ISSN:
1436-5081
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 ɛ〉0, the number of integersn such that $$|n^{ - 1} \mathop {\mathop \sum \limits_{i \le n} }\limits^n X_i ---EX_1 | 〉 \varepsilon $$ ,N ɛ is finite a. s. It is known thatEN ɛ〈∞ iffEX 1 2 〈∞ and that ε2 ENε → var X1 as ɛ→0, ifE X 1 2 〈∞. Here we consider the asymptotic behaviour ofEN ɛ(n) asn→∞, whereN ɛ(n) is the number of integersk≤n such that $$|k^{ - 1} \mathop {\mathop \sum \limits_{i \le 1} }\limits^k X_i ---EX_1 | 〉 \varepsilon $$ andE N 1 2 =∞.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01538031
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