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.
Statement of the inequality.
Let "X"1, ..., "X""n" : Ω → R be independent random variables defined on a common probability space (Ω, "F", Pr), with expected value E["X""k"] = 0 and variance Var["X""k"]  ; +∞ for "k" = 1, ..., "n". Then, for each λ  ; 0,
formula_1
where "S""k" = "X"1 + ... + "X""k".
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 employs discrete martingales. 
As argued in the discussion of Doob's martingale inequality, the sequence formula_2 is a martingale.
Define formula_3 as follows. Let formula_4, and
formula_5
for all formula_6.
Then formula_3 is also a martingale. 
For any martingale formula_8 with formula_9, we have that 
formula_10
Applying this result to the martingale formula_11, we have
formula_12
where the first inequality follows by Chebyshev's inequality.
This inequality was generalized by Hájek and Rényi in 1955.
References.
"This article incorporates material from Kolmogorov's inequality on PlanetMath, which is licensed under the ."