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

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

Then

$\mathbb{P} \Bigl( \max_{1 \leq k \leq n} | S_{k} | \geq 3 \alpha \Bigr) \leq 3 \max_{1 \leq k \leq n} \mathbb{P} \bigl( | S_{k} | \geq \alpha \bigr).$

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:

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

References

• (Theorem 22.5)