Geodesic
Power moments of first passage times for some oscillating perturbed random walks
Teoriâ slučajnyh processov, Tome 23 (2018) no. 1, pp. 93-97.

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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. 
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     author = {B. Rashytov},
     title = {Power moments of first passage times for some oscillating perturbed random walks},
     journal = {Teori\^a slu\v{c}ajnyh processov},
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     publisher = {mathdoc},
     volume = {23},
     number = {1},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2018_23_1_a7/}
}
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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/

[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