Wednesday, 11 March 2015

Lecture 24

The overhead slides I used for our last lecture are here (best downloaded and viewed with a pdf reader, rather than as a web page).

I showed you a bit of Large deviation theory and Chernoff's upper bound. The lower bound  was quoted without proof (except that in Example 24.1 we did the lower bound for the special case of $B(1,p)$). If perhaps some time mid-summer you have a moment and are curious to see the lower bound proved (which you have learned enough to understand), and more about large deviations, see these slides from a talk I once gave, pages 5–9. This note on the Cramer-Chernoff theorem is also good. Interestingly, Chernoff says in an interview with John Bather that the Chernoff bound should really be named after someone else!

I have used large deviation theory in my research to analyse buffer overflows in queues. But I know almost nothing about the subject of Random matrices. I prepared the content for Section 24.2 because I was intrigued by the results, wanted to learn, and thought it would be something entertaining with which to conclude. I did some of my reading in Chapter 2 of An Introduction to Random Matrices, by Anderson, Guiomet and Zeotouni, and then simplified their treatment to make it suitable for IA.

I thought it was nice that in showing you these two advanced topics, I could bring into play so many of the ideas we have had in our course: Markov and Chebyshev inequalities, moment generating function, sums of Bernoulli r.vs, Stirling’s formula, normal distribution, gambler’s ruin, Dyke words, generating functions, and the Central limit theorem.

So I hope you will consider the applicable courses in your next year of study. In Appendix H I have written some things about applicable mathematics courses in IB.