Given a sequence of non-negative martingales we find a sequence of convex combinations and a limiting process such that the convex combinations converge in probability to the limiting process at all finite stopping times. The limiting process then is an optional strong supermartingale. A counter-example reveals that the convergence in probability cannot be replaced by almost sure convergence in this statement. We also give similar convergence results for sequences of optional strong supermartingales, their left limits and their stochastic integrals and explain the relation to the notion of the Fatou limit.
The talk is based on joint work with Christoph Czichowsky.