Customer Reviews

Theory of Probability

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This is a non-measure theoretic treatment of probability whose scope is similar to the first volume of Feller. Although each topic begins with a few simple but insightful examples, what they result is somewhat in-depth discussion. There are minor problems in writing and editing, but I found none so serious. This is another amazing Russian mathematics book that begins at an elementary level but ends with some real treatment, and this book is totally accessible to those who did not rigorously study analysis or measure theory. Highly recommend this to those who seriously desire to study probability.

What is also apparent as a feature of this book is that Gnedenko often notes the details of history. He spends a 70-page-appendix for this purpose besides many little footnotes. The author often integrates historical evidences as a part of the treatment. I believe one still benefits a lot from this historical notes, even if already familiar with the subject.

This book has chapters on the author's specialty topics: limit theorems and infinitely divisible distributions. This book also has a chapter on stochastic processes and elements of statistics, the latter which Feller's first volume does not touch. However, these brief treatments are only for introductions to each subject.

I bought my Chesea copy for about 5 bucks while in grad. school, look at it more and more these days. E.g., the derivation of Kolmogorov's 1st pde, the backward time pde for general variable diffusion coefficients, therefore not assuming space or time translational invariance, is hard to find elsewhere in readable form. That equation is important: to within a trivial time transformation it's the (generalized) Black-Scholes pde of 'fair' option pricing in finance. What's lacking is an example of a nontrivial example solved in closed form, one where the transition density g(x,t:x',t') is not symmetric. Let the reader work out the Green function for the drift-free lognormal model for herself in order to see an example that provides insight. In any case, I've looked at Risken and Gardiner and did not find them useful for research. I refer even more often to Stratonovich (Vol. I) and Wax, the classics in the field.