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, ..., 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 . {\displaystyle S_{k}=X_{1}+\cdots +X_{k}.\,}

Then

Pr ( max 1 ≤ k ≤ n | S k | ≥ 3 α ) ≤ 3 max 1 ≤ k ≤ n Pr ( | S k | ≥ α ) . {\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 )}.}

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:

Pr ( max 1 ≤ k ≤ n | S k | ≥ α ) ≤ 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}).}

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. JSTOR 25050536. MR 0844022.