Etemadi's inequality

In

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

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

Then

${\displaystyle \Pr {\Bigl (}\max _{1\leq k\leq n}|S_{k}|\geq 3\alpha {\Bigr )}\leq 3\max _{1\leq k\leq n}\Pr {\bigl (}|S_{k}|\geq \alpha {\bigr )}.}$

Remark

Suppose that the random variables 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:

${\displaystyle \Pr {\Bigl (}\max _{1\leq k\leq n}|S_{k}|\geq \alpha {\Bigr )}\leq {\frac {27}{\alpha ^{2}}}\operatorname {var} (S_{n}).}$

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.
  MR 0844022
  .