Kolmogorov's inequality

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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.[citation needed]

Statement of the inequality[edit]

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 convenience of this result is that we can bound the worst case deviation of a random walk at any point of time using its value at the end of time interval.

Proof[edit]

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

by Chebyshev's inequality.

This inequality was generalized by Hájek and Rényi in 1955.

See also[edit]

References[edit]

  • Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc. ISBN 0-471-00710-2.  (Theorem 22.4)
  • Feller, William (1968) [1950]. An Introduction to Probability Theory and its Applications, Vol 1 (Third ed.). New York: John Wiley & Sons, Inc. xviii+509. ISBN 0-471-25708-7. 

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