Inequality in probability theory
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 [ edit ] Let X 1 , ..., X n probability space , and let α ≥ 0. Let S k
S
k
=
X
1
+
⋯
+
X
k
.
{\displaystyle S_{k}=X_{1}+\cdots +X_{k}.\,}
Then
Pr
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max
1
≤
k
≤
n
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S
k
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≥
3
α
)
≤
3
max
1
≤
k
≤
n
Pr
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S
k
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≥
α
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{\displaystyle \Pr {\Bigl (}\max _{1\leq k\leq n}|S_{k}|\geq 3\alpha {\Bigr )}\leq 3\max _{1\leq k\leq n}\Pr {\bigl (}|S_{k}|\geq \alpha {\bigr )}.}
Suppose that the random variables X k 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
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max
1
≤
k
≤
n
|
S
k
|
≥
α
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≤
27
α
2
var
(
S
n
)
.
{\displaystyle \Pr {\Bigl (}\max _{1\leq k\leq n}|S_{k}|\geq \alpha {\Bigr )}\leq {\frac {27}{\alpha ^{2}}}\operatorname {var} (S_{n}).}
Billingsley, Patrick (1995). Probability and Measure . New York: John Wiley & Sons, Inc. ISBN 0-471-00710-2 Etemadi, Nasrollah (1985). "On some classical results in probability theory". Sankhyā Ser. A . 47 (2): 215– 221. JSTOR 25050536 . MR 0844022 . 
