ANOTHER CLASSIFICATION OF QUASI-MARTINGALES IN THE LIMIT

Tran Quang Vinh

Abstract

           Given a stochastic basic (A_n), a sequence (X_n) of integrable random variables, adapted to (A_n) is said to be a quasi-martingale in the limit if for every ε > 0, there exists p ∈ N such that for every m ≥  p there exists p_m ≥ m such that for all n ≥  p_m we have

P(

sup|Xq(n)-Xq| > ε p≤q≤m

) < ε

The main aim of this note is to prove that the class of all quasi-martingales in the limit would be classified into a nondecreasing directed family of subclasses whose smallest element is just the class of mils introduced by M.Talagrand (1985).