Small time Chung-type LIL for Lévy processes
aa r X i v : . [ m a t h . P R ] J a n Bernoulli (1), 2013, 115–136DOI: 10.3150/11-BEJ395 Small time Chung-type LIL for L´evy processes
FRANK AURZADA , LEIF D ¨ORING and MLADEN SAVOV Technische Universit¨at Berlin, Institut f¨ur Mathematik, Sekr. MA 7-4, Straße des 17. Juni 136,10623 Berlin, Germany. E-mail: [email protected] Department of Statistics, University of Oxford, 1, South Parks Road, Oxford OX1 3TG, UK.E-mail: [email protected] New College, University of Oxford, Holywell Street, Oxford OX1 3BN, UK.E-mail: [email protected]
We prove Chung-type laws of the iterated logarithm for general L´evy processes at zero. Inparticular, we provide tools to translate small deviation estimates directly into laws of theiterated logarithm.This reveals laws of the iterated logarithm for L´evy processes at small times in many concreteexamples. In some cases, exotic norming functions are derived.
Keywords: law of the Iterated Logarithm; L´evy process; small ball problem; small deviations
1. Introduction
A classical question in stochastic process theory is to understand the asymptotic behaviorof a given stochastic process X = ( X t ) t ≥ on the level of paths. In the present work, weconsider general L´evy processes and find Chung-type LIL (laws of the iterated logarithm)at zero; that is, given the L´evy process X , we aim at characterizing a norming function b , satisfying lim inf t → k X k t b ( t ) = 1 , where k X k t := sup ≤ s ≤ t | X s | . (1.1)The topic of large and small time fluctuations of L´evy processes has been studied exten-sively in the past (see, e.g., Doney [10] for an overview and Bertoin [3], Sato [19], Bertoin,Doney and Maller [5]).It is well known that, via the Borel–Cantelli lemma, Chung-type LIL for a generalstochastic process are connected to the so-called small deviation rate of the process, thatis, − log P ( k X k t ≤ ε ) , as ε → t → . (1.2)The main motivation for this paper originates from the recent work Aurzada and Dereich[2], where a framework for obtaining the small deviation rate (1.2) for general L´evy This is an electronic reprint of the original article published by the ISI/BS in
Bernoulli ,2013, Vol. 19, No. 1, 115–136. This reprint differs from the original in pagination andtypographic detail. (cid:13)
F. Aurzada, L. D¨oring and M. Savov processes (but fixed t ) is provided. The difficulty in passing over from the small deviationestimate to the respective LIL concerns circumventing the independence assumption ofthe Borel–Cantelli lemma.In this paper we show how the asymptotics of (1.2) imply explicit LIL. We stress thatit is not sufficient to have estimates for (1.2) for fixed t , which usually are referred to assmall deviation estimates.Small deviation problems are studied independently of LIL and have connections toother fields, such as the approximation of stochastic processes, coding problems, the pathregularity of the process, limit laws in statistics and entropy numbers of linear operators.We refer to the surveys Li and Shao [12], Lifshits [15], for an overview of the field, andto Lifshits [14], for a regularly updated list of references, which also includes referencesto laws of the iterated logarithm of Chung type. The papers of Taylor [25], Mogul’ski˘ı[18], Borovkov and Mogul’ski˘ı [7], Simon [23, 24], Linde and Shi [16], Lifshits and Simon[13], Linde and Zipfel [17], Shmileva [21], Shmileva [22] provide a good source for earlierresults on small deviations of L´evy processes.We now discuss LIL for special L´evy processes that have already appeared in the liter-ature. The norming function b ( t ) = p π t/ (8 log | log t | ) for a standard Brownian motioncan be derived from the large time LIL, proved by Chung [9], via time inversion. Forany L´evy process with non-trivial Brownian component, the recent result of Buchmannand Maller [8] shows that (1.1) holds with the same norming function as for a standardBrownian motion. If X is an α -stable L´evy process, (1.1) holds with norming function b ( t ) = ( c α t/ log | log t | ) /α , which goes back to Taylor [25]. The question was studied forsubordinators already in [11]; there, the norming function can be obtained from theLaplace transform.Of course, it is natural to ask for the general structure of the norming function forarbitrary L´evy processes not having the special features of the examples mentioned sofar. LIL for more general L´evy processes were obtained in Wee [26]; see Wee [27] for moreexamples. It was shown that if, for some positive constant θ , P ( X t > ≥ θ and P ( X t < ≥ θ for all t sufficiently small, (1.3)holds, then upper and lower bounds in the LIL hold in the following sense: for λ suffi-ciently small and λ sufficiently large,1 ≤ lim inf t → k X k t b λ ( t ) and lim inf t → k X k t b λ ( t ) ≤ b λ given by b λ ( t ) := f − (cid:18) log | log t | λt (cid:19) , where f is given by some explicit, but complicated expression depending on the L´evytriplet.Although the results of Wee are quite general, there are some points which we aim toimprove in the present work. First, we try to demonstrate and explain clearly how the LIL hung-type LIL for L´evy processes at zero λ and λ above, which can influence the normingfunction essentially (see (3.2) below for an example of influence on the exponential level)in the case when b λ is not regularly varying at zero. In our approach, we keep track ofthe appearing constants in an optimal way. This allows us, in the case of known strongsmall deviation order, to transfer the constant in the strong small deviation order to thelimiting constant in the LIL. Third, we provide alternative conditions to (1.3) which areexplicit in terms of the L´evy triplet. We believe our conditions to be weaker than (1.3),but, as necessary and sufficient conditions for the latter in terms of the L´evy triplet seemto be unknown in general, it is difficult to verify our claim, although our examples hintat this direction.This paper is structured as follows. In Section 2, we give the main results that managethe transfer between small deviations and LIL. Several examples of LIL for concrete L´evyprocesses are collected in Section 3. The proofs are given in Section 4.Let us finally fix some notation. In this paper we let X be a L´evy process with char-acteristic triplet ( γ, σ , Π), where γ ∈ R , σ ≥
0, and the L´evy measure Π has no atomat zero and satisfies Z (1 ∧ x )Π(d x ) < ∞ . For basic definitions and properties of L´evy processes we refer to Bertoin [3], Sato [19].As we are interested only in the behavior for small times, we discard all jumps biggerthan 1 in absolute value and assume such truncation throughout the paper. Hence, thecharacteristic exponent, E e i zX t =: e tψ ( z ) , has the form ψ ( z ) = i γz − σ z Z − (e i zx − − i zx )Π(d x ) , z ∈ R . For later use we denote by Φ the Laplace exponent of a subordinator A , E e − uA = e − Φ( u ) ,Φ( u ) = uγ A + Z ∞ (1 − e − ux )Π A (d x ) . Further, we use the standard notation ¯Π( ε ) := Π([ − ε, ε ] c ) for the two-sided tail of theL´evy measure.In the following, we denote by f ∼ g the strong asymptotic equivalence, that is,lim f /g = 1, and by f ≈ g the weak asymptotic equivalence, that is, 0 < lim inf f /g ≤ lim sup f /g < ∞ .
2. Main results
Our first theorem manages the transfer from small deviation rates to LIL under minimalloss of constants.
F. Aurzada, L. D¨oring and M. Savov
Theorem 2.1.
Let X be a L´evy process (without loss of generality assume that X hasjumps smaller than in absolute value). Let F be a function increasing to infinity atzero, such that with some < λ ≤ λ < ∞ λ F ( ε ) t ≤ − log P ( k X k t < ε ) ≤ λ F ( ε ) t for all ε < ε and t < t . (2.1) Further, define b λ ( t ) := F − (cid:18) log | log t | λt (cid:19) for λ > , and assume that, as n → ∞ , ( n + 1) − ( n +1) β (cid:12)(cid:12)(cid:12)(cid:12)Z | x | >b λ ′ ( n − nβ ) x Π(d x ) − γ (cid:12)(cid:12)(cid:12)(cid:12) (2.2)= o( b λ ′ ( n − n β )) for all β > and λ ′ > λ .Then the LIL ≤ lim inf t → k X k t b λ ′ ( t ) and lim inf t → k X k t b λ ′ ( t ) ≤ hold almost surely for any λ ′ < λ and λ ′ > λ . Remark 2.1.
It is important to note the role of (2.2). It ensures that the process doesnot become too asymmetric when one continues to cut off more and more smaller jumps.Only in this case is it possible to expect an estimate of type (2.1) to follow from theframework given in Aurzada and Dereich [2]. Corollary 2.4 below and, in particular,(2.11) give a sufficient condition when this is the case.
Remark 2.2.
Let us relate our condition (2.2) with the condition of Wee [26]. Notethat (2.2) is analytic, that is, in terms of the L´evy triplet, whereas Wee’s condition (1.3)is probabilistic. It seems that (1.3) cannot always be checked from the L´evy triplet. Tounderstand the difficulty, it may be instructive to look at Theorems 4 and 5 in Andrew[1], which reformulate (1.3) in terms of other probabilistic quantities.It is crucial that there is almost no loss of constants in the transfer from the smalldeviations to the LIL as in cases when b λ is not regularly varying, the constants λ ′ , λ ′ may influence the rate function drastically; see (3.2) for an extreme example.If instead b λ only depends on λ via a multiplicative constant, our approach allows tostrengthen the previous theorem to the optimal limiting constants. Such examples occur,for instance, if the small deviation rate function F is regularly varying. hung-type LIL for L´evy processes at zero Corollary 2.1.
