Probability theory. (English) Zbl 0980.60002

Varadhan, S. R. S.

Courant Lecture Notes in Mathematics. 7. Providence, RI: American Mathematical Society (AMS). New York, NY: New York Univ., Courant Institute of Mathematical Sciences, vii, 167 p. (2001).

The book is based on lectures given for first-year graduate students at the Courant Institute of Mathematical Sciences. There are seven chapters, with exercises. Chapter 1 introduces the concept of probability and provides background material from measure theory: the construction of measures, integration and relevant properties, distributions and expectation. In Chapter 2 characteristic and moment generating functions, as well as the notion of weak convergence of probability distributions, are discussed. In Chapter 3 the concept of independence of random variables and the standard limit theorems for sums and series of independent random variables are presented. This chapter concludes with the topics of infinitely divisible distributions and the law of the iterated logarithm. Chapter 4 covers conditional expectation, conditional probability and Markov chains. In Chapter 5 martingales are treated, the topics include martingale convergence theorems, the Doob decomposition theorem and Doob’s inequalities. The connection between Markov chains and martingales is illustrated with the aid of a set of examples. Chapter 6 deals with stationary stochastic processes and ergodic theorems. The last short chapter introduces the topics optimal control, optimal stopping and filtering.
With this book a succinct treatment of probability theory, essentially covering discrete time processes, has appeared. It does not consider permutation-coin-die-urn problems, it starts with the measure theoretic foundations. It is a solid and clearly written mathematical textbook, with the advantage of providing intuition into the subject. The book can be highly recommended to students as well as to lecturers.

Reviewer: Evelyn Buckwar (Berlin)

MSC:

60-01

Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory

Keywords:

measure theory; probability theory; limit theorems; Markov chains; martingales; stochastic processes