Supriya Ghosh (Editor)

Etemadi's inequality

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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.

Contents

Statement of the inequality

Let X1, ..., Xn be independent real-valued random variables defined on some common probability space, and let α ≥ 0. Let Sk denote the partial sum

S k = X 1 + ⋯ + X k .

Then

P ( max 1 ≤ k ≤ n | S k | ≥ 3 α ) ≤ 3 max 1 ≤ k ≤ n P ( | S k | ≥ α ) .

Remark

Suppose that the random variables Xk 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:

P ( max 1 ≤ k ≤ n | S k | ≥ α ) ≤ 27 α 2 V a r ( S n ) .

References

Etemadi's inequality Wikipedia
(Text) CC BY-SA


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