A Conditional Local Limit Theorem for Recurrent Random Walk

Abstract

Let $S_n, n = 1, 2, 3, \cdots$ denote the recurrent random walk formed by the partial sums of i.i.d. lattice random variables with mean zero and finite variance. Let $T_{\{x\}} = \min \lbrack n \geqq 1 \mid S_n = x \rbrack$ with $T \equiv T_{\{0\}}$. We obtain a local limit theorem for the random walk conditioned by the event $\lbrack T > n \rbrack$. This result is applied then to obtain an approximation for $P\lbrack T_{\{x\}} = n \rbrack$ and the asymptotic distribution of $T_{\{x\}}$ as $x$ approaches infinity.