Convergence of weighted averages of noncommutative martingales

Let x =(xn)n1 be a martingale on a noncommutative probability space(M,τ) and(wn)n1 a sequence of positive numbers such that Wn=∑nk=1wk→∞asn→∞.We prove that x =(xn)n1 converges bilaterally almost uniformly(b.a.u.) if and only if the weighted average(σn(x))n1 of x converges b.a.u.to the same limit under some condition,where σn(x) is given by σn(x)=W1n∑nk=1wkxk,n=1,2,...Furthermore,we prove that x=(xn)n1 converges in Lp(M) if and only if(σn(x))n1 converges in Lp(M),where 1p<∞.We also get a criterion of uniform integrability for a family in L1(M).

Abstract Let x = （xn）n≥1 be a martingale on a noncommutative probability space （М,τ） and （Wn）n≥1 a sequence of positive numbers such that Wn =∑^n_k=1 wk→∞ as n→∞. We prove that x = （Xn）n≥1 converges bilaterally almost uniformly （b.a.u.） if and only if the weighted average （σan（x））n≥1 of x converges b.a.u, to the same limit under some condition, where σn（x） is given by σn（x）=1/Wn ^n∑_k=1 wkxk,n=1,2,… Furthermore, we prove that x = （xn）n≥1 converges in Lp（М） if and only if （σ＇n（x））n≥1 converges in Lp（М）, where 1 ≤p 〈 ∞ .We also get a criterion of uniform integrability for a family in L1（М）.