Small ball probabilities for jump Lévy processes from the Wiener domain of attraction

Shmileva, Elena

doi:10.1016/j.spl.2006.04.037

Cited by 4 publications

(3 citation statements)

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“…A related result for scaled Poisson processes connected to the Strassen law is shown in [1]. Further, similar results are obtained in [22] for symmetric stable Lévy processes with 1 < α < 2. Moreover, Simon [20] investigated the small deviation problem for Lévy processes under p-variation norm in contrast to the present considerations.…”

Section: Introduction and Resultssupporting

confidence: 72%

Small deviations of general Lévy processes

Аурзада¹,

Dereich²

2009

Ann. Probab.

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We study the small deviation problem log P(sup t∈[0,1] |Xt| ≤ ε), as ε → 0, for general Lévy processes X. The techniques enable us to determine the asymptotic rate for general real-valued Lévy processes, which we demonstrate with many examples.As a particular consequence, we show that a Lévy process with nonvanishing Gaussian component has the same (strong) asymptotic small deviation rate as the corresponding Brownian motion.F. AURZADA AND S. DEREICH Further, f ∼ g means that lim f /g = 1. We also use f gThis paper is organized as follows. In Section 1.2, we review results of Simon, who studied the question when the problem (1) actually makes sense for Lévy processes. Section 1.3 contains our main results, which are illustrated by several examples in Section 2. The proofs are postponed to Sections 3 and 4 (proofs of the main results) and Section 5 (proof of the explicit rates in the examples).1.2. The small deviation property. In [19], the following question was studied: for which Lévy processes does the small deviation problem make sense? Namely, one says that a stochastic process X = (X t ) t∈[0,1] possesses the small deviation property if P supSimon investigated this property for R d -valued Lévy processes. For realvalued Lévy processes, it reduces to the following, easily verifiable, equivalent characterization [19].Proposition 1.1. A (ν, σ 2 , b)-Lévy process X possesses the small deviation property if and only if it is not of type (I) or if it is of type (I) and, for c := b − |x|≤1 xν(dx), we have:• c = 0, or • c > 0 and ν{−ε ≤ x < 0} = 0 for all ε > 0, or • c < 0 and ν{0 < x ≤ ε} = 0 for all ε > 0.Let us visualize this fact with a simple example.Example 1.2. Let us consider an α-stable subordinator with drift: X t + µt, where X has the characteristic function:

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Section: Introduction and Resultssupporting

confidence: 72%

Small deviations of general Lévy processes

Аурзада¹,

Dereich²

2009

Ann. Probab.

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“…We refer to the surveys Li and Shao [12], Lifshits [15], for an overview of the field, and to Lifshits [14], for a regularly updated list of references, which also includes references to laws of the iterated logarithm of Chung type. The papers of Taylor [25], Mogul'skiȋ [18], Borovkov and Mogul'skiȋ [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 earlier results on small deviations of Lévy processes.…”

Section: Introductionmentioning

confidence: 99%

Small time Chung-type LIL for Lévy processes

Аурзада¹,

Doering²,

Savov³

2013

Bernoulli

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“…The latter has a good overview of small ball research up to 2001, including applications to Chung's other law. See also [2,23,[33][34][35].) At the end of this section we supply some further technical comments relevant to Theorem 2.1 in Remark 4.1.…”

Section: Preliminaries To the Proofsmentioning

confidence: 99%

The small-time Chung-Wichura law for Lévy processes with non-vanishing Brownian component

Buchmann

Maller

2009

Probab. Theory Relat. Fields

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We give a "small time" functional version of Chung's "other" law of the iterated logarithm for Lévy processes with non-vanishing Brownian component. This is an analogue of the "other" law of the iterated logarithm at "large times" for Lévy processes and random walks with finite variance, as extended to a functional version by Wichura. As one of many possible applications, we mention a functional law for a two-sided passage time process.Keywords Lévy process · Local behaviour · Almost sure convergence · Iterated logarithm laws · Other law of the iterated logarithm · Cluster sets · Functional limit theorem Mathematics Subject Classification (2000) 60G51 · 60F15 · 60F17 · 60F05 · 60J65 · 60J75

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Small ball probabilities for jump Lévy processes from the Wiener domain of attraction

Cited by 4 publications

References 4 publications

Small deviations of general Lévy processes

Small deviations of general Lévy processes

Small time Chung-type LIL for Lévy processes

The small-time Chung-Wichura law for Lévy processes with non-vanishing Brownian component

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