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
n⋃k=1Ak={max1≤k≤n|Sk|≥λ}. |
Let IA be the indicator function of event A. Since A1, A2,…,An are disjoint, we have
0≤n∑k=1IAk≤1. |
Hence, we obtain
n∑k=1Var[Xk]=E[S2n]≥n∑k=1E[S2nIAk]. |
After replacing S2n by S2k+2Sk(Sn-Sk)+(Sn-Sk)2, we get
n∑k=1Var[Xk] | ≥n∑k=1E[(S2k+2Sk(Sn-Sk)+(Sn-Sk)2)IAk] | ||
≥n∑k=1E[(S2k+2Sk(Sn-Sk))IAk] | |||
=n∑k=1E[S2kIAk]+2n∑k=1E[Sn-Sk]E[SkIAk] | |||
=n∑k=1E[S2kIAk] | |||
≥λ2n∑k=1E[IAk] | |||
=λ2n∑k=1Pr(Ak) | |||
=λ2Pr(n⋃k=1Ak) | |||
=λ2Pr(max1≤k≤n|Sk|≥λ), |
where in the third line, we have used the assumption that Sn-Sk is independent of SkIAk.
Title | proof of Kolmogorov’s inequality![]() |
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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 |