The Annals of Probability

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

Tze Leung Lai

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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.

Article information

Source
Ann. Probab., Volume 4, Number 1 (1976), 51-66.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996180

Digital Object Identifier
doi:10.1214/aop/1176996180

Mathematical Reviews number (MathSciNet)
MR391265

Zentralblatt MATH identifier
0351.60062

JSTOR
links.jstor.org

Subjects
Primary: 60J15
Secondary: 60K05: Renewal theory

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

Citation

Lai, Tze Leung. Asymptotic Moments of Random Walks with Applications to Ladder Variables and Renewal Theory. Ann. Probab. 4 (1976), no. 1, 51--66. doi:10.1214/aop/1176996180. https://projecteuclid.org/euclid.aop/1176996180


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