An exact asymptotics for the moment of crossing a curved boundary by an asymptotically stable random walk

V. I. Wachtel; Denis Denisov

doi:10.1137/S0040585X97T987740

An exact asymptotics for the moment of crossing a curved boundary by an asymptotically stable random walk

V. I. Wachtel, Denis Denisov

Universität Augsburg

Research output: Contribution to journal › Article › peer-review

Abstract

Suppose that {Sn, n ≧ 0} is an asymptotically stable random walk. Let g be a positive function and T_g be the first time when Sn leaves [-g(n), ∞). In this paper we study asymptotic behavior of T_g. We provide integral tests for function g that guarantee P(T_g > n) ∼ V (g)P(T0 > n), where T0 is the first strict descending ladder epoch of {S_n}.

Original language	English
Pages (from-to)	481-500
Number of pages	20
Journal	Theory of Probability and Its Applications
Volume	60
Issue number	3
DOIs	https://doi.org/10.1137/S0040585X97T987740
Publication status	Published - 2015

Keywords

Exit time
First passage time
Moving boundary
Nonlinear boundary

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abstract = "Suppose that {Sn, n ≧ 0} is an asymptotically stable random walk. Let g be a positive function and Tg be the first time when Sn leaves [-g(n), ∞). In this paper we study asymptotic behavior of Tg. We provide integral tests for function g that guarantee P(Tg > n) ∼ V (g)P(T0 > n), where T0 is the first strict descending ladder epoch of {Sn}.",

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N2 - Suppose that {Sn, n ≧ 0} is an asymptotically stable random walk. Let g be a positive function and Tg be the first time when Sn leaves [-g(n), ∞). In this paper we study asymptotic behavior of Tg. We provide integral tests for function g that guarantee P(Tg > n) ∼ V (g)P(T0 > n), where T0 is the first strict descending ladder epoch of {Sn}.

AB - Suppose that {Sn, n ≧ 0} is an asymptotically stable random walk. Let g be a positive function and Tg be the first time when Sn leaves [-g(n), ∞). In this paper we study asymptotic behavior of Tg. We provide integral tests for function g that guarantee P(Tg > n) ∼ V (g)P(T0 > n), where T0 is the first strict descending ladder epoch of {Sn}.

KW - Exit time

KW - First passage time

KW - Moving boundary

KW - Nonlinear boundary

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JO - Theory of Probability and Its Applications

JF - Theory of Probability and Its Applications

SN - 0040-585X

IS - 3

ER -

An exact asymptotics for the moment of crossing a curved boundary by an asymptotically stable random walk

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Keywords

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