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April 2019 Etemadi and Kolmogorov Inequalities in Noncommutative Probability Spaces
Ali Talebi, Mohammad Sal Moslehian, Ghadir Sadeghi
Michigan Math. J. 68(1): 57-69 (April 2019). DOI: 10.1307/mmj/1541667627

Abstract

Based on a maximal inequality-type result of Cuculescu, we establish some noncommutative maximal inequalities such as the Hajék–Penyi and Etemadi inequalities. In addition, we present a noncommutative Kolmogorov-type inequality by showing that if x1,x2,,xn are successively independent self-adjoint random variables in a noncommutative probability space (M,τ) such that τ(xk)=0 and sksk1=sk1sk, where sk=j=1kxj, then, for any λ>0, there exists a projection e such that

1(λ+max 1knxk)2k=1nvar(xk)τ(e)τ(sn2)λ2. As a result, we investigate the relation between the convergence of a series of independent random variables and the corresponding series of their variances.

Citation

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Ali Talebi. Mohammad Sal Moslehian. Ghadir Sadeghi. "Etemadi and Kolmogorov Inequalities in Noncommutative Probability Spaces." Michigan Math. J. 68 (1) 57 - 69, April 2019. https://doi.org/10.1307/mmj/1541667627

Information

Received: 1 February 2017; Revised: 25 September 2017; Published: April 2019
First available in Project Euclid: 8 November 2018

zbMATH: 07155458
MathSciNet: MR3934604
Digital Object Identifier: 10.1307/mmj/1541667627

Subjects:
Primary: 46L53
Secondary: 46L10, 47A30, 60F99

Rights: Copyright © 2019 The University of Michigan

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Vol.68 • No. 1 • April 2019
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