In the setting of Theorem 2.1, assume additionally that F is regularlyvarying at zero with non-positive exponent. Then the following LIL hold almost surely: ≤ lim inf t → k X k t b λ ( t ) and lim inf t → k X k t b λ ( t ) ≤ . (2.3) In particular, if there is λ > such that (2.1) holds for all λ < λ and all λ > λ , then lim inf t → k X k t b λ ( t ) = 1 a.s. In the setting of a regularly varying rate function, say F is regularly varying at zerowith exponent − α , α >
0, one can express (2.3) aslim inf t → k X k t b ( t ) ∈ [ λ /α , λ /α ] , a.s.This shows that only the quality of the small deviation estimate (2.1) matters in order toobtain the limiting constant in the LIL. Recall that the Blumenthal zero–one law impliesthat the limit is almost surely equal to a deterministic constant, which in this case canbe specified.Theorem 2.1 reduces the question of the right norming function for the LIL to thequestion of small deviations which is known precisely for many examples. For generalL´evy processes, those have been obtained in Aurzada and Dereich [2] (their results werestated for t = 1 only, but hold, in general, as we discuss in Proposition 2.1 below). Inparticular, for symmetric L´evy processes, their main result states that the rate functionis given by F ( ε ) = ε − U ( ε ) , (2.4)where U ( ε ) is the variance of X with jumps larger than ε replaced by jumps of size ε , U ( ε ) := ε ¯Π( ε ) + σ + Z ε − ε x Π(d x ) . (2.5)From these specific small deviations we can deduce the following corollary for symmetricprocesses. Corollary 2.2.
Let X be a symmetric L´evy process; then there are < λ ≤ λ < ∞ such that, almost surely, ≤ lim inf t → k X k t b λ ( t ) and lim inf t → k X k t b λ ( t ) ≤ , with b λ ( t ) := F − (cid:18) log | log t | λt (cid:19) F. Aurzada, L. D¨oring and M. Savovand F defined in (2.4). If, additionally, F is regularly varying at zero with exponent − α , α > , then the following general bounds hold:
112 12 α ≤ λ ≤ λ ≤ α . The loss of constants in the corollary is only due to the general formulation. For someexamples we will see below that the small deviations are known in the strong asymptoticsense so that Theorem 2.1 gives the precise law.In the sequel we call “strongly non-symmetric” L´evy processes the processes for which(2.2) does not hold. Their study requires different assumptions on b λ ; see (2.8). For thiscase, we provide a different link between small deviation rates and LIL. The next resultdoes not require (2.2) and thus allows us to study the “strongly non-symmetric” L´evyprocesses as well as other cases when (2.2) is difficult to verify. The latter is substitutedby the seemingly easier (2.8) at the expense of the strength of the result; that is, wemanage to keep track of the constants in the norming function in an optimal way, butlose the limiting constant. We have tried unsuccessfully to find a suitable relation between(2.2) and (2.8). We strongly suspect that neither one follows from the other. Theorem 2.2.
Let X be a L´evy process with jumps smaller than in absolute value,and let F be a function increasing to infinity at zero such that for < λ ≤ λ < ∞ λ F ( ε ) t ≤ − log P ( k X k t < ε ) ≤ λ F ( ε ) t for all ε < ε and t < t . (2.6) Furthermore, set b λ ( t ) := F − (cid:18) log | log t | λt (cid:19) , (2.7) and suppose that there is a constant C > such that Cb λ ( t ) ≤ b λ ( t/ , < t ≤ t , λ ∈ ( λ / , λ ) . (2.8) Then the LIL < lim inf t → k X k t b λ ′ ( t ) and lim inf t → k X k t b λ ′ ( t ) < ∞ hold almost surely for all λ ′ < λ and λ < λ ′ . Again, if the rate function F is regularly varying, then we can strengthen the result.Recall that the L´evy processes that appear in the formulation of the next sequence ofresults have jumps smaller than 1 in absolute value. Corollary 2.3.
In the setting of Theorem 2.2, assume additionally that F is regularlyvarying at zero with negative exponent. Then the following LIL holds almost surely: lim inf t → k X k t b ( t ) ∈ (0 , ∞ ) . hung-type LIL for L´evy processes at zero Corollary 2.4.
Let X be a L´evy process with triplet ( γ, σ , Π) . Assume that u ε is thesolution of the equation Λ ′ ε ( u ) = 0 , where Λ ε is the following log Laplace transform: Λ ε ( u ) = σ u + (cid:18) γ − Z [ − , \ [ − ε,ε ] x Π(d x ) (cid:19) u + Z ε − ε (e ux − − ux )Π(d x ) . (2.9) Set F ( ε ) := ε − U ε ( ε ) − Λ ε ( u ε ) , U ε ( ε ) := ε ¯Π( ε ) + σ + Z ε − ε x e − u ε x Π(d x ) , (2.10) and assume F is increasing to infinity as ε → . Define b as in (2.7), and assume that b satisfies (2.8). If, furthermore, ε | u ε | = o(log log F ( ε )) , as ε → , (2.11) is satisfied, then we have, for some λ , λ > , < lim inf t → k X k t b λ ( t ) and lim inf t → k X k t b λ ( t ) < ∞ a.s. Let us explain the quantities appearing in Corollary 2.4 in more detail. The mainobservation is that the proof for the small deviation estimates in Aurzada and Dereich[2] (Theorem 1.5) can be used directly for any t >
Proposition 2.1.
