On the convergence in mean of martingale difference sequences. (English) Zbl 0986.46017
Summary: In [Proc. Am. Math. Soc. 103, No. 1, 234-236 (1988; Zbl 0667.46019)] F. J. 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 \(\nabla_3\) has arithmetic means that converge in norm to \(0\). Then based on a theorem in W. F. Stout [“Almost sure convergence” (1974; Zbl 0321.60022), 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 \(\Delta_2\) functions to have means that converge to \(0\) a.s. Finally, we conclude with some expository comments including an easy proof of J. Komlós’ theorem [Acta Math. Acad. Sci. Hung. 18, 217-229 (1967; Zbl 0228.60012)] for \(L_p[0,1]\), \(1< p< \infty\).
MSC:
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 60F25 | \(L^p\)-limit theorems |
| 28A20 | Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence |
| 60F15 | Strong limit theorems |
