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A remark on the strong law of large numbers

Authors:	Dr J Hüsler

Institution:	(1) Institut für Mathematische Statistik und Versicherungslehre der Universität Bern, Sidlerstraße 5, CH-3012 Bern, Switzerland

Abstract:	The strong law of large numbers for independent and identically distributed random variablesX _i,i=1, 2, 3,... with finite expectationE\|X ₁\| can be stated as, for any >0, the number of integersn such that $\|n^{ - 1} \mathop {\mathop \sum \limits_{i \le n} }\limits^n X_i ---EX_1 \| > \varepsilon$ ,N is finite a. s. It is known thatEN < iffEX ₁ ² < and that ² EN var X₁ as 0, ifE X ₁ ² <. Here we consider the asymptotic behaviour ofEN (n) asn, whereN (n) is the number of integerskn such that $\|k^{ - 1} \mathop {\mathop \sum \limits_{i \le 1} }\limits^k X_i ---EX_1 \| > \varepsilon$ andE N ₁ ² =.

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