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State Kolmogorov's inequality

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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 \(S_{k}=X_{1}+\cdots+X_{k}\)
by Platinum
(106,844 points)

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