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This article is cited in 3 scientific papers (total in 3 papers)
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.
Received: 12.03.2018 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
Linking options:
https://www.mathnet.ru/eng/tvp5181https://doi.org/10.4213/tvp5181 https://www.mathnet.ru/eng/tvp/v63/i4/p755
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