Estimates of the Rates of Convergence in Limit Theorems for the First Passage Times of Random Walks

Abstract

Let $T_r$ be the time of first passage to the level $r > 0$ by a random walk with independent and identically distributed steps and mean $\nu \geqq 0$. Estimates are given for the rate at which the distribution of $T_r$, suitably scaled and normalized, converges to the stable distribution with index $\frac{1}{2}$ when $\nu = 0$ and to the normal distribution when $\nu > 0$ as $r \rightarrow \infty$.