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.

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