On the Moments and Limit Distributions of Some First Passage Times

Abstract

Let $S_n, n = 1,2,3, \cdots$, denote the partial sums of i.i.d. random variables with positive, finite mean. The first passage times $\min \{n; S_n > c\}$ and $\min \{n; S_n > c \cdot a(n)\}$, where $c \geqq 0$ and $a(y)$ is a positive, continuous function on $\lbrack 0, \infty)$, such that $a(y) = o(y)$ as $y \uparrow \infty$, 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 $c \rightarrow \infty$. Convergence to the normal and stable distributions is established when $c \rightarrow \infty$. Finally, some of the results are generalized to a class of random processes.