Friday, 7 February 2014

Lecture 10

I mentioned that there is a 318 page book completely devoted to the Cauchy-Schwarz inequality. It is The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Its author, Michael J. Steele, is a past President of the Institute of Mathematical Statistics. One reviewer wrote "... this is one of that handful of mathematics books that you can read almost like a novel." Chapter 1 is free to download and really interesting to read. I can also recommend his highly entertaining web site, which includes material for advanced courses in probability and finance, and also Semi-Random Rants, Videos and Favorite Quotes.

There are many proofs of this inequality. Here are two in addition to the one I did in lectures.

1. Consider E(tXY)2]=t2E[X2]2tE[XY]+E[Y2]0, and >0 unless tX=Y. This quadratic has most one real roots and so the discriminant must be nonpositive. This is essentially the proof that was given by Schwarz and is a textbook favorite.

2. We started the lecture with Jensen's inequality. It is natural to ask if the Cauchy-Schwarz inequality can be deduced from it. One version of the Cauchy-Schwarz inequality is

(iaibi)2ia2iib2i,

for all numbers (a1,,an) and (b1,,bn). WLOG we may assume all aj,bj>0. Consider a random variable X for which P(X=aj/bj)=bj/ib2i. Apply Jensen's inequality with the choice of convex function f(x)=x2. Ef(X)f(EX) gives the result.

This paper presents 12 different proofs of the Cauchy-Schwarz inequality, and this web page has 10.

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