Basic stochastic processes. The Mark Kac lectures.

*(English)*Zbl 0681.60035
New York: Macmillan Publishing Company; London: Collier Macmillan Publishers. xiv, 258 p. £36.75 (1988).

The book is based on the lectures given by the late professor Marc Kac in a graduate course given to students in engineering, mathematics and physics. It does not use abstract mathematical concepts, such as e.g. measure theory. The text is entirely self-contained, including a preliminary chapter where the basics of probability theory are given, and including an appendix consisting of the relevant background material in linear algebra. Chapters one and two are also of preparatory character; they discuss standard Fourier series and Fourier techniques. Both the power and simplicity of this approach are made evident later throughout the text.

The main body of the text covers Poisson processes, shot noise, Gaussian processes including stationary ones, Ornstein-Uhlenbeck processes and Brownian motion and Markov chains related to Brownian motion. Included are descriptions of Bochner’s theorem, linear systems and the spectral density of stationary processes. Birth and death processes, martingales, branching processes, stochastic differential equations, and renewal processes are briefly mentioned.

There is an abundant set of problems (or rather, exercises) attached to each chapter including the zeroth one and naturally excluding the Appendix. The semi-intuitive method of the authors makes it possible to cover a surprisingly large material (e.g. even stochastic differential equations are mentioned).

Although the book is a real gain of the literature on applied mathematics, several critical remