Stochastic processes. Theory for applications. (Q2879237)

This is a textbook on discrete and continuous stochastic processes for engineers and for graduate students. Its author is a professor emeritus at MIT and this text has been prepared over some 20 years.NEWLINENEWLINEChapter 1 is introductory and contains a review of basic results in probability theory. The introduction is oriented to fundamentals, including the law of large numbers, not to combinatorial probability. Chapters 2, 3, and 4 describe three of the simplest and most important classes of stochastic processes, namely, Poisson processes, Gaussian processes, and finite-state Markov chains. Chapter 5 on renewal processes is a basic tool for the remaining parts of the book. It contains such topics as stopping times for repeated experiments, expected number of renewals, renewal-rewards processes, and delayed renewal processes. A generalization of Markov chains to countable state spaces and continuous time is described in Chapters 6 and 7. Finally, Chapters 8, 9, 10 describe decision making, random walks, large deviations, martingales, and estimation.NEWLINENEWLINEThe book contains the theory which is needed for applications in engineering, operations research, physics, biology, economics, finance, statistics and other sciences. There are numerous illustrative examples and exercises.NEWLINENEWLINEThe main difference between this book and most other textbooks is that the author tries to use an intuitive and approachable style. The explanations may be longer but understandable for engineers. Sometimes this is a quite difficult task because knowledge of measure theory is neither assumed nor explained. As a result, some formulations suffer from oversimplification. For example, the author writes on p.\,13: ``If CDF \(F_X(x)\) of \(\mathrm{rv} X\) has a finite derivative at \(x\), the derivative is called the density\dots.'' It is not clear if the author means that the derivative should exist for some \(x\) or for all \(x\). A strange assertion can be found on p.\,10: ``\dots for any events \(A_1, A_2\) and any scalars \(\alpha_1, \alpha_2\), we have \(\mathrm{Pr}\{\alpha_1 A_1+ \alpha_2 A_2|B\} = \alpha_1 \mathrm{Pr}\{A_1|B\}+ \alpha_ 2 \mathrm{Pr}\{A_2|B\}\).'' The operation \(\alpha_1 A_1\) is not defined and an appropriate choice of \(\alpha_1, \alpha_2\) would lead to negative probability or to probability larger than 1. Nevertheless, such flaws are quite exceptional. The chosen style of explanation, when a concept or a proof are intuitively justified and then rigorously explained, should be attractive for a broad audience. The book may be recommended to teachers who will find here interesting motivation for definitions, theorems, and examples.