Kolmogorov's inequality

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Kolmogorov's inequality is a mathematical inequality appearing in probability theory, named after the Russian mathematician Andrey Kolmogorov. The statement of the inequality is as follows

Let $X_1,\dots, X_n$ be independent random variables in a probability space, such that $\operatorname{E} X_k=0$ and $\operatorname{Var} \, X_k <\infty$ for $k=1,\dots, n.$ Then, for each $λ > 0$ ,

$P\left(\max_{1\leq k\leq n} | S_k|\geq\lambda\right)\leq \frac{1}{\lambda^2} \operatorname{Var}\,S_n = \frac{1}{\lambda^2}\sum_{k=1}^n \operatorname{Var}\,X_k,$