The Michigan Mathematical Journal

Etemadi and Kolmogorov Inequalities in Noncommutative Probability Spaces

Ali Talebi, Mohammad Sal Moslehian, and Ghadir Sadeghi

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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.

Article information

Source
Michigan Math. J., Volume 68, Issue 1 (2019), 57-69.

Dates
Received: 1 February 2017
Revised: 25 September 2017
First available in Project Euclid: 8 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1541667627

Digital Object Identifier
doi:10.1307/mmj/1541667627

Mathematical Reviews number (MathSciNet)
MR3934604

Subjects
Primary: 46L53: Noncommutative probability and statistics
Secondary: 46L10: General theory of von Neumann algebras 47A30: Norms (inequalities, more than one norm, etc.) 60F99: None of the above, but in this section

Citation

Talebi, Ali; Moslehian, Mohammad Sal; Sadeghi, Ghadir. Etemadi and Kolmogorov Inequalities in Noncommutative Probability Spaces. Michigan Math. J. 68 (2019), no. 1, 57--69. doi:10.1307/mmj/1541667627. https://projecteuclid.org/euclid.mmj/1541667627


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