On the convergence in mean of martingale difference sequenceS

Abstract

In [6] Freniche proved that any weakly null martingale difference sequence in L ₁ [0, 1] has arithmetic means that converge in norm to 0. We show any weakly null martingale difference sequence in an Orlicz space whose N-function belongs to ∇₃ has arithmetic means that converge in norm to 0. Then based on a theorem in Stout [13, Theorem 3.3.9 (i) and (iii)], we give necessary and sufficient conditions for a bounded martingale difference sequence in an Orlicz space whose N-function belongs to a large class of ∇₂ functions to have means that converge to 0 a.s. Finally, we conclude with some expository comments including an easy proof of Komlos' theorem [9] for L _p[0, 1], 1 < p < ∞.