Etemadi's inequality
From HandWiki
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, ..., Xn be independent real-valued random variables defined on some common probability space, and let α ≥ 0. Let Sk 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 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:
- [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.
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