CONVERGENCE OF WEIGHTED AVERAGES OF MARTINGALES IN NONCOMMUTATIVE BANACH FUNCTION SPACES

Let x = (xn)(n >= 1) be a martingale on a noncommutative probability space (M, T) and (w(n))(n >= 1) a sequence of positive numbers such that W-n =Sigma(n)(k=1) w(k)-> infinity as n ->infinity. We prove that x = (x(n))(n >= 1) converges in E(M) if and only if (sigma(n)(x))(n >= 1) converges in E(M), where E(M) is a noncommutative rearrangement invariant; Banach function space with the Fatou property and sigma(n)(x) is given by sigma(n)(x)=1/W-n Sigma(n)(k=1) w(k)x(k), n = 1, 2,... If in addition, E(M) has absolutely continuous norm, then, (sigma(n)(x))(n >= 1) converges in E(M) if and only if x = (x(n))(n >= 1) is uniformly integrable and its limit in measure topology x(infinity) epsilon E(M).