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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