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 condit...
