Boundary Crossing Probabilities for the Wiener Process and Sample Sums

Abstract

1. Introduction and summary. Let W(t) denote a standard Wiener process for 0 ≦ t 0 (or for some t > 0) for a certain class of functions g(t), including functions which are ~ (2t log log t)½ as y → ∞. We also prove an invariance theorem which states that this probability is the limit as m → ∞ of the probability that S n ≦m ½ g(n/m) for some n ≦ τm (or for some n ≦ 1), where S n is the nth partial sum of any sequence x 1, x 2, … of independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1.