- Etemadi's inequality
In
probability theory , Etemadi's inequality is a so-called "maximal inequality", aninequality that gives a bound on theprobability that thepartial sum s of afinite collection ofindependent random variables exceed some specified bound. The result is due toNasrollah Etemadi .tatement of the inequality
Let "X"1, ..., "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} left( max_{1 leq k leq n} | S_{k} | geq 3 alpha ight) leq 3 max_{1 leq k leq n} mathbb{P} left( | S_{k} | geq alpha ight).
Remark
Suppose that the random variables "X""k" have common
expected value zero. ApplyChebyshev's inequality to the right-hand side of Etemadi's inequality and replace "α" by "α" / 3. The result isKolmogorov's inequality with an extra factor of 27 on the right-hand side::mathbb{P} left( max_{1 leq k leq n} | S_{k} | geq alpha ight) leq frac{27}{alpha^{2 mathrm{Var} (S_{n}).
References
* (Theorem 22.5)
*
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