measure in the ‘origin’ X (0) toobtain a stationary measure on paths. From Q X we can derive a stationary version ofthe occupation measure and then disintegrate to obtain Q ℓ . Note however, that Q ℓ and P ℓ need not be interpreted in this way – it is only necessary that the distribution of localtime as a random measure is a Palm distribution.Finally, we are in the position to prove Proposition 3.1 and thus conclude the proofof Theorem 2.1. Proof of Proposition 3.1.
The representation (10) follows immediately from Proposition 5.1upon showing / − H -bi-scale-invariance of ˆ N . This is, in turn, equivalent to / − H -scale-invariance and stationarity of ˆ ℓ . The scale-invariance has already been established inthe opening paragraph of Section 3. We now show that L has stationary incrementsand thus ˆ ℓ is a stationary random measure. By Proposition 6.1, ℓ is Palm-distributed.17or x ∈ R set t ( x ) := inf { t : ℓ ( t ) > x } = L x . Fix x ∈ R and consider a finite familyof points x , . . . , x n with x < x < · · · < x n and the corresponding random times t ( x i ) , i = 1 , . . . , n. Almost surely, { t ( x i ) , i = 1 , . . . , n } ⊂ supp ( ℓ ) and because ℓ is Palm-distributed, we may apply Lemma 4.1, to obtain (cid:0) t ( x i ) − t ( x ) (cid:1) ni =1 d = (cid:0) t ( x i − x ) − t (0) (cid:1) ni =1 = (cid:0) L x i − x (cid:1) ni =1 and thus L has stationary increments. Consequently, ˆ ℓ is / − H -scale-invariant and sta-tionary and this concludes the proof of the Proposition 3.1.
7. Related work and historical remarks
As already noted, the key part of our approach is Zähle’s construction of the local timeas a Palm distribution. His main goal was to show that scale-invariance of randommeasures already determines their carrying Hausdorff dimension. He did however al-ready notice the relation between index of self-similarity and tails of the ‘gap lengths’in fractal sets, see [Zäh88, Theorem 5.3] and even mentioned the zero set of Brownianmotion as an explicit example. However, he did not pursue this investigation further inthe more general set up of [Zäh90, Zäh91]. In fact [Zäh88, Theorem 5.3] prompted thepresent author’s investigation of Palm distributions as a means to derive persistence ex-ponents from invariance properties and be regarded as a precursor of Proposition 3.1.One great advantage of Zähle’s theory is that it works in arbitrary space and time di-mension, and it would certainly be fruitful to try and extend the discussion presentedhere to higher dimensions.The ‘Palm duality’ mentioned in Section 4 is actually also true in the context of ran-dom measures. The invariance relation expressed in Lemma 4.1 holds in a more generalsetting [Mec75]. A converse statement of equal generality was proved in [HL05], whereit is also shown that Palm distributions are characterised by the duality principle evenin the random measure setting. The random time change approach we have chosen istaken from [GH73]. In fact, its importance for studying local times was already pointedout there. Earlier uses of the same concept can be found in [Tot66, Mar65].Generalising the EMPP representation from completely random measures to station-ary random measures was first suggested by Vere-Jones in [VJ05], see also [DVJ07]. Itis also conjectured there, that it might prove fruitful to pursue this generalisation in thesetting of quasi-distributions introduced in [Zäh88, Zäh90, Zäh91] and this view wasjustified as the results of this article illustrate.
Acknowledgement.
The author would like to express his gratitude to Frank Aurzadafor introducing him to the subject of persistence probabilities, discussing with him18any aspects of the present work and offering helpful comments on the manuscript.Furthermore, he wishes to thank Safari Mukeru for an interesting exchange on the sub-ject of inverse local times of fractional Brownian motion.
A. Some auxiliary statements used in the text
Lemma A.1.
Let ℓ be a non-zero local time of an H -self-similar process satisfying AL , then { ℓ t , t ∈ R } = R . Proof.
From the path-wise representation of ℓ , it follows immediately that ( ℓ t ) t ≥ is an − H -self-similar process. Assume that P ( ℓ t ∈ [0 , K ] for all t ) > δ > for some K < ∞ , δ > . Fix b > , then we also have P ( ℓ T b ∈ [0 , K ]) > δ for any T and by self-similarity, this means P ( ℓ b ∈ [0 , KT − (1 − H ) ]) > δ for any T > . Thus ℓ (0 , b ] = 0 with probability exceeding δ . Since b was arbitrary, this means ℓ t ≡ on [0 , ∞ ) . Asimilar argument works for negative b and we obtain a contradiction to ℓ being non-zero. Hence the range of ( ℓ t ) t ≥ is a.s. not bounded on either side. By AL , i.e. continuityof ℓ t , the range must cover all of R . Lemma A.2.
The definition of the Palm measure P Q of Q does not depend on the choice of theset A .Proof. For probability distributions this is shown in [Kal06, Lemma 11.2]. The prooftransfers easily to quasi-distributions. Let Q be stationary and non-zero. Fix G ∈ B ( M ) ,let g ( ν ) = { ν ∈ G } and consider the measures ν g ( B ) = Z B g ( θ − s ν ) ν ( d s ) , B ∈ B ( R ) , then the definition of the Palm measure reads P Q ( G ) = R ν g ( A ) Q ( d ν ) R A d s . (16)We claim that θ − t ν g = ( θ − t ν ) g , then we have for any measurable function h , Z h ( θ − t ν g ) d Q = Z h (( θ − t ν ) g ) d Q = Z h ( ν g ) d Q by stationarity of Q . This means that stationarity of Q is preserved under the opera-tion ( · ) g , hence the numerator in (16) does not change under translations of A , i.e. is amultiple of Lebesgue measure. To prove the claim we note that, for any Borel set B , θ − t ν g ( B ) = ν g ( B + t ) = Z B + t g ( θ − s ν ) ν ( d s )= Z { s − t ∈ B } g ( θ − s ν ) ν ( d s ) = Z { u ∈ B } g ( θ − u − t ν ) ν ( d u + t )= Z B g ( θ − u θ − t ν ) θ − t ν ( d u ) = ( θ − t ν ) g ( B ) . eferences [AB18] F. Aurzada and M. Buck. Persistence probabilities of two-sided (integrated)sums of correlated stationary Gaussian sequences. Journal of StatisticalPhysics , 170(4):784–799, 2018.[AGP15] F. Aurzada and N. Guillotin-Plantard. Persistence exponent for discrete-time, time-reversible processes. arXiv preprint arXiv:1502.06799 , 2015.[AGPP18] F. Aurzada, N. Guillotin-Plantard, and F. Pene. Persistence probabilitiesfor stationary increment processes.
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