Let Λ ε be as defined in (2.9) and assume that u ε is the solution of Λ ′ ε ( u ε ) = 0 . Then, with F as in (2.10), we have, for all t > and all ε < , tF (2 ε ) − ε | u ε | − ≤ − log P ( k X k t ≤ ε ) ≤ tF (cid:18) ε (cid:19) + ε | u ε/ | + 3 . (2.12)The term ¯Π(2 ε ) in (2.12) (included in the F term) comes from the requirement thatthere should be no jumps larger than 2 ε . After removing these jumps, the process maydrift out of the interval [ − ε, ε ], which is prevented by applying an Esscher transform tothe process, whose “price” is given by the term − Λ ε ( u ε ). The quantity u ε is the driftthat has to be subtracted in order to make the process a martingale. Then the remainingprocess is treated as in the symmetric case, and the same term ε − U ε ( ε ) appears as in(2.4), but, this time, with respect to the L´evy measure transformed by the change ofmeasure. F. Aurzada, L. D¨oring and M. Savov
Note that (2.12) is almost the required estimate in (2.6), except for the term ε | u ε | ,which may spoil the estimate. It is exactly condition (2.11) that ensures that the term ε | u ε | can be neglected.We stress that in some cases ε | u ε | does give an order that is larger than tF ( ε ) so thatthe function b from (2.7) is not the right norming function. This effect can be observedin some examples below. In particular, this happens for processes of bounded variationwith non-zero drift. Proposition 2.2.
Let X be a L´evy process with bounded variation and non-vanishingeffective drift, that is, R [ − , | x | Π(d x ) < ∞ and c := γ − R − x Π(d x ) = 0 . Then lim t → k X k t t = | c | a.s. The proof of this proposition is based on classical arguments rather than any connectionto small deviations.
3. Explicit LIL for L´evy processes
In this section, we collect concrete L´evy processes for which we can transform smalldeviation results to an LIL. As we have seen, understanding the small deviation rates iscrucial.In this section we keep in mind that our processes in all proofs have no jumps biggerthan 1 in absolute value. However, without loss of generality, in some statements we use“stable L´evy processes” and others which presuppose unbounded jumps.The first corollary gives us a useful variance domination principle for LIL that worksfor many examples.
Corollary 3.1.
Suppose X and X are independent symmetric L´evy processes, then X + X and X fulfill precisely the same LIL if lim ε → U X ( ε ) U X ( ε ) = 0 . Proof.
This follows directly from Corollary 2.2 noticing that U X + X = U X + U X . (cid:3) In the same spirit, the following corollary (recovering (3.2) in Buchmann and Maller[8]) displays the intuitive fact that a non-zero Brownian component dominates the jumpsof a L´evy process.
Corollary 3.2. If X is a L´evy process with σ = 0 , then lim inf t → k X k t p t/ log | log t | = π σ √ a.s.hung-type LIL for L´evy processes at zero Proof.
Following precisely the proof of Corollary 2.6 of Aurzada and Dereich [2], one canshow that the small deviation rates of L´evy processes with non-zero Brownian componentare given by − log P ( k X k t < ε ) ∼ π σ ε − t, as ε → t → b ( t ) = p t π / (8 log | log t | ) and R | x | >ε | x | Π(d x ) = o( ε − ), it remains to be seen that a n +1 ≤ cb ( a n ) = a n / log | log a n | for a n = n − n β and β >
1. This can be verified by simple computations. (cid:3)
Similarly to L´evy processes with non-zero Brownian component, symmetric processesof smaller small deviation order (e.g., stable processes of smaller index) are dominatedby stable L´evy processes.
Corollary 3.3.
Let X be a symmetric α -stable L´evy process with α ∈ (0 , , and let Y be symmetric with U Y ( x ) = o( x − α ) . Then there is a constant < c α < ∞ such that lim inf t → k X + Y k t ( t/ log | log t | ) /α = lim inf t → k X k t ( t/ log | log t | ) /α = c /αα a.s. Proof.
The small deviation rate is given by − log P ( k X k t < ε ) ∼ c α ε − α t, as ε → t → c α > (cid:3) Remark 3.1.
The constant c α in the LIL of stable L´evy processes is the unknownconstant of the small deviations for respective α -stable L´evy processes (see Taylor [25]and Proposition 3 and Theorem 6 in Chapter VIII of Bertoin [3]). The results of Aurzadaand Dereich [2] entail the following concrete bounds:2 C α (cid:18) α + 112(2 − α ) (cid:19) < c α < α · C (cid:18) α + 102 − α (cid:19) , where C is the constant in the L´evy measure: Π(d x ) = C | x | − (1+ α ) d x . This implies c α ∼ C/α , as α →
0. We remark that, contrary to the symmetric case, the constant c α isknown explicitly for completely asymmetric stable L´evy processes; see Bertoin [4].Let us study the case when Π behaves as a regularly varying function at zero and issymmetric. Then the following LIL are satisfied.0 F. Aurzada, L. D¨oring and M. Savov
Corollary 3.4.
Let X be a L´evy process with triplet (0 , , Π) with Π being symmetricand ¯Π( ε ) ≈ ε − α | log ε | − γ , as ε → ,with < α < or α = 2 , γ > . Then lim inf t → k X k t b ( t ) ∈ (0 , ∞ ) a.s.with b ( t ) = (cid:18) t | log t | − γ log | log t | (cid:19) /α , < α < , (cid:18) t | log t | − γ log | log t | (cid:19) / , α = 2 , γ > . Proof.
