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 ${\tau }^{(a)}=\min \{k\ge 1:S_k^{(a)}<0\}$. We study the asymptotic, as $a\to 0$, 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 ${\tau }^{(a)}$.

27 pages