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

\mathbb {P} {\Bigl (}\max _{1\leq k\leq n}|S_{k}|\geq 3\alpha {\Bigr )}\leq 3\max _{1\leq k\leq n}\mathbb {P} {\bigl (}|S_{k}|\geq \alpha {\bigr )}.

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} {\Bigl (}\max _{1\leq k\leq n}|S_{k}|\geq \alpha {\Bigr )}\leq {\frac {27}{\alpha ^{2}}}\mathrm {Var} (S_{n}).

References

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

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