The corollary follows from Theorem 2.1. The required small deviation estimate, − log P ( k X k t < ε ) ≈ (cid:26) ε − α | log ε | − γ t, < α < ε − | log ε | − γ t, α = 2 , γ > ε → t →
0, is obtained from Proposition 2.1 (cf. Example 2.2 in Aurzada andDereich [2] for t = 1). Since we deal with a symmetric process, condition (2.11) is triviallysatisfied due to u ε = 0. (cid:3) Having discussed the α -stable like cases, we now consider L´evy processes with polyno-mial tails near zero of different exponents. The technique used for this example can beextended to any case with essentially regularly varying L´evy measure at zero. Let X bea L´evy process with triplet ( γ, , Π), where Π is given byΠ(d x )d x = C (0 , ( x ) x α + C [ − , ( x )( − x ) α , (3.1)with 2 > α ≥ α and C , C ≥ C + C = 0. We now analyze the pathwise behavior atzero in the cases when α > α = 1, and 0 < α <
1, respectively. The second exponent α can be even negative. Corollary 3.5.
Let X be a L´evy process with triplet ( γ, , Π) with Π as in (3.1). Thenthe following holds: If α ≥ α , C = 0 , and α > , then lim inf t → k X k t ( t/ log | log t | ) /α ∈ (0 , ∞ ) a.s.hung-type LIL for L´evy processes at zero If α = α = 1 and C = C , then lim inf t → k X k t t/ log | log t | ∈ (0 , ∞ ) a.s. If > α ≥ α and the effective drift does not vanish, then lim t → k X k t t = | c | a.s. Proof.
Parts 1 and 2 follow from Theorem 2.1. The required small deviation estimates, − log P ( k X k t < ε ) ≈ ε − α t for ε → t →
0, are obtained from Proposition 2.1 (cf. Corollary 2.7, 2.8 and 2.9 ofAurzada and Dereich [2] for t = 1; note that u ε ≈ ε − in all cases). One can easily checkcondition (2.11).In part 3 the process is of bounded variation, so that the claim is included in Propo-sition 2.2. (cid:3) We now come to L´evy processes obtained from Brownian motion by subordination,that is, X t = σB A t , where B is a Brownian motion independent of the subordinator A .In this case, the resulting L´evy process is symmetric and the small deviation asymptoticsis governed by the truncated variance U from (2.5). Corollary 3.6.
Let B be a Brownian motion independent of the subordinator A , where A has Laplace exponent Φ . For λ > we set b λ ( t ) := F − ( log | log t | λt ) with F ( ε ) := Φ( σ ε − ) + γ A σ ε − . Then, for some λ , λ > , ≤ lim inf t → k X k t b λ ( t ) and lim inf t → k X k t b λ ( t ) ≤ a.s.In particular, if γ A = 0 and Φ is regularly varying with positive exponent, we have lim inf t → k X k t (Φ − (log | log t | /t )) − / ∈ (0 , ∞ ) a.s. Proof.
The corollary follows from Theorem 2.1 with the small deviation estimate fromProposition 2.1, − log P ( k X k t ≤ ε ) ≈ (Φ( σ ε − ) + γ A σ ε − ) t, as ε → t → t = 1 and note themisprint there). Condition (2.2) is trivially fulfilled as the process is symmetric. (cid:3) F. Aurzada, L. D¨oring and M. Savov
For a more specific example, in particular, exhibiting exotic small time behavior, wechoose the subordinator A to be a Gamma process. Then one defines the so calledVariance-Gamma process as X t = σB A t + µA t for some constants σ = 0 and µ ∈ R . Corollary 3.7.
Let X be a Variance-Gamma process; then for µ = 0 there are someconstants < λ ≤ λ < ∞ such that ≤ lim inf t → k X k t e − λ log | log t | /t and lim inf t → k X k t e − λ log | log t | /t ≤ a.s. , (3.2) whereas for µ = 0 lim inf t → k X k t t = | µ | E ( A ) a.s. Proof.
The second part is included in Proposition 2.2, since the process is of boundedvariation with non-zero effective drift. In the first part, the effective drift is zero, and theclaim follows from Theorem 2.1. The small deviation estimate, − log P ( k X k t ≤ ε ) ≈ t | log ε | , as ε → t → , follows from Proposition 2.1 (cf. Example 2.12 of Aurzada and Dereich [2] for t = 1). (cid:3) In the first case of the previous corollary, the dependence of good small deviation esti-mates and good LIL becomes transparant. The fact that we cannot specify the constants λ , λ in (3.2) is only caused by the weak asymptotics for the small deviation estimateas we do not lose any further constants in the transfer of small deviations to the LIL. Ifone does not have more control on the constants λ , λ , the understanding of the precisesmall time behavior of X is far from optimal as the error enters exponentially.
4. Proofs
We start with a lemma which shows that the small deviation order is at least as large asthe term induced by the variance, defined in (2.5).
Lemma 4.1.
Let ε > , and let X be a L´evy process with L´evy measure concentrated on [ − ε, ε ] , then P ( k X k t ≤ ε/ ≤ exp (cid:18) − ε − (cid:18)Z ε − ε x Π(d x ) + σ (cid:19) t /12 + 1 (cid:19) for t ≥ .hung-type LIL for L´evy processes at zero Proof.
