Etemadi and Kolmogorov inequalities in noncommutative probability spaces

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 $x_1,x_2,\dots ,x_n$ 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$s_k = \sum_{j=1}^k x_j$,thenforany$\lambda > 0$ there exists a projection $e$ such that $$1-\frac{(\lambda +\underset{1\le k\le n}{\max }\mathrm{\parallel }{x}_k\mathrm{\parallel }{)}^2}{\sum _{k=1}^n\mathrm{v}\mathrm{a}\mathrm{r}(x_k)}\le \tau (e)\le \frac{\tau (s_n^2)}{{\lambda }^2}\mathrm{.}$$ As a result, we investigate the relation between convergence of a series of independent random variables and the corresponding series of their variances.