PROBABILITY DISTRIBUTIONS RELATED TO THE LAW OF THE ITERATED LOGARITHM

Abstract

Let W(t) denote a standard Wiener process for 0 ≤ t < ∞. We compute the probability that W(t) ≥ t ^½ A(t) for some t ≥ 1 (or for some t ≥ 0) for a certain class of functions A(t), including functions which are ∼ (2 log log t)^½ as t → ∞. We also give an invariance principle which states that this probability is the limit as m → ∞ of the probability that s_n ≥ n ^½ A(n/m) for some n ≥ m (or for some n ≥ 1), where s_n is the sum of n independent and identically distributed random variables with mean 0 and variance 1.