# Etemadi and Kolmogorov inequalities in noncommutative probability spaces

Research paper by **Ali Talebi, Mohammad Sal Moslehian, Ghadir Sadeghi**

Indexed on: **23 Sep '17**Published on: **23 Sep '17**Published in: **arXiv - Mathematics - Operator Algebras**

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#### 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.