We proceed similarly to Lemma 4.2 in Aurzada and Dereich [2]. Let τ be thefirst exit time of X out of [ − ε, ε ]. Then, by Wald’s identity,4 ε ≥ lim sup t →∞ E [ X t ∧ τ ] ≥ lim sup t →∞ var[ X t ∧ τ ]= lim sup t →∞ (cid:18)Z ε − ε x Π(d x ) + σ (cid:19) E [ t ∧ τ ] = (cid:18)Z ε − ε x Π(d x ) + σ (cid:19) E [ τ ] . Therefore, P (cid:18) τ ≥ ε / (cid:18)Z ε − ε x Π(d x ) + σ (cid:19)(cid:19) ≤ ( R ε − ε x Π(d x ) + σ ) E [ τ ]8 ε ≤ . Let n := ⌊ t ( R ε − ε x Π(d x ) + σ ) / (8 ε ) ⌋ , and set t i := 8 iε / ( R ε − ε x Π(d x ) + σ ), i = 0 , . . . , n .Then P ( k X k t ≤ ε/ ≤ P (cid:16) ∀ i = 0 , . . . , n −
1: sup s ∈ [ t i ,t i +1 ) | X s − X t i | ≤ ε (cid:17) = P ( τ ≥ t ) n ≤ − n . (cid:3) This shows that the small deviation order is always at least as large as the term inducedby the truncated variance process. This fact will be needed later on.
Lemma 4.2.
Let F be a function that increases to infinity at zero. If, for some L´evyprocess X , for t ≤ t and ε < ε , − log P ( k X k t ≤ ε ) ≤ F ( ε ) t, then, for some absolute constant c > and all ε > small enough, ε − U ( ε ) ≤ c ( F ( ε ) + 1) . Proof.
We use the assumption together with the fact that if k X k t ≤ ε , then X must nothave jumps larger than 2 ε and the previous lemma,e − F ( ε ) t ≤ P ( k X k t ≤ ε ) = e − ¯Π(2 ε ) t P ( k X ′ k t ≤ ε ) ≤ e − ¯Π(2 ε ) t e − (2 ε ) − ( R ε − ε x Π(d x )+ σ ) t/ , where X ′ has L´evy measure Π restricted to [ − ε, ε ]. Noting that Lemma 5.1 of Au-rzada and Dereich [2] implies that U ( ε ) /ε ≈ U (2 ε ) / (2 ε ) , the statement of the lemmais proved. (cid:3) The lower bound in the LIL comes from the following lemma.
Lemma 4.3.
Let F be a function that increases to infinity at zero such that, for all t ≤ t and ε ≤ ε , λF ( ε ) t ≤ − log P ( k X k t ≤ ε ) , F. Aurzada, L. D¨oring and M. Savovand, for λ > , we set b λ ( t ) := F − ( log | log t | λt ) . Then, for any λ ′ < λ , ≤ lim inf t → k X k t b λ ′ ( t ) a.s. Proof.
For any λ ′ < λ , we can find 0 < r < < λr/λ ′ . Note that X n P ( k X k r n +1 ≤ b λ ′ ( r n )) < ∞ since − log P ( k X k r n +1 ≤ b λ ′ ( r n )) ≥ λF ( b λ ′ ( r n )) r n r = λ rλ ′ log | log r n | = log n rλ/λ ′ + const . (4.1)Hence, by the Borel–Cantelli lemma, { n : k X k r n +1 ≤ b λ ′ ( r n ) } is almost surely a finite set. Thus, for each path ω , we have that, for any n ≥ n ( ω ) andany t ∈ [ r n +1 , r n ), k X k t b λ ′ ( t ) ≥ k X k r n +1 b λ ′ ( r n ) ≥ , as b λ ′ is an increasing function. We take lim inf t → to obtain the statement. (cid:3) The proof of the upper bound in the LIL requires the following lemma.
Lemma 4.4.
Let F be a function that increases to infinity at zero such that for all t ≤ t and ε ≤ ε − log P ( k X k t ≤ ε ) ≤ λF ( ε ) t and, for λ > , set b λ ( t ) := F − ( log | log t | λt ) . Assume that lim sup n →∞ k X k ( n +1) − ( n +1) β b λ ( n − n β ) = 0 a.s. (4.2) for all β > . Then, for any λ ′ > λ , lim inf t → k X k t b λ ′ ( t ) ≤ a.s. (4.3) Proof.
