Description
Book SynopsisProbability theory has been extraordinarily successful at describing a variety of phenomena, from the behaviour of gases to the transmission of messages, and is, besides, a powerful tool with applications throughout mathematics. At its heart are a number of concepts familiar in one guise or another to many: Gauss' bell-shaped curve, the law of averages, and so on, concepts that crop up in so many settings they are in some sense universal. This universality is predicted by probability theory to a remarkable degree. This book explains that theory and investigates its ramifications. Assuming a good working knowledge of basic analysis, real and complex, the author maps out a route from basic probability, via random walks, Brownian motion, the law of large numbers and the central limit theorem, to aspects of ergodic theorems, equilibrium and nonequilibrium statistical mechanics, communication over a noisy channel, and random matrices. Numerous examples and exercises enrich the text.
Trade Review'… packs a great deal of material into a moderate-sized book, starting with a synopsis of measure theory and ending with a taste of current research into random matrices and number theory. The book ranges more widely than the title might suggest … There are numerous exercises sprinkled throughout the book. Most of these are exhortations to fill in details left out of the main discussion or illustrative examples. The exercises are a natural part of the book, unlike the exercises in so many books that were apparently grafted on after-the-fact at a publisher's insistence. McKean has worked in probability and related areas since obtaining his PhD under William Feller in 1955. His book contains invaluable insights from a long career.' John D. Cook, MAA Reviews
'The scope is wide, not restricted to 'elementary facts' only. There is an abundance of pretty details … This book is highly recommendable …' Jorma K. Merikoski, International Statistical Review
Table of ContentsPreface; 1. Preliminaries; 2. Bernoulli trials; 3. The standard random walk; 4. The standard random walk in higher dimensions; 5. LLN, CLT, iterated log, and arcsine in general; 6. Brownian motion; 7. Markov chains; 8. The ergodic theorem; 9. Communication over a noisy channel; 10. Equilibrium statistical mechanics; 11. Statistical mechanics out of equilibrium; 12. Random matrices; Bibliography; Index.
