On the Moments and Limit Distributions of Some First Passage Times

Abstract

Let S_n, n = 1, 2, 3, ... , denote the partial sums of i.i.d. random variables with positive, finite mean . The first passage times min {n; S_n > c} and min {n; S_n > c * a(n)}, where c ≥ 0 and a(y) is a positive, continuous function on [0, ∞), such that a(y) = o(y) as y ↑ ∞, are investigated. Necessary and sufficient conditions for finiteness of their moments and moment generating functions are given. Under some further assumptions on a(y), asymptotic expressions for the moments and the excess over the boundary are obtained when c → ∞. Convergence to the normal and stable distributions is established when c → ∞. Finally, some of the results are generalized to a class of random processes.