The Annals of Applied Probability

Inequalities for the Overshoot

Joseph T. Chang

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Let $X_1, X_2, \ldots$ be independent and identically distributed positive random variables with $S_n = X_1 + \cdots + X_n$, and for nonnegative $b$ define $R_b = \inf\{S_n - b: S_n > b\}$. Then $R_b$ is called the overshoot at $b$. In terms of the moments of $X_1$, Lorden gave bounds for the moments of $R_b$ that hold uniformly over all $b$. Using a coupling argument, we establish stochastic ordering inequalities that imply the moment inequalities of Lorden. In addition to simple new proofs of Lorden's inequalities, we provide new inequalities for the tail probabilities $P\{R_b > x\}$ and moments of $R_b$ that improve upon those of Lorden. We also present conjectures for sharp moment inequalities and describe an application to the first ladder height of random walks.

Article information

Ann. Appl. Probab., Volume 4, Number 4 (1994), 1223-1233.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60K05: Renewal theory
Secondary: 60E15: Inequalities; stochastic orderings 60J15

Renewal theory residual lifetime excess over the boundary total lifetime length-biased sampling inspection paradox ladder height coupling stochastic ordering


Chang, Joseph T. Inequalities for the Overshoot. Ann. Appl. Probab. 4 (1994), no. 4, 1223--1233. doi:10.1214/aoap/1177004913.

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