Open Access
April, 1970 On Excess Over the Boundary
Gary Lorden
Ann. Math. Statist. 41(2): 520-527 (April, 1970). DOI: 10.1214/aoms/1177697092


A random walk, $\{S_n\}^\infty_{n=0}$, having positive drift and starting at the origin, is stopped the first time $S_n > t \geqq 0$. The present paper studies the "excess," $S_n - t$, when the walk is stopped. The main result is an upper bound on the mean of the excess, uniform in $t$. Through Wald's equation, this gives an upper bound on the mean stopping time, as well as upper bounds on the average sample numbers of sequential probability ratio tests. The same elementary approach yields simple upper bounds on the moments and tail probabilities of residual and spent waiting times of renewal processes.


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Gary Lorden. "On Excess Over the Boundary." Ann. Math. Statist. 41 (2) 520 - 527, April, 1970.


Published: April, 1970
First available in Project Euclid: 27 April 2007

zbMATH: 0212.49703
MathSciNet: MR254981
Digital Object Identifier: 10.1214/aoms/1177697092

Rights: Copyright © 1970 Institute of Mathematical Statistics

Vol.41 • No. 2 • April, 1970
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