Abstract: For a probability space and a filtration in we consider the
sequences of random variables satisfying the condition
In general, the process is not required to be 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 be a uniformly integrable
pseudo-martingale with Then weakly in where
approximation results for -fields are obtained with implications to pseudo-martingales. A
number of examples is collected.