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.