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

Abstract

For a family of random walks {S (a)} satisfying E S 1 (a)=-a<0, we consider ladder epochs τ(a)=min {k≥1: S k (a)<0}. We study the asymptotic behaviour, as a⇒0, of P (τ(a)>n) in the case when n=n(a)→∞. As a consequence, we also obtain the growth rates of the moments of τ(a).