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

[math]\displaystyle{ S_k = X_1 + \cdots + X_k.\, }[/math]

Then

[math]\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). }[/math]

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:

[math]\displaystyle{ \Pr \Bigl( \max_{1 \leq k \leq n} | S_k | \geq \alpha \Bigr) \leq \frac{27}{\alpha^2} \operatorname{var} (S_n). }[/math]

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.