In probability theory, Kolmogorov's inequality is a so-called "maximal 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 inequality is named after the Russian mathematician Andrey Kolmogorov.
Statement of the inequality
Let X1, ..., Xn : Ω → R be independent random variables defined on a common probability space (Ω, F, Pr), with expected value E[Xk] = 0 and variance Var[Xk] < +∞ for k = 1, ..., n. Then, for each λ > 0,
where Sk = X1 + ... + Xk.
The following argument is due to Kareem Amin and employs discrete martingales. As argued in the discussion of Doob's martingale inequality, the sequence is a martingale. Without loss of generality, we can assume that and for all . Define as follows. Let , and
for all . Then is also a martingale. Since is independent and mean zero,
The same is true for . Thus
This inequality was generalized by Hájek and Rényi in 1955.
- Chebyshev's inequality
- Etemadi's inequality
- Landau–Kolmogorov inequality
- Markov's inequality
- Bernstein inequalities (probability theory)
- Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc. ISBN 0-471-00710-2. (Theorem 22.4)
- Feller, William (1968) . An Introduction to Probability Theory and its Applications, Vol 1 (Third Edition ed.). New York: John Wiley & Sons, Inc. xviii+509. ISBN 0-471-25708-7.