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 Hajék--Penyi inequality and Etemadi inequality. 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 $\tau \left(x_k\right)=0$ 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 $$1-\frac{(\lambda +{\max }_{1\le k\le n}\Vert x_k\Vert {)}^2}{{\sum }_{k=1}^n\mathrm{var}(x_k)}\le \tau (e)\le \frac{\tau (s_n^2)}{{\lambda }^2}.$$ As a result, we investigate the relation between convergence of a series of independent random variables and the corresponding series of their variances.

17 pages; to appear in Michigan Math. J