Etemadi and Kolmogorov inequalities in noncommutative probability spaces. (English) Zbl 1442.46050

Talebi, Ali; Moslehian, Mohammad Sal; Sadeghi, Ghadir

Mich. Math. J. 68, No. 1, 57-69 (2019).

Summary: Based on a maximal inequality-type result of I. Cuculescu [J. Multivariate Anal. 1, 17–27 (1971; Zbl 0254.60031)], we establish some noncommutative maximal inequalities such as the Hajék-Renyi and Etemadi inequalities [J. Hájek and A. Rényi, Acta Math. Acad. Sci. Hung. 6, 281–283 (1955; Zbl 0067.10701); N. Etemadi, Sankhyā, Ser. A 47, 215–221 (1985; Zbl 0581.60019)]. In addition, we present a noncommutative Kolmogorov-type inequality by showing that if \(x_1,x_2,\dots,x_n\) are successively independent self-adjoint random variables in a noncommutative probability space \((\mathfrak{M},\tau)\) such that \(\tau(x_k)=0\) and \(s_ks_{k-1}=s_{k-1}s_k \), where \(s_k=\sum_{j=1}^kx_j \), then, for any \(\lambda> 0\), 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.

MSC:

46L53	Noncommutative probability and statistics
46L10	General theory of von Neumann algebras
60F99	Limit theorems in probability theory
47A30	Norms (inequalities, more than one norm, etc.) of linear operators

Keywords:

noncommutative maximal inequalities; noncommutative Kolmogorov-type inequality

Citations:

Zbl 0254.60031; Zbl 0067.10701; Zbl 0581.60019

References:

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