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In [[probability theory]], '''Etemadi's inequality''' is a so-called "maximal inequality", an [[inequality (mathematics)|inequality]] that gives a bound on the [[probability]] that the [[partial sum]]s of a [[Finite set|finite]] collection of [[independent random variables]] exceed some specified bound. The result is due to [[Nasrollah Etemadi]].

==Statement of the inequality==

Let ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> be independent real-valued random variables defined on some common [[probability space]], and let ''α'' ≥ 0. Let ''S''<sub>''k''</sub> denote the partial sum

:<math>S_{k} = X_{1} + \cdots + X_{k}.\,</math>

Then

:<math>\mathbb{P} \left( \max_{1 \leq k \leq n} | S_{k} | \geq 3 \alpha \right) \leq 3 \max_{1 \leq k \leq n} \mathbb{P} \left( | S_{k} | \geq \alpha \right).</math>

==Remark==

Suppose that the random variables ''X''<sub>''k''</sub> 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:

:<math>\mathbb{P} \left( \max_{1 \leq k \leq n} | S_{k} | \geq \alpha \right) \leq \frac{27}{\alpha^{2}} \mathrm{Var} (S_{n}).</math>

==References==

* {{cite book | last=Billingsley | first=Patrick | title=Probability and Measure | publisher=John Wiley & Sons, Inc. | location=New York | year=1995 | isbn=0-471-00710-2}} (Theorem 22.5)
* {{cite journal | last=Etemadi | first=Nasrollah | title=On some classical results in probability theory | journal=[[Sankhya (journal)|Sankhyā]] Ser. A | volume=47 | year=1985 | pages=215&ndash;221 |mr=0844022 | jstor = 25050536 | issue=2 }}

[[Category:Probabilistic inequalities]]
[[Category:Statistical inequalities]]