On moment conditions for the supremum of normed sums

AbstractLet {Sn, n⩾1} denote the partial sum of a sequence (Xn) of i.i.d. random variables with mean 0. The equivalent statements of E(supn|Sn|α/cn)< t8 for certain sequences (cn) and positive constants α are investigated. We find a useful sufficient condition of E(supn|Sn|α/cn)< ∞ for a sequence (Xn) of martingale differences. From this result we prove the equivalence of E(supn|Sn|p/npq) < ∞, E(supn|Xn|p/npq) < ∞ and E|X|q < ∞ for 0 < p < q < 2