Let $(\xi_1,\eta_1)$, $(\xi_2, \eta_2),\ldots$ be a sequence of i.i.d. random vectors taking values in $\mathbb{R}^2$, and let $S_0:=0$ and $S_n:=\xi_1+\ldots+\ldots\xi_n$ for $n\in\mathbb{N}$. The sequence $(S_{n-1}+\eta_n)_{n\in\mathbb{N}}$ is then called perturbed random walk. For real $x$, denote by $\tau(x)$ the first time the perturbed random walk exits the interval $(-\infty, x]$. We consider a rather intricate case in which $S_n$ drifts to the left, yet the perturbed random walk oscillates because of occasional big jumps to the right of the perturbating sequence $(\eta_n)_{n\in{\mathbb N}}$. Under these assumptions we provide necessary and sufficient conditions for the finiteness of power moments of $\tau(x)$, there by solving an open problem posed by Alsmeyer, Iksanov and Meiners in [2].
Keywords:
First passage time, perturbed random walk, power moment.
B. Rashytov. Power moments of first passage times for some oscillating perturbed random walks. Teoriâ slučajnyh processov, Tome 23 (2018) no. 1, pp. 93-97. http://geodesic.mathdoc.fr/item/THSP_2018_23_1_a7/
@article{THSP_2018_23_1_a7,
author = {B. Rashytov},
title = {Power moments of first passage times for some oscillating perturbed random walks},
journal = {Teori\^a slu\v{c}ajnyh processov},
pages = {93--97},
year = {2018},
volume = {23},
number = {1},
language = {en},
url = {http://geodesic.mathdoc.fr/item/THSP_2018_23_1_a7/}
}
TY - JOUR
AU - B. Rashytov
TI - Power moments of first passage times for some oscillating perturbed random walks
JO - Teoriâ slučajnyh processov
PY - 2018
SP - 93
EP - 97
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IS - 1
UR - http://geodesic.mathdoc.fr/item/THSP_2018_23_1_a7/
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%J Teoriâ slučajnyh processov
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[1] G. Alsmeyer, D. Buraczewski, A. Iksanov, “Null recurrence and transience of random difference equations in the contractive case”, J. Appl. Probab., 54 (2017), 1089–1110 | DOI | MR | Zbl
[2] G. Alsmeyer, A. Iksanov, M. Meiners, “Power and exponential moments of the number of visits and related quantities for perturbed random walks”, J. Theoret. Probab., 28 (2015), 1–40 | DOI | MR | Zbl
[3] S. Janson, “Moments for first-passage and last-exit times, the minimum, and related quantities for random walks with positive drift”, Adv. Appl. Probab., 18 (1986), 865–879 | DOI | MR | Zbl
[4] A. Iksanov, Renewal theory for perturbed random walks and similar processes, Birkhäuser, 2016 | MR | Zbl
