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 Teor. Veroyatnost. i Primenen., 2018, Volume 63, Issue 4, Pages 755–778 (Mi tvp5181)

First-passage times over moving boundaries for asymptotically stable walks

D. Denisova, A. Sakhanenkob, V. Wachtelc

a School of Mathematics, University of Manchester, Oxford Road, UK
b Novosibirsk State University
c Institut für Mathematik, Universität Augsburg, Augsburg, Germany

Abstract: Let $\{S_n, n\geq1\}$ be a random walk with independent and identically distributed increments, and let $\{g_n, n\geq1\}$ be a sequence of real numbers. Let $T_g$ denote the first time when $S_n$ leaves $(g_n,\infty)$. Assume that the random walk is oscillating and asymptotically stable, that is, there exists a sequence $\{c_n, n\geq1\}$ such that $S_n/c_n$ converges to a stable law. In this paper we determine the tail behavior of $T_g$ for all oscillating asymptotically stable walks and all boundary sequences satisfying $g_n=o(c_n)$. Furthermore, we prove that the rescaled random walk conditioned to stay above the boundary up to time $n$ converges, as $n\to\infty$, towards the stable meander.

Keywords: random walk, stable distribution, first-passage time, overshoot, moving boundary.

 Funding Agency Grant Number Russian Science Foundation 17-11-01173 The research of A. Sakhanenko and V. Wachtel has been supported by RSF research grant № 17-11-01173.

DOI: https://doi.org/10.4213/tvp5181

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English version:
Theory of Probability and its Applications, 2019, 63:4, 613–633

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Accepted:21.06.2018

Citation: D. Denisov, A. Sakhanenko, V. Wachtel, “First-passage times over moving boundaries for asymptotically stable walks”, Teor. Veroyatnost. i Primenen., 63:4 (2018), 755–778; Theory Probab. Appl., 63:4 (2019), 613–633

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp5181
• https://doi.org/10.4213/tvp5181
• http://mi.mathnet.ru/eng/tvp/v63/i4/p755

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This publication is cited in the following articles:
1. Caravenna F., Doney R., “Local Large Deviations and the Strong Renewal Theorem”, Electron. J. Probab., 24 (2019), 72, 1–48
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