Kolmogorov's inequality

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 inequalityEdit

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.

ProofEdit

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. Define   as follows. Let  , and

 

for all  . Then   is also a martingale.

For any martingale   with  , we have that

 

Applying this result to the martingale  , we have

 

where the first inequality follows by Chebyshev's inequality.


This inequality was generalized by Hájek and Rényi in 1955.

See alsoEdit

ReferencesEdit

  • 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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