Martingale Property of Empirical Processes ¤

It is shown that for a large collection of almost independent martingales in a suitable framework, the martingale property is preserved on the empirical processes almost surely. Under the assumptions of almost independence and essentially identical finite dimensional distributions, it is proven that a large collection of stochastic processes are martingales essentially if and only if so are the empirical processes. These two results shed some light on the testability of the martingale property in scientific modeling. Extensions to submartingales and supermartingales are given. The proofs are based on the exact law of large numbers obtained recently. ∗AMS subject classification: Primary 60G42, 60G44; Secondary 03H05, 28E05, 60F15.