Asymptotic Moments of Random Walks with Applications to Ladder Variables and Renewal Theory

Abstract

Let $X_1, X_2, \cdots$ be i.i.d. random variables with $EX_1 = 0, EX_1^2 = 1$ and let $S_n = X_1 + \cdots + X_n$. In this paper, we study the ladder variable $S_N$ where $N = \inf \{n \geqq 1: S_n > 0\}$. The well-known result of Spitzer concerning $ES_N$ is extended to the higher moments $ES_N^k$. In this connection, we develop an asymptotic expansion of the one-sided moments $E\lbrack(n^{-\frac{1}{2}}S_n)^-\rbrack^\nu$ related to the central limit theorem. Using a truncation argument involving this asymptotic expansion, we obtain the absolute convergence of Spitzer's series of order $k - 2$ under the condition $E|X_1|^k < \infty$, extending earlier results of Rosen, Baum and Katz in connection with $ES_N$. Some applications of these results to renewal theory are also given.