Strong supermartingales and limits of non-negative martingales

Given a sequence (M n ) ∞ n=1 (Mn)n=1∞ of nonnegative martingales starting at M n 0 =1 M0n=1, we find a sequence of convex combinations (M ~ n ) ∞ n=1 (M~n)n=1∞ and a limiting process X X such that (M ~ n τ ) ∞ n=1 (M~τn)n=1∞ converges in probability to X τ Xτ, for all finite stopping times τ τ. The limiting process X X then is an optional strong supermartingale. A counterexample 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 (X n ) ∞ n=1 (Xn)n=1∞, their left limits (X n − ) ∞ n=1 (X−n)n=1∞ and their stochastic integrals (∫φdX n ) ∞ n=1 (∫φdXn)n=1∞ and explain the relation to the notion of the Fatou limit