ON MOMENT CONDITIONS FOR SUPREMUM OF NORMED SUMS OF MARTINGALE DIFFERENCES

Abstract

Let {S(n), n greater-than-or-equal-to 1} denote the partial sum of sequence (X(n)) of identically distributed martingale differences. it is shown that E\X1\q(lg\X1\)r < infinity implies E(sup ((lg n)pr/q/n(p/q)) \S(n)\p) < infinity, where 1 < p < 2, p < q, r is-an-element-of R and lg x = max{1, log+ x} For the independent identically distributed case, the converse of the above statement holds.