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On excess over the boundary

Lorden, Gary (1970) On excess over the boundary. Annals of Mathematical Statistics, 41 (2). pp. 520-527. ISSN 0003-4851. http://resolver.caltech.edu/CaltechAUTHORS:LORams70

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Abstract

A random walk, {Sn}∞n = 0 , having positive drift and starting at the origin, is stopped the first time Sn > t ≧ 0. The present paper studies the "excess," Sn - 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.


Item Type:Article
Additional Information:The author is grateful to Professor J. Sacks for helpful discussions in the course of this research. Research sponsored in part by the National Foundation under grant NSF GP-4972.
Record Number:CaltechAUTHORS:LORams70
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:LORams70
Alternative URL:http://links.jstor.org/sici?sici=0003-4851%28197004%2941%3A2%3C520%3AOEOTB%3E2.0.CO%3B2-W
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1647
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:03 May 2006
Last Modified:26 Dec 2012 08:45

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