Power moments of first passage times for some oscillating perturbed random walks

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].