Review of book ‘Probability Theory and Stochastic Processes’
Virgilio Gómez‐Rubio · Journal of the Royal Statistical Society Series A (Statistics in Society) · 2025
This book provides a comprehensive introduction to stochastic processes from a very mathematical perspective. In this regard, the book tries to be self-contained by providing an introduction to formal probability as well as some other advanced topics. The book is structured in three parts that cover an introduction to probability theory, a description of standard stochastic processes, and some other advanced topics. First of all, I find the introduction to probability theory quite appealing. Not only does it provide a solid summary of probability and measure theory (with a focus on the Lebesgue integral) but also these contents are framed within the context of stochastic processes. I particularly enjoyed the chapter on convergence, in which different types of convergence are nicely assorted and discussed. Regarding the main part on stochastic processes, the book dives into the most frequent stochastic processes (Markov processes, Brownian motion, and other). Here, I would highlight the introduction of some specific processes, such as spatial processes and queuing processes. Finally, the section on advanced topics includes an introduction to martingales, stochastic integration, and ergodic processes, among other topics. The book is written in an easy to follow style (mathematical complexity aside). While most of the contents follow the typical structure of typical mathematical texts (i.e. definitions followed by theorems and their proofs, etc.) the author has included plenty of notes and remarks to provide insight of some particular important points. In addition, the exercises included may point to relevant comments about the main content or simply develop relevant examples and case studies. Furthermore, each chapter ends with its own list of exercises. Altogether, I find these type of contents very useful when preparing lectures and exercises for my own students. No data is used in this book review.