Transition phenomena for ladder epochs of random walks with small negative drift

Abstract

For a family of random walks $\{S^{(a)}\}$ satisfying $\mathbf{E}S_1^{(a)}=-a<0$ we consider ladder epochs $\tau^{(a)}=\min\{k\geq1: S_k^{(a)}<0\}$. We study the asymptotic, as $a\to0$, behaviour of $\mathbf{P}(\tau^{(a)}>n)$ in the case when $n=n(a)\to\infty$. As a consequence we obtain also the growth rates of the moments of $\tau^{(a)}$.