## The Annals of Mathematical Statistics

### On Excess Over the Boundary

Gary Lorden

#### Abstract

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

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

Dates
First available in Project Euclid: 27 April 2007

http://projecteuclid.org/euclid.aoms/1177697092

Digital Object Identifier
doi:10.1214/aoms/1177697092

Mathematical Reviews number (MathSciNet)
MR254981

Zentralblatt MATH identifier
0212.49703

JSTOR