On the Convergence in Mean of Martingale Difference Sequences
نویسندگان
چکیده
In [6] Freniche proved that any weakly null martingale difference sequence in L1[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 Lp[0, 1], 1 < p <∞. Mathematics Subject Classification. 46E30, 60F25, 28A20.
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