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

Buchmann, Boris; Maller, Ross

doi:10.1007/s00440-009-0255-1

Cited by 8 publications

(7 citation statements)

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“…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 jumps of a Lévy process. Proof.…”

Section: Explicit Lil For Lévy Processesmentioning

confidence: 81%

“…The norming function b(t) = π 2 t/(8 log | log t|) for a standard Brownian motion can be derived from the large time LIL, proved by Chung [9], via time inversion. For any Lévy process with non-trivial Brownian component, the recent result of Buchmann and Maller [8] shows that (1.1) holds with the same norming function as for a standard Brownian motion. If X is an α-stable Lévy process, (1.1) holds with norming function b(t) = (c α t/ log | log t|) 1/α , which goes back to Taylor [25].…”

Section: Introductionmentioning

confidence: 94%

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Small time Chung-type LIL for Lévy processes

Аурзада¹,

Doering²,

Savov³

2013

Bernoulli

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Section: Explicit Lil For Lévy Processesmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 94%

Small time Chung-type LIL for Lévy processes

Аурзада¹,

Doering²,

Savov³

2013

Bernoulli

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“…Then Wee [87] succeeded in obtain liminf LILs for numerous non-symmetric Lévy processes in R 1 . See also [2,17] and the references therein. Recently, Knopova and Schilling [68] extended liminf LIL at zero to non-symmetric Lévy-type processes in R 1 .…”

Section: Introduction and General Resultsmentioning

confidence: 99%

General Law of iterated logarithm for Markov processes: Limsup law

Cho¹,

Kim²,

Lee³

2021

Preprint

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In this paper, we discuss general criteria and forms of both liminf and limsup laws of iterated logarithm (LIL) for continuous-time Markov processes. We consider minimal assumptions for both LILs to hold at zero (at infinity, respectively) in general metric measure spaces. We establish LILs under local assumptions near zero (near infinity, respectively) on uniform bounds of the first exit time from balls in terms of a function φ and uniform bounds on the tails of the jumping measure in terms of a function ψ. One of the main results is that a simple ratio test in terms of the functions φ and ψ completely determines whether there exists a positive nondecreasing function Ψ such that lim sup |Xt|/Ψ(t) is positive and finite a.s., or not. We also provide a general formulation of liminf LIL, which covers jump processes whose jumping measures have logarithmic tails. Our results cover a large class of subordinated diffusions, jump processes with mixed polynomial local growths, jump processes with singular jumping kernels and random conductance models with long range jumps.

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“…A multidimensional version of Proposition 1.1 (2) was first proved by Taylor in [39], and then a refined version of Proposition 1.1 (2) for (non-symmetric) Lévy processes was established by Wee in [40]. We refer the reader to [1,10,11,37] and the references therein. Recently the results in Proposition 1.1 have been extended to some class of Feller processes (see [29] and the references therein).…”

Section: Introduction and Settingmentioning

confidence: 99%