## The Annals of Probability

### Boundary Crossing Probabilities for Sample Sums and Confidence Sequences

Tze Leung Lai

#### Abstract

By making use of the martingale $\int^\infty_0 \exp (yW(t) - (t/2)y^2) dF(y)$, Robbins and Siegmund have evaluated the probability that the Wiener process $W(t)$ would ever cross certain moving boundaries. In this paper, we study this class of boundaries and make use of certain moment generating function martingales to obtain boundary crossing probabilities for sums of i.i.d. random variables. Invariance theorems for these boundary crossing probabilities are proved, and some applications to confidence sequences and power-one tests are also given.

#### Article information

Source
Ann. Probab., Volume 4, Number 2 (1976), 299-312.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176996135

Digital Object Identifier
doi:10.1214/aop/1176996135

Mathematical Reviews number (MathSciNet)
MR405578

Zentralblatt MATH identifier
0344.60024

JSTOR