Convergence of weighted averages of martingales in Banach function spaces

Let f = (f(n))(n greater than or equal to 1) be a martingale and (w(n))(n greater than or equal to 1) a sequence of positive numbers such that W-n = Sigma(k-1)(n)w(k) --> infinity. Kazamaki and Tsuchikura proved that f converges in L-P (1 < p < infinity) if and only if the weighted average (sigma(n)(f))(n greater than or equal to 1) of f converges in L-P, where sigma(n)(f) are given by sigma(n)(f) = 1/W-n (k=1)Sigma(n) w(k)f(k), n = 1, 2, ... . We shall investigate the convergence of f and sigma(n)(f) in general Banach function spaces X. Our main result can be applied to the case where X is a rearrangement-invariant space, or X is a weighted L-P-space with a weight function satisfying the condition A(p) introduced by Izumisawa and Kazamaki. (C) 2000 Academic Press.