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

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.


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
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