Acta mathematica scientia, Series B >
CONVERGENCE OF WEIGHTED AVERAGES OF MARTINGALES IN NONCOMMUTATIVE BANACH FUNCTION SPACES
Received date: 2010-10-02
Revised date: 2011-03-14
Online published: 2012-03-20
Supported by
This research was supported by the National Natural Science Foundation of China (11071190).
Let x = (xn)n≥1 be a martingale on a noncommutative probability space (M, τ ) and (wn)n≥1 a sequence of positive numbers such that Wn =
∑nk=1 wk →∞ as n →1. We prove that x = (xn)n≥1 converges in E(M) if and only if (σn(x))n≥1 converges in E(M), where E(M) is a noncommutative rearrangement invariant Banach function space with the Fatou property and σn(x) is given by
σn(x) =1/Wn∑nk=1wkxk, n = 1, 2, · · · . If in addition, E(M) has absolutely continuous norm, then, (σn(x))n≥1 converges in E(M) if and only if x = (xn)n≥1 is uniformly integrable and its limit in measure topology x∞ ∈E(M).
ZHANG Chao , HOU You-Liang . CONVERGENCE OF WEIGHTED AVERAGES OF MARTINGALES IN NONCOMMUTATIVE BANACH FUNCTION SPACES[J]. Acta mathematica scientia, Series B, 2012 , 32(2) : 735 -744 . DOI: 10.1016/S0252-9602(12)60053-8
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