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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 5, Fasc. 1,
pages 91 - 97
 

ON THE RATE OF CONVERGENCE FOR THE WEAK LAW OF LARGE NUMBERS

Robert Bartoszyński
Prem S. Puri

Abstract: Let X,X ,X ,... 1 2 be i.i.d. random variables with the common distribution F. Further, let (c ) n be a sequence of positive numbers, and (b ) n be a strictly increasing sequence of positive integers. The paper considers the convergence of the series

sum oo cnP (| X1 + ...+ Xbn|> ebn) n=1
under the interplay of three types of conditions:

(i) convergence of this series,

(ii) an appropriate moment condition on X ,

(iii) a condition imposing constraints on the behavior of the sequences (cn) and (bn).

Three theorems have been proven; in each of these two among (i)-(iii) implying the third, with one of the theorems being valid for the general case, where the random variables involved are not necessarily i.i.d.

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -

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