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.

Statement of the inequality

Let X₁, ..., 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

$\mathrm{Pr}(\underset{1\le k\le n}{\max }|S_k|\ge 3\alpha )\le 3\underset{1\le k\le n}{\max }\mathrm{Pr}(|S_k|\ge \alpha ).$

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:

$\mathrm{Pr}(\underset{1\le k\le n}{\max }|S_k|\ge \alpha )\le \frac{27}{{\alpha }^2}\mathrm{var}(S_n).$

References

• Template:Cite book (Theorem 22.5)
• Template:Cite journal