Asymptotic expansions for first passage times

Abstract

Let F be a strongly non-lattice distribution function with a positive mean, a positive variance, and a finite third moment. Let X1, X2,... be i.i.d. with common distribution function F; and let Sn=X1+...+Xn, and ta = inf{n[ges]1:Sn > a} for n [ges] 1 and a >0. The main result reported here is a two term asymptotic expansion for Ha(n, z) = P{ta n, Sn - a [les] z } as a --> [infinity]. Assuming higher moments, a three term expansion for P{ta [les] n} and refined estimates for the probability of ruin in finite time are obtained as simple corollaries. A key tool is an asymptotic expansion in Stone's formulation of the local limit theorem.