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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=1nXi − μ| > ∊, N(∊), is finite a.s. It is known, furthermore, that EN(∊) < ∞ if and only if EX12 < ∞. Here it is shown that if EX12 < ∞ then ∊2EN(∊)→ var X1 as ∊ → 0.

Keywords

SUMS OF INDEPENDENT RANDOM VARIABLESSTRONG LAWEXCEPTIONAL EVENTS
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

[1] Erdös, P. (1949) On a theorem of Hsu and Robbins. Ann. Math. Statist. 20, 286–291.CrossRefGoogle Scholar
[2] Fuk, D. Kh. and Nagaev, S. V. (1971) Probability inequalities for sums of independent random variables. Theor. Probability Appl. 16, 643–660.CrossRefGoogle Scholar
[3] Slivka, J. and Severo, N. C. (1970) On the strong law of large numbers. Proc. Amer. Math . Soc. 24, 729–734.Google Scholar
[4] Wu, C. F. (1973) A note on the convergence rate of the strong law of large numbers. Bull. Inst. Math. Acad. Sinica 1, 121–124.Google Scholar