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

Auteur: Walsh, John B. - AuteurType de document: MonographieCollection: Graduate studies in mathematics ; 139Langue: anglaisPays: Etats UnisÉditeur: Providence (R.I.) : American Mathematical Society, cop. 2012Description: 1 vol. (XVI-421 p.) ; 26 cm ISBN: 9780821885321 ; rel. Note: 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)Bibliographie: Bibliogr. p. 413-414. Index. Sujets MSC: 60-01 Probability theory and stochastic processes -- Instructional exposition (textbooks, tutorial papers, etc.)
60A05 Probability theory and stochastic processes -- Foundations of probability theory -- Axioms; other general questions
60E05 Probability theory and stochastic processes -- Distribution theory -- Distributions: general theory
60F05 Probability theory and stochastic processes -- Limit theorems -- Central limit and other weak theorems
60J10 Probability theory and stochastic processes -- Markov processes -- Markov chains (discrete-time Markov processes on discrete state spaces)
60J65 Probability theory and stochastic processes -- Markov processes -- Brownian motion
En-ligne: Zentralblatt | AMS
Location Call Number Status Date Due
Salle R 05858-01 / 60 WAL (Browse Shelf) Available

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

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