- Kolmogorov's inequality
In
probability theory , Kolmogorov's inequality is a so-called "maximalinequality " that gives a bound on the probability that thepartial sum s of afinite collection ofindependent random variables exceed some specified bound. The inequality is named after theRussia nmathematician Andrey Kolmogorov .Fact|date=May 2007tatement of the inequality
Let "X"1, ..., "X""n" : Ω → R be independent
random variable s defined on a commonprobability space (Ω, "F", Pr), withexpected value E ["X""k"] = 0 andvariance Var ["X""k"] < +∞ for "k" = 1, ..., "n". Then, for each λ > 0,:
where "S""k" = "X"1 + ... + "X""k".
Proof
The following argument is due to
Kareem Amin and employs discrete martingales. As argued in the discussion ofDoob'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 a also a martingale. Since for all and by thelaw of total expectation ,:The same is true for . Thus:byChebyshev's inequality .ee also
*
Chebyshev's inequality
*Doob's martingale inequality
*Etemadi's inequality
*Landau-Kolmogorov inequality
*Markov's inequality References
* (Theorem 22.4)
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