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Two basic probabilistic proofs[zz]

Repost from http://djalil.chafai.net/blog/2011/08/23/two-basic-probabilistic-proofs/

Very elegant proof!!

Johan Jensen (mathematician)

I enjoy basic beautiful mathematical proofs. I see them like small jewels, that I collect from time to time. In this spirit, this post proposes probabilistic proofs of a couple of basic results.

Jensen inequality. The Jensen inequality states that if X is an integrable random variable on Rn and φ:Rn→R a convex function such that φ(X) is integrable, then

 

φ(E(X))≤E(φ(X)).

 

To prove it, we start by using the convexity of φ, which gives, for every integer n≥1 and every sequence x1,…,xn in Rn,

 

φ(x1+⋯+xnn)≤φ(x1)+⋯+φ(xn)n.

 

Now, we use the integrability of X and φ(X): we take x1,x2,… random independent and distributed as X, we use the strong law of large numbers for both sides, the fact that φ is continuous for the left hand side, and the fact that if P(A)=P(B)=1 then A∩B≠∅. I also appreciate the proof based on the equality for affine functions, the variational expression of a convex function as the envelope of its tangent hyperplanes, together with the fact that the supremum of expectations is less than or equal to the expectation of the supremum. Read more…

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