Etemadi's inequality explained

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} + … + X _{k.}

Then

\Pr l ( max _{1} | S _{k} | \geq 3 \alpha r ) \leq 3 max _{1} \Pr l ( | S _{k} | \geq \alpha r ).

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:

\Pr l ( max _{1} | S _{k} | \geq \alpha r ) \leq (27)/(\alpha ^{2}) \operatorname{var} (S _{n).}

References

• Book: Billingsley, Patrick . Probability and Measure . John Wiley & Sons, Inc. . New York . 1995 . 0-471-00710-2. (Theorem 22.5)
• Etemadi . Nasrollah . On some classical results in probability theory . Sankhyā Ser. A . 47 . 1985 . 215 - 221 . 0844022 . 25050536 . 2 .