CONVERGENCE OF WEIGHTED AVERAGES OF MARTINGALES IN NONCOMMUTATIVE BANACH FUNCTION SPACES

doi:10.1016/S0252-9602(12)60053-8

Acta mathematica scientia,Series B ›› 2012, Vol. 32 ›› Issue (2): 735-744.doi: 10.1016/S0252-9602(12)60053-8

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CONVERGENCE OF WEIGHTED AVERAGES OF MARTINGALES IN NONCOMMUTATIVE BANACH FUNCTION SPACES

ZHANG Chao^1,2, HOU You-Liang¹

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
2. Departamento de Matem´aticas, Facultad de Ciencias, Universidad Aut´onoma de Madrid, Madrid 28049, Spain

Received:2010-10-02 Revised:2011-03-14 Online:2012-03-20 Published:2012-03-20
Supported by:
This research was supported by the National Natural Science Foundation of China (11071190).

Abstract

Abstract:

Let x = (x_n)_n≥1 be a martingale on a noncommutative probability space (M, τ ) and (w_n)_n≥1 a sequence of positive numbers such that W_n =
∑ⁿ_k₌₁w_k →∞ as n →1. We prove that x = (x_n)_n≥1 converges in E(M) if and only if (σ_n(x))_n≥1 converges in E(M), where E(M) is a noncommutative rearrangement invariant Banach function space with the Fatou property and σ_n(x) is given by
σ_n(x) =1/W_n∑ⁿ_k₌₁w_kx_k, 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 = (x_n)_n≥1 is uniformly integrable and its limit in measure topology x_∞ ∈E(M).

Key words: Weighted average, noncommutative martingales, noncommutative Banach function spaces, uniform integrability

CLC Number:

46L52

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.

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CONVERGENCE OF WEIGHTED AVERAGES OF MARTINGALES IN NONCOMMUTATIVE BANACH FUNCTION SPACES

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