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Acta Mathematica Vietnamica

ON THE ALMOST SURE CONVERGENCE OF TWO-PARAMETER MARTINGLES AND THE STRONG LAW OF LARGE NUMBERS IN BANACH SPACES

NGUYEN VAN HUNG, NGUYEN DUY TIEN

Abstract

Let (Ω,F,P) be a probability space, N2=N×N denote the set of parameters with the partial order defined by (m1,n1)(m2,n2) if and only if m1m2 and n1n2 (m1,n1,m2,n2N). Let Fmn be an increasing family of sub-δ-fields of F satisfying the usual condition (F4) and (Mmn,Fmn) a two-parameter martingale taking values in a Banach space (B,). In this paper we investigate the interrelation between geometric properties of Banach spaces and Martingale convergence theorems. Moreover we also study Marcinkiewicz-Zygmund's type strong law of large numbers for two-parameter Banach-valued martingales and the integrability of two-parameter Banach-valued martingale maximal functions.