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Discrete Stochastic Processes and Applications

Jean-François Collet
Publication Date: 
Number of Pages: 
[Reviewed by
John K. McSweeney
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This text serves as an introduction to stochastic processes which are discrete in space but may be discrete or continuous in time. The style is approachable and conversational, but the material is presented with full rigor, using a good deal of mathematical sophistication and elegance. The same result may be arrived at via different mathematical approaches (for example, the Markov chain ergodic theorem is initially proved entirely without the language of linear algebra.) The discussion of convexity is presented in an intuitive fashion which naturally unites the geometric viewpoint, the related algebraic manipulations, and the probabilistic intuition required to introduce the idea of entropy. Each chapter has a good number of exercises of varying levels of difficulty.
The ideal audience for this text would be students or practitioners in that sweet spot where mathematical rigor is important, but who don't have a foundation in measure theory. The lack of a measure-theoretic framework means that the concepts of martingales and stopping times are wholly absent; it's unfortunate that a text like this with such a nice blend of rigor and intuition couldn't fit in a treatment of, for example, the optional stopping theorem (a result that is at once totally intuitive from a gambling standpoint, but is very delicate to state and prove properly.) 
A solid undergraduate-level foundation in probability and linear algebra are essential for making the most of this text; it could support a second probability course for students interested in signals, coding, and an introduction to viewing probability from an information-theoretic viewpoint. However, an advanced undergraduate student taking a "terminal" probability course might find more suitable a text with less rigor and more applications along the lines of queuing theory, such as Sheldon Ross's Introduction to Probability Models. Additionally, a reader hungry for an application-focused treatment will find this text lacking, despite the title. There are indeed standard applications such as random walks and birth/death processes mixed in throughout the text and a short dedicated chapter on binary coding. However, due to the amount of space required to rigorously present the theory and develop the required mathematical intuition, there is necessarily a limited amount of practical material that might interest, for example, a master's student in electrical engineering.
That being said, this is a text that faculty should find very handy as a reference; it's got the most comprehensible introduction to the notion of (probabilistic) entropy that this reviewer has seen, and will be an excellent reference for Markov Chain theory for an instructor struggling to determine how much rigor to introduce into a course on Markov chains.


John K. McSweeney is an associate professor in the department of mathematics at the Rose-Hulman Institute of Technology in Terre Haute, Indiana. He teaches courses across the undergraduate spectrum but has an affinity for problems in biology and recreational mathematics where probability can be brought to bear.

See the table of contents in the publisher's webpage.