# arrow_back State Kolmogorov's inequality

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State Kolmogorov's inequality

Let $X_{1}, \ldots, X_{n}$ be independent random variables in a probability space, such that $\mathrm{E}\left[X_{k}\right]=0$ and $\operatorname{Var}\left[X_{k}\right]<\infty$ for $k=1, \ldots, n .$ Then, for each $\lambda>0$
$P\left(\max _{1 \leq k \leq n}\left|S_{k}\right| \geq \lambda\right) \leq \frac{1}{\lambda^{2}} \operatorname{Var}\left[S_{n}\right]=\frac{1}{\lambda^{2}} \sum_{k=1}^{n} \operatorname{Var}\left[X_{k}\right]$
where $S_{k}=X_{1}+\cdots+X_{k}$
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