## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F99: None of the above, but in this section

Secondary: 62L10: Sequential analysis

**Keywords**

Confidence sequences boundary crossing probabilities Robbins-Siegmund boundaries sample sums Wiener process moment generating function martingales

#### Citation

Lai, Tze Leung. Boundary Crossing Probabilities for Sample Sums and Confidence Sequences. Ann. Probab. 4 (1976), no. 2, 299--312. doi:10.1214/aop/1176996135. https://projecteuclid.org/euclid.aop/1176996135