In probability theory, Etemadi's inequality is a so-called "maximal inequality", an inequality that gives a bound on the probability that the partial sums of a finite collection of independent random variables exceed some specified bound. The result is due to Nasrollah Etemadi.
Statement of the inequality.
Let "X"1, ..., "X""n" be independent real-valued random variables defined on some common probability space, and let "α" ≥ 0. Let "S""k" denote the partial sum
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Then
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Remark.
Suppose that the random variables "X""k" have common expected value zero. Apply Chebyshev's inequality to the right-hand side of Etemadi's inequality and replace "α" by "α" / 3. The result is Kolmogorov's inequality with an extra factor of 27 on the right-hand side:
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