Convergence of Weighted Averages of Martingales in Banach Function Spaces

Let f=(f_n)_n≥1 be a martingale and (w_n)_n≥1 a sequence of positive numbers such that W_n=∑ⁿ_k=1w_k→∞. Kazamaki and Tsuchikura proved that f converges in L^p (1<p<∞) if and only if the weighted average (σ_n(f))_n≥1 of f converges in L^p, where σ_n(f) are given byσ_nf=1W_n∑k=1nw_kf_k,n=1,2,....We shall investigate the convergence of f and σ_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.