Asymptotic Expansions for the Distributions of Stopped Random Walks and First Passage Times

Abstract

Let $S_n = X_1 + \cdots + X_n, n \geq 1$, be a $d$-dimensional random walk and let $T_a = \inf\{n \geq n_a: ng(S_n/n) \geq a\}$, where $n_a = o(a)$. Let $\theta = g(EX_1), \hat{\theta}_n = g(S_n/n)$ and $\Delta_a = T_a\hat{\theta}_{T_a} - a$. Edgeworth-type expansions are developed for $P\{T_a = n, y_1 \leq \Delta_a \leq y_2\}$ and for the distribution functions of $T_a$ and of $\sqrt T_a(h(\hat{\theta}_{T_a}) - h(\theta))$, where $h$ is a real-valued function such that $h'(\theta) \neq 0$.