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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 16, Fasc. 1,
pages 113 - 126
 

ON THE KOLMOGOROV QUASIMARTINGALE PROPERTY

Bernard Heinkel

Abstract: Let (X ) k be a sequence of real-valued random variables (r.v.), which are centered, square integrable and independent. A well-known result, due to Kolmogorov, states that if

sum E(X2k) --k2--< +o o , k>1
(i)

then (Sn/n) converges almost surely (a.s.) to 0, where Sn = X1 + ...+ Xn.

This paper is devoted to the interpretation of condition (i). For instance, it is shown that if the r.v. Xk are weighted Rademacher r.v., then (i) is equivalent to the fact that 2 ((Sn/n) ,Gn) is a quasimartingale (Gn being the natural filtration associated with the sequence (Xn) ).

The problem of the interpretation of (i) for Banach space valued r.v. Xk is also studied.

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

Key words and phrases: -

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