A supplement to the strong law of large numbers

Abstract

The strong law of large numbers for independent and identically distributed random variables X i , i = 1,2,3, …, with finite mean µ can be stated as, for any ∊ > 0, the number of integers n such that | n −1 Σ i =1 n X i − μ| > ∊ , N ( ∊ ) , is finite a.s. It is known, furthermore, that EN ( ∊ ) < ∞ if and only if EX 1 2 < ∞. Here it is shown that if EX 1 2 < ∞ then ∊ 2 EN ( ∊ ) → var X 1 as ∊ → 0.