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February, 1976 Asymptotic Moments of Random Walks with Applications to Ladder Variables and Renewal Theory
Tze Leung Lai
Ann. Probab. 4(1): 51-66 (February, 1976). DOI: 10.1214/aop/1176996180

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.

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Tze Leung Lai. "Asymptotic Moments of Random Walks with Applications to Ladder Variables and Renewal Theory." Ann. Probab. 4 (1) 51 - 66, February, 1976. https://doi.org/10.1214/aop/1176996180

Information

Published: February, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0351.60062
MathSciNet: MR391265
Digital Object Identifier: 10.1214/aop/1176996180

Subjects:
Primary: 60J15
Secondary: 60K05

Keywords: asymptotic moments , expected overshoot , Ladder epoch , ladder variable , renewal theory , Spitzer's series , Tauberian theorem

Rights: Copyright © 1976 Institute of Mathematical Statistics

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Vol.4 • No. 1 • February, 1976
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