An extension of the law of the iterated logarithm

Abstract

1. Let x1, x2, …, xn, … be a set of independent variables each with a uniform probability distribution in 0 ≤ x ≤ 1. If 0 ≤ α < β ≤ 1 we denote by FN (α, β) the number of x1, …, xN which satisfy α < x ≤ β,and put RN(α, β) = FN(α, β) − N(β − α).