The Annals of Probability

A Simple proof of a Known Result in Random Walk Theory

Austin J. Lemoine

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Abstract

Let $\{X_n, n \geqq 1\}$ be a stationary independent sequence of real random variables, $S_n = X_1 + \cdots + X_n$, and $\alpha_A$ the hitting time of the set $A$ by the process $\{S_n, n \geqq 1\}$, where $A$ is one of the half-lines $(0, \infty), \lbrack 0, \infty), (-\infty, 0 \rbrack$ or $(-\infty, 0)$. This note provides a simple proof of a known result in random walk theory on necessary and sufficient conditions for $E\{\alpha_A\}$ to be finite. The method requires neither generating functions nor moment conditions on $X_1$.

Article information

Source
Ann. Probab., Volume 2, Number 2 (1974), 347-348.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996718

Digital Object Identifier
doi:10.1214/aop/1176996718

Mathematical Reviews number (MathSciNet)
MR356244

Zentralblatt MATH identifier
0278.60043

JSTOR
links.jstor.org

Subjects
Primary: 60J15
Secondary: 60K25: Queueing theory [See also 68M20, 90B22]

Keywords
Random walks hitting times

Citation

Lemoine, Austin J. A Simple proof of a Known Result in Random Walk Theory. Ann. Probab. 2 (1974), no. 2, 347--348. doi:10.1214/aop/1176996718. https://projecteuclid.org/euclid.aop/1176996718


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