Etemadi and Kolmogorov inequalities in noncommutative probability spaces

TL;DR

This paper extends classical inequalities to noncommutative probability spaces, establishing maximal inequalities and a Kolmogorov-type inequality, which deepen understanding of convergence and independence in noncommutative settings.

Contribution

It introduces noncommutative versions of Hajék--Penyi, Etemadi, and Kolmogorov inequalities, expanding the theoretical framework of noncommutative probability.

Findings

Established noncommutative maximal inequalities based on Cuculescu's result.

Proved a noncommutative Kolmogorov inequality relating variances and projections.

Analyzed the connection between series convergence and variances in noncommutative probability.

Abstract

Based on a maximal inequality type result of Cuculescu, we establish some noncommutative maximal inequalities such as Haj\'ek--Penyi inequality and Etemadi inequality. 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 $\tau\left(x_k\right) = 0$ and $s_k s_{k-1} = s_{k-1} s_k$, where $s_k = \sum_{j=1}^k x_j$, then for any $\lambda > 0$ there exists a projection $e$ such that $$1 - \frac{(\lambda + \max_{1 \leq k \leq n} \|x_k\|)^2}{\sum_{k=1}^n {\rm var}(x_k)}\leq \tau(e)\leq \frac{\tau(s_n^2)}{\lambda^2}.$$ As a result, we investigate the relation between convergence of a series of independent random variables and the corresponding series of their variances.