## The Annals of Probability

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

### 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.

#### 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