The Michigan Mathematical Journal

Etemadi and Kolmogorov Inequalities in Noncommutative Probability Spaces

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 $x_{1},x_{2},\ldots,x_{n}$ are successively independent self-adjoint random variables in a noncommutative probability space $(\mathfrak{M},\tau)$ such that $\tau(x_{k})=0$ and $s_{k}s_{k-1}=s_{k-1}s_{k}$, where $s_{k}=\sum_{j=1}^{k}x_{j}$, then, for any $\lambda\gt 0$, there exists a projection $e$ such that

$$1-\frac{(\lambda+\max_{1\leq k\leq n}\Vert x_{k}\Vert )^{2}}{\sum_{k=1}^{n}\operatorname{var}(x_{k})}\leq\tau(e)\leq\frac{\tau(s_{n}^{2})}{\lambda^{2}}.$$ 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
Revised: 25 September 2017
First available in Project Euclid: 8 November 2018

https://projecteuclid.org/euclid.mmj/1541667627

Digital Object Identifier
doi:10.1307/mmj/1541667627

Mathematical Reviews number (MathSciNet)
MR3934604

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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