For λ ′ > λ , we choose β > λ ′ > λβ . First note that (4.2) implieslim sup n →∞ k X k ( n +1) − ( n +1) β b λ ′ ( n − n β ) = 0 a.s. , (4.4) hung-type LIL for L´evy processes at zero b λ ( t ) is an increasing function in λ for fixed t ≥
0. Using the L´evy property, we seethe following: X n P (cid:16) sup ( n +1) − ( n +1) β ≤ t 1. The Borel–Cantelli lemma showsthat the sequence of independent events A n = n sup ( n +1) − ( n +1) β ≤ t Now we are in position to prove Theorem 2.1. For a detailed analysis of the lim supcase, we refer to Savov [20]. Proof of Theorem 2.1. The claim follows from Lemmas 4.3 and 4.4. To verify the useof Lemma 4.4 we still need to check that condition (4.2) holds for all β > β > λ ′ > λ . Since λ ′ is fixed, we set b := b λ ′ in order to increasereadability. We define the auxiliary function h ( t ) = b ( φ ( t )) , where φ ( t ) is chosen such that φ (( tt +1 ) (( t +1) /t ) β ) = t /t β and φ (0) = 0. Note that φ isincreasing and that φ ( s − s β ) = ( s − − ( s − β . We also do not record that φ and h dependon β and λ ′ . Step 1 : We show that Z / ¯Π( h ( t )) d t < ∞ . (4.5)6 F. Aurzada, L. D¨oring and M. Savov First, by the definition of h and a change of variables, we obtain Z / ¯Π( h ( t )) d t = Z C ( β )0 ¯Π( b ( s s − β )) d( s/ ( s + 1)) (( s +1) /s ) β d s = Z C ( β )0 ¯Π( b ( s s − β )) (cid:18) ss + 1 (cid:19) (( s +1) /s ) β (cid:18) s + 1 s (cid:19) β − s − (1 − β log(1 − ( s + 1) − )) d s, which can be estimated from above by C Z C ( β )0 b ( s s − β ) ¯Π( b ( s s − β )) b ( s s − β ) (cid:18) ss + 1 (cid:19) (( s +1) /s ) β s − − β | log s | d s ≤ C Z C ( β )0 U ( b ( s s − β )) b ( s s − β ) (cid:18) ss + 1 (cid:19) (( s +1) /s ) β s − − β | log s | d s ≤ C ′ Z C ( β )0 F ( b ( s s − β )) (cid:18) ss + 1 (cid:19) (( s +1) /s ) β s − − β | log s | d s = C ′ λ Z C ( β )0 log | log s s − β | s s − β (cid:18) ss + 1 (cid:19) (( s +1) /s ) β s − − β | log s | d s ≤ C ′ λ Z C ( β )0 s − − β (log | log s s − β | ) s (( s +1) /s ) β − /s β | log s | d s < ∞ , where we have used x ¯Π( x ) ≤ x ¯Π( x ) + R x − x y Π(d y ) + σ = U ( x ) ≤ cx F ( x ) for someabsolute c > b . Step 2 : We denote by A n := { there is at least one jump with modulus > b ( n − n β ) up to time ( n + 1) − ( n +1) β } (4.6)and show that X n P ( A n ) < ∞ . (4.7)This comes from (4.5). Indeed, note that h inherits the monotonicity of b and φ , andhence (4.5) implies that X n (( n + 1) − ( n +1) β − ( n + 2) − ( n +2) β ) ¯Π( h (( n + 1) − ( n +1) β )) (4.8) ≤ X n Z ( n +1) − ( n +1) β ( n +2) − ( n +2) β ¯Π( h ( t )) d t < ∞ . hung-type LIL for L´evy processes at zero n + 1) − ( n +1) β − ( n + 2) − ( n +2) β ∼ ( n + 1) − ( n +1) β ,b ( n − n β ) = h (( n + 1) − ( n +1) β ) , and that the sequence ( n + 1) − ( n +1) β ¯Π( h (( n + 1) − ( n +1) β )) tends to zero by (4.8), weobtain that P ( A n ) = 1 − e − ( n +1) − ( n +1) β ¯Π( b ( n − nβ )) ∼ ( n + 1) − ( n +1) β ¯Π( h (( n + 1) − ( n +1) β ))is summable. Therefore (4.7) is proved. Step 3 : Let us now show how to use (4.7) to deduce (4.2). Obviously, it suffices to showthat lim sup n →∞ k X k ( n +1) − ( n +1) β b ( n − n β ) < ε a.s.for any ε > X n P ( k X k ( n +1) − ( n +1) β > εb ( n − n β )) < ∞ . Separating jumps of absolute value larger or smaller than b ( n − n β ), and, using the defi-nition of A n in (4.6), we obtain that X n P ( k X k ( n +1) − ( n +1) β > εb ( n − n β ))= X n P ( k X k ( n +1) − ( n +1) β > εb ( n − n β ); A cn ) + X n P ( k X k ( n +1) − ( n +1) β > εb ( n − n β ); A n ) , which is bounded from above by X n P ( k X k ( n +1) − ( n +1) β > εb ( n − n β ) | A cn ) · P ( A cn ) + X n P ( A n ) . The second term is finite by (4.7); and the first term is bounded by X n P ( k X k ( n +1) − ( n +1) β > εb ( n − n β ) | A cn ) . (4.9)To estimate this sum note that conditionally on A cn , X t d = X t ( n ), where X ( n ) differs from X only by removing jumps of size larger than | b ( n − n β ) | . Clearly, by Wald’s identity,var( X t ( n )) = t (cid:18)Z b ( n − nβ ) − b ( n − nβ ) y Π(d y ) + σ (cid:19) ≤ tU ( b ( n − n β )) . (4.10)8 F. Aurzada, L. D¨oring and