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 &alpha; &ge; 0. Let Sk denote the partial sum


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

Then


 * $$\mathbb{P} \left( \max_{1 \leq k \leq n} | S_{k} | \geq 3 \alpha \right) \leq 3 \max_{1 \leq k \leq n} \mathbb{P} \left( | S_{k} | \geq \alpha \right).$$

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 &alpha; by &alpha; / 3. The result is Kolmogorov's inequality with an extra factor of 27 on the right-hand side:


 * $$\mathbb{P} \left( \max_{1 \leq k \leq n} | S_{k} | \geq \alpha \right) \leq \frac{27}{\alpha^{2}} \mathrm{Var} (S_{n}).$$