Stochastic Processes

Here we have yet another Dover reissue of a classic textbook, this one from 1962. One reason it became a classic in its day was that it made an attempt to present this material at a lower mathematical level than earlier works on the subject. As a result, we often see the phrase “it may be shown” accompanied by a reference to one of those more advanced works. Even so, the prerequisites are non-trivial and might reasonably be considered to be the entire calculus sequence and a mathematical statistics course.

Of course, since then there have been newer textbooks as well as new results in the field. Parzen is no longer the most elementary introduction nor the most up-to-date. Still, it is hard to present this material at a level which is much lower than what we find here, and a glance at the Wikipedia article on stochastic processes suggests that the basics have not changed much in the past 55 years. In addition, Parzen’s text has the advantage of lucid English prose as well as a huge number of examples of applications.

At the same time, sometimes one wonders what learning objectives the author had in mind. The applications are presented very briefly in words, and then we jump to a theorem-proof presentation of the mathematics. In the Preface the author lists three goals for his book. The first and third have already been discussed here implicitly, and with those he succeeds. The second goal is to teach the reader probability model-building. It seems less certain that that goal has been attained. There is too large a gap between the simple examples and the theory. We do not see the steps by which certain aspects in a practical situation are abstracted and modeled. Generally, we get a lot of examples, and then the model is presented fully formed and we prove theorems about it. Rarely are we told the implications of those theorems in the application.

The exercises are extensive and interesting, with a reasonable balance between computations and proofs, but applications there are quite rare in the exercises as well. Hence the many examples of applications seem only half successful. They serve well the purpose of convincing the reader that the theorems can be used, but they (and the exercises) are less helpful in showing how.

A useful question to ask oneself in writing a book review is what sort of reader would be the best match for the book at hand. Here I think that matching reader would be someone interested in applications who already knows that their process of interest is a Wiener or Poisson or Markov process and so they can then turn to this book to learn more about same.

After a few years in industry, Robert W. Hayden (bob@statland.org) taught mathematics at colleges and universities for 32 years and statistics for 20 years. In 2005 he retired from full-time classroom work. He contributed the chapter on evaluating introductory statistics textbooks to the MAA's Teaching Statistics.