M. Savov Note that | E X ( n +1) − ( n +1) β ( n ) | = ( n + 1) − ( n +1) β (cid:12)(cid:12)(cid:12)(cid:12)Z | x | >b ( n − nβ ) x Π(d x ) − γ (cid:12)(cid:12)(cid:12)(cid:12) . Therefore, by assumption (2.2), taking also into account that | E X t ( n ) | = t | E X ( n ) | , weobtain sup t ≤ ( n +1) − ( n +1) β | E X t ( n ) | = | E X ( n +1) − ( n +1) β ( n ) | = o( b ( n − n β )) . Using the previous relation (first step), Doob’s martingale inequality (second step), (4.10)(third step), Lemma 4.2 (fourth step) and the definition of b (fifth step), we are led tothe upper bound of the term in (4.9), X n P ( k X ( n ) k ( n +1) − ( n +1) β > εb ( n − n β )) ≤ X n P (cid:18) k X ( n ) − E X ( n ) k ( n +1) − ( n +1) β > εb ( n − n β ) (cid:19) ≤ X n E | X ( n +1) − ( n +1) β ( n ) − E X ( n +1) − ( n +1) β ( n ) | ( ε/ b ( n − n β ) ≤ X n n + 1) − ( n +1) β U ( b ( n − n β ))( ε/ b ( n − n β ) ≤ X n n + 1) − ( n +1) β C · F ( b ( n − n β ))( ε/ = C ′ λε X n ( n + 1) − ( n +1) β log | log n − n β | n − n β < ∞ , where we used the definition of b in the last step. Thus, the term in (4.9) is finite, asrequired. (cid:3) Proof of Corollary 2.1. If F is regularly varying so is b λ ; see Bingham, Goldie andTeugels [6], Proposition 1.5.7. Now note that if F is regularly varying with exponent − α < 0, we have b λ ( t ) = F − (log | log t | /λt ) ∼ λ /α F − (log | log t | /t )= λ /α b ( t ) . hung-type LIL for L´evy processes at zero λ ′ ) /α ≤ lim inf t → k X k t b ( t ) ≤ ( λ ′ ) /α a.s.for all λ ′ < λ and λ ′ > λ . Taking the limits on both sides, we obtain( λ ) /α ≤ lim inf t → k X k t b ( t ) ≤ ( λ ) /α a.s.Applying the regular variation argument in the reverse direction yields the claim. (cid:3) Proof of Corollary 2.2. This follows directly from Theorem 2.1. The bounds on theconstants can be obtained from the absolute constants in Proposition 2.1. (cid:3) Proof of Theorem 2.2. Lemma 4.3 gives the lower LIL of the theorem. Unfortunately,the arguments for the proof of Theorem 2.1 do not apply here. Hence, for the reversedirection, we show more directly that the given norming function of the LIL implies therate function of the small deviations. The following arguments go back to Kesten. Theproof is via contradiction, assuming thatlim inf t → k X k t b λ ′ ( t ) > C + δ (4.11)for some δ > λ ′ > λ . We show that under this assumption we can derive, forsufficiently large l , the estimates1 ≥ X n ≥ l P (cid:18) k X k r j − r n b λ ′ ( r j − r n ) > C ; for all l ≤ j ≤ n − (cid:19) P ( k X k r n ≤ b λ ′ ( r n )) (4.12) ≥ X n ≥ l P ( k X k r n ≤ b λ ′ ( r n )) (4.13)which is a contradiction as, by the choice of b λ ′ and the small deviation rate (2.6), thesum in (4.13) is infinite. First, let us derive estimate (4.12) for which Assumption (4.11)is not needed. For any fixed integer l partitioning the probability space, we obtain1 ≥ X n ≥ l P ( k X k r j > b λ ′ ( r j ) for all l ≤ j ≤ n − k X k r n ≤ b λ ′ ( r n )) ≥ X n ≥ l P (cid:16) sup r n ≤ s Proof of Corollary 2.3. This is completely analogous to the proof of Corollary 2.1. (cid:3) Proof of Corollary 2.4. We use Proposition 2.1 and Theorem 2.1. In order to do so,we have to see that the term εu ε in (2.12) has no influence on the order. We applyLemma 4.3 and follow the proof of Theorem 2.2 with the scaling t = r n and ε = b ( r n )and with the sequence n − n β , respectively. Therefore, it is sufficient to show that εu ε = o( tF ( ε )) hung-type LIL for L´evy processes at zero t and ε . Since ε = b ( t ) and thus t ∼ F ( ε ) − log log F ( ε ), weneed to show that εu ε = o(log log F ( ε )) . As this is precisely what we stated in condition (2.11), the proof is complete. (cid:3) Proof of Proposition 2.2. As X is of bounded variation, the representation X t = A t − A t + ct holds with two independent pure jump subordinators A , A . Next, we use the simpleobservation | X t | t ≤ k X k t t ≤ k A k t + k A k t + | c | tt = A t t + A t t + | c | to conclude the proof. The left-hand side converges to | c | , as X has bounded variation(see Theorem 39 of Doney [10]). 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