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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 21, Fasc. 2,
pages 321 - 328
 

ON THE APPROXIMATION OF A RANDOM VARIABLE BY A CONDITIONING OF A GIVEN SEQUENCE

Krzysztof Kaniowski

Abstract: Let (_O_,F,P) be a non-atomic probability space. If (X ) n is a sequence of r.v.’s satisfying X -- > 0 n a.s. (respectively, in probability) as n --> oo and EX+ --> oo , n EX - --> oo n as n --> oo , then for any r.v. Y there exists a sequence (U ) n of s -fields such that E(X | U | )-- > Y n n a.s. (respectively, in probability) as n --> oo .

1991 AMS Mathematics Subject Classification: 60A10.

Key words and phrases: Conditional expectation, almost sure convergence, stochastic convergence.

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