The Annals of Mathematical Statistics

On Excess Over the Boundary

Gary Lorden

Full-text: Open access


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.

Article information

Ann. Math. Statist., Volume 41, Number 2 (1970), 520-527.

First available in Project Euclid: 27 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier



Lorden, Gary. On Excess Over the Boundary. Ann. Math. Statist. 41 (1970), no. 2, 520--527. doi:10.1214/aoms/1177697092.

Export citation