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 RussianmathematicianAndrey Kolmogorov.[citation needed]
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[]
The following argument is due to and employs discrete martingales.
As argued in the discussion of Doob's martingale inequality, the sequence is a martingale.
Define as follows. Let , and
Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc. ISBN0-471-00710-2. (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. ISBN0-471-25708-7.
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