This is a lively introduction to the subject, focusing almost completely on Markov processes and omitting martingales and Brownian motion. The writing is very clear, and the book is loaded with examples and with exercises (selected exercises have answers in the back). The book is aimed at upper-division undergraduates, and assumes a modest knowledge of probability and a moderate knowledge of calculus, but no measure theory. The present volume is a 2013 unaltered reprint of the 1975 Prentice-Hall volume.
It starts with a brief summary of probability theory, and then two fairly in-depth chapters studying Bernoulli processes and Poisson processes as introductions to the general subject of stochastic processes. Then it covers Markov processes in much more depth. It ends with some general material on renewal theory and regenerative processes. There is also some queueing theory scattered throughout.
The book is not comprehensive, and picks and chooses which theorems to prove, but it does cover all the useful and interesting parts of the subject. I think this development works well, building up a good collection of specific examples and distributions which it then parlays into general theorems.
Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.
|1. Probability Spaces and Random Variables|
|2. Expectations and Independence|
|3. Bernoulli Processes and Sums of Independent Random Variables|
|4. Poisson Processes|
|5. Markov Chains|
|6. Limiting Behavior and Applications of Markov Chains|
|7. Potentials, Excessive Functions, and Optimal Stopping of Markov Chains|
|8. Markov Processes|
|9. Renewal Theory|
|10. Markov Renewal Theory|
|Appendix. Non-Negative Matrices|
|Answers to Selected Exercises|
|Index of Notations|