Knowing the odds : an introduction to probability / John B. Walsh

This textbook is ambitious since it proceeds from basic principles of probability theory to advanced stochastic processes, as can be seen from the table of contents: 1. Probability spaces; 2. Random variables; 3. Expectations II: The general case; 4. Convergence; 5. Laws of large numbers; 6. Convergence of distributions and the CLT; 7. Markov chains and random walks; 8. Conditional expectations; 9. Discrete-parameter martingales; 10. Brownian motion.

Up to Chapter 7 the author tries to avoid measure and integration theory, but the price to be paid is a rather uncomfortable extension of the expectation of discrete random variables to that of integrable general random variables. It is indicated that the expectation is a Lebesgue integral, but this remains vague since the Lebesgue integral is not defined in this book. The author’s approach of avoiding measure and integration theory breaks down in Chapter 8 which, with regard to martingales and Brownian motion, introduces the conditional expectation in the usual way but without proving its existence.

Reviewer’s remark: The presentation suffers from several deficiencies. For example, the discussion of distributions is incomplete since the joint distribution of a finite family of random variables is used in Section 2.8, while the joint distribution of two only random variables is defined later in Section 3.6 and also since the distribution of a stochastic process referred to in Theorem 7.2 is not defined at all. The discussion of independence is incomplete as well since it does not provide all of the results needed in the chapters on stochastic processes.

I would not dare to give a lecture on the basis of this book. Nevertheless, the book may very well serve as a valuable complementary source since the author provides a lot of interesting comments and examples, in particular with regard to financial mathematics. (Zentralblatt)

Bibliogr. p. 413-414. Index