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November, 1994 Inequalities for the Overshoot
Joseph T. Chang
Ann. Appl. Probab. 4(4): 1223-1233 (November, 1994). DOI: 10.1214/aoap/1177004913


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.


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Joseph T. Chang. "Inequalities for the Overshoot." Ann. Appl. Probab. 4 (4) 1223 - 1233, November, 1994.


Published: November, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0812.60021
MathSciNet: MR1304783
Digital Object Identifier: 10.1214/aoap/1177004913

Primary: 60K05
Secondary: 60E15 , 60J15

Keywords: coupling , excess over the boundary , inspection paradox , ladder height , length-biased sampling , renewal theory , Residual lifetime , stochastic ordering , total lifetime

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.4 • No. 4 • November, 1994
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