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


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


Hence, we obtain


After replacing Sn2 by Sk2+2Sk(Sn-Sk)+(Sn-Sk)2, we get

k=1nVar[Xk] k=1nE[(Sk2+2Sk(Sn-Sk)+(Sn-Sk)2)IAk]

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