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proof of Kolmogorov’s inequality


For k=1,2,,n, let Ak be the event that |Sk|λ but |Si|<λ for all i=1,2,,k-1. Note that the events A1, A2,,An are disjoint, and

nk=1Ak={max1kn|Sk|λ}.

Let IA be the indicator functionPlanetmathPlanetmath of event A. Since A1, A2,,An are disjoint, we have

0nk=1IAk1.

Hence, we obtain

nk=1Var[Xk]=E[S2n]nk=1E[S2nIAk].

After replacing S2n by S2k+2Sk(Sn-Sk)+(Sn-Sk)2, we get

nk=1Var[Xk] nk=1E[(S2k+2Sk(Sn-Sk)+(Sn-Sk)2)IAk]
nk=1E[(S2k+2Sk(Sn-Sk))IAk]
=nk=1E[S2kIAk]+2nk=1E[Sn-Sk]E[SkIAk]
=nk=1E[S2kIAk]
λ2nk=1E[IAk]
=λ2nk=1Pr(Ak)
=λ2Pr(nk=1Ak)
=λ2Pr(max1kn|Sk|λ),

where in the third line, we have used the assumptionPlanetmathPlanetmath that Sn-Sk is independent of SkIAk.

Title proof of Kolmogorov’s inequalityMathworldPlanetmath
Canonical name ProofOfKolmogorovsInequality
Date of creation 2013-03-22 17:48:35
Last modified on 2013-03-22 17:48:35
Owner kshum (5987)
Last modified by kshum (5987)
Numerical id 4
Author kshum (5987)
Entry type Proof
Classification msc 60E15