Asymptotic Expansions in Boundary Crossing Problems

Abstract

Let $S_n, n \geq 1$, be a random walk and $t = t_a = \inf\{n \geq 1: ng(S_n/n) > a\}$. The main results of this paper are two-term asymptotic expansions as $a \rightarrow \infty$ for the marginal distributions of $t_a$ and the normalized partial sum $S^\ast_t = (S_t - t\mu)/\sigma\sqrt t$. To leading order, $S^\ast_t$ has a standard normal distribution. The effect of the randomness in the sample size $t$ on the distribution of $S^\ast_t$ appears in the correction term of the expansion.