CONVERGENCE OF WEIGHTED AVERAGES OF MARTINGALES IN NONCOMMUTATIVE BANACH FUNCTION SPACES

Let x （xn）≥1 be a martingale on a noncommutative probability space n （M, r） and （wn）n≥1 a sequence of positive numbers such that Wn = ∑ k=1^n wk →∞ as n →∞ We prove that x = （x.）n≥1 converges in E（M） if and only if （σn（x）n≥1 converges in E（.hd）, where E（A//） is a noncommutative rearrangement invaria...

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Vydáno v:Acta mathematica scientia Ročník 32; číslo 2; s. 735 - 744
Hlavní autor: 张超 侯友良
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier Ltd 01.03.2012
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid,Madrid 28049, Spain%School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
Témata:
46L52
46L53
60G42
Banach函数空间
Convergence
Function space
Martingales
noncommutative
noncommutative Banach function spaces
Norms
Topology
uniform integrability
Weighted average
一致可积
加权平均
收敛性
概率空间
绝对连续
非交换
鞅
46L52
Weighted average
noncommutative Banach function spaces
60G42
martingales
noncommutative
uniform integrability
46L53
noncommutative martingales
ISSN:0252-9602, 1572-9087
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Shrnutí:Let x （xn）≥1 be a martingale on a noncommutative probability space n （M, r） and （wn）n≥1 a sequence of positive numbers such that Wn = ∑ k=1^n wk →∞ as n →∞ We prove that x = （x.）n≥1 converges in E（M） if and only if （σn（x）n≥1 converges in E（.hd）, where E（A//） is a noncommutative rearrangement invariant Banach function space with the Fatou property and σn（x） is given by σn（x） = 1/Wn ∑k=1^n wkxk, n=1, 2, .If in addition, E（Ad） has absolutely continuous norm, then, （an（x））≥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）.
Bibliografie:Weighted average; noncommutative martingales; noncommutative BanachfunCtion spaces; uniform integrability
Zhang Chao Hou Youliang 1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China 2. Departamento de Matemdticas, Facultad de Ciencias, Universidad Autdnoma de Madrid, Madrid 28049, Spain
Let x （xn）≥1 be a martingale on a noncommutative probability space n （M, r） and （wn）n≥1 a sequence of positive numbers such that Wn = ∑ k=1^n wk →∞ as n →∞ We prove that x = （x.）n≥1 converges in E（M） if and only if （σn（x）n≥1 converges in E（.hd）, where E（A//） is a noncommutative rearrangement invariant Banach function space with the Fatou property and σn（x） is given by σn（x） = 1/Wn ∑k=1^n wkxk, n=1, 2, .If in addition, E（Ad） has absolutely continuous norm, then, （an（x））≥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）.
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ISSN:0252-9602
1572-9087
DOI:10.1016/S0252-9602(12)60053-8