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A supplement to the strong law of large numbers

Published online by Cambridge University Press:  14 July 2016

C. C. Heyde
C. C. Heyde*
Affiliation:
The Australian National University
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Abstract

The strong law of large numbers for independent and identically distributed random variables Xi, i = 1,2,3, …, with finite mean µ can be stated as, for any ∊ > 0, the number of integers n such that |n −1 Σi=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.

Keywords

SUMS OF INDEPENDENT RANDOM VARIABLESSTRONG LAWEXCEPTIONAL EVENTS

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 12 , Issue 1 , March 1975 , pp. 173 - 175
Copyright
Copyright © Applied Probability Trust 1975 

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References

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