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Publisher:

Springer Verlag

Publication Date:

2007

Number of Pages:

210

Format:

Paperback

Series:

Universitext

Price:

49.95

ISBN:

978-0-387-71938-2

Category:

Textbook

[Reviewed by , on ]

Kenneth A. Ross

02/25/2008

A more accurate title for this book would have been, "A Concise Graduate Level Course in Probability Theory." The mentioned prerequisites are exposure to measure theory and analysis. Three appendices (29 pages) provide a brief but thorough introduction to the measure theory and functional analysis that is needed.

I contend the reader will also need a good deal of mathematical maturity and an undergraduate probability course. For example, the terms "moment" and "induced topology" are used, but never defined.

The extent of this book can be gleaned from some of the thirteen chapter titles: III: Martingales and Stopping Times; VI: Fourier Series, Fourier Transform, and Characteristic Functions; VIII: Laplace Transforms and Tauberian Theorem; and XII: Skorohod Embedding and Donsker's Invariance Principle. Chapter XI is titled: Brownian Motion: The LIL and Some Fine-Scale Properties. LIL refers to the Law of the Interated Logarithm (for Brownian motion). This chapter is five pages long, six if you include the exercises.

How do they accomplish so much? With the help of the publisher, they cram more than 40 lines of small print into 35 square inches per page. The proofs are complete, but often terse, and a lot of theory is in the exercises, especially in Chapter VI. This well-written book is full of wonderful probability theory. But to this old reader heading over the hill, it looks more like a very handy reference book than a text for a basic probability course.

Kenneth A. Ross (ross@math.uoregon.edu) taught at the University of Oregon from 1965 to 2000. His research area of interest was commutative harmonic analysis, especially where it has a probabilistic flavor. He is the author of the book Elementary Analysis: The Theory of Calculus (1980, now in 14th printing), co-author of Discrete Mathematics (with Charles R.B. Wright, 2003, fifth edition), and, as Ken Ross, the author of A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans (2004).

Random Maps, Distribution, and Mathematical Expectation.- Independence, Conditional Expectation.- Martingales and Stopping Times.- Classical Zero-One Laws, Laws of Large Numbers and Large Deviations.- Weak Convergence of Probability Measures.- Fourier Series, Fourier Transform, and Characteristic Functions.- Classical Central Limit Theorems.- Laplace Transforms and Tauberian Theorem.- Random Series of Independent Summands.- Kolmogorov's Extension Theorem and Brownian Motion.- Brownian Motion: The LIL and Some Fine-Scale Properties.- Skorokhod Embedding and Donsker's Invariance Principle.- A Historical Note on Brownian Motion.- References.- Index.- Symbol Index.

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