Geodesic
First-passage times over moving boundaries for asymptotically stable walks
Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 4, pp. 755-778

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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, first-passage time, overshoot, moving boundary.
Mots-clés : stable distribution 
@article{TVP_2018_63_4_a6,
     author = {D. Denisov and A. Sakhanenko and V. Wachtel},
     title = {First-passage times over moving boundaries for asymptotically stable walks},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {755--778},
     publisher = {mathdoc},
     volume = {63},
     number = {4},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2018_63_4_a6/}
}
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D. Denisov; A. Sakhanenko; V. Wachtel. First-passage times over moving boundaries for asymptotically stable walks. Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 4, pp. 755-778. http://geodesic.mathdoc.fr/item/TVP_2018_63_4_a6/