Etemadi's inequality

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.

tatement 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

:$S\_\{k\}\; =\; X\_\{1\}\; +\; cdots\; +\; X\_\{k\}.,$

Then

:$mathbb\{P\}\; left(\; max\_\{1\; leq\; k\; leq\; n\}\; |\; S\_\{k\}\; |\; geq\; 3\; alpha\; ight)\; leq\; 3\; max\_\{1\; leq\; k\; leq\; n\}\; mathbb\{P\}\; left(\; |\; S\_\{k\}\; |\; geq\; alpha\; ight).$

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:

:$mathbb\{P\}\; left(\; max\_\{1\; leq\; k\; leq\; n\}\; |\; S\_\{k\}\; |\; geq\; alpha\; ight)\; leq\; frac\{27\}\{alpha^\{2\; mathrm\{Var\}\; (S\_\{n\}).$

References

* (Theorem 22.5)
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