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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 25, Fasc. 2,
pages 393 - 403
 

ON THE APPROXIMATION OF A RANDOM VARIABLE BY A CONDITIONAL EXPECTATION OF ANOTHER RANDOM VARIABLE

Krzysztof Kaniowski

Abstract: Let X and Y be R -valued random variables on a non-atomic probability space (_O_,F, P). We give conditions under which Y can be approximated by a conditional expectation of X. In particular, we prove the following theorem:

Let X be an R -valued random variable such that EX+ = EX -= oo . Then for each random variable Y and arbitrary e > 0 there exist B (- F and a sub-s -field S of F such that P(B) < e and E(X |S) = Y a.s. on Bc.

We also review some facts on the conditional expectation of unintegrable random variables.

2000 AMS Mathematics Subject Classification: 60A10.

Key words and phrases: Conditional expectation.

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