A generalization of martingale theory to self-averaging processes

We introduce and study a generalization of martingales with the following self-averaging property: at each time, the conditional expectation of future random variables given the past, is a weighted average of all the random
variables comprising the past. We assume only that more recent random variables are weighted no less than older random variables. We investigate conditions under which important properties satisfied by martingales, such as maximal inequalities and convergence, are present in an appropriate form.