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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 19, Fasc. 1,
pages 181 - 201
 

PSEUDO-MARTINGALES

R. Jajte
A. Paszkiewicz

Abstract: For a probability space (_O_, F,P ) and a filtration (Un) in Q, we consider the sequences (Xn) of random variables satisfying the condition

E(Xn+1  - Xn|Un) = 0, n = 1,2,...

In general, the process (Xn) is not required to be (Un) adapted and it is called a pseudo-martingale. We indicate simple and natural conditions implying a good asymptotic behaviour of pseudo-martingales. For example: let (Xn,Un) be a uniformly integrable pseudo-martingale with Un  /^  F. Then Xn --> X weakly in L1  where

X = nli-->mo o E(Xn |Un).
Some approximation results for s -fields are obtained with implications to pseudo-martingales. A number of examples is collected.

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

Key words and phrases: -

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