A Convergence Theorem for Non Negative Almost Supermartingales and Some Applications

The purpose of this paper is to give a unified treatment of a number of almost sure convergence theorems by exploiting the fact that the processes involved possess a common “almost supermartingale” properties To be precise, let (Ω,F,P) be a probability space and F1 ⊂ F2 ⊂ … a sequence of sub-σ-algebras of F. For each n= 1,2, … let zn, βn, ξn, and ζn be non-negative Fn -measurable random variables such that $$E({z_{n + 1}}|{F_n}) \leqslant {z_n}(1 + {\beta _n}) + {\xi _n} - {\zeta _n}.$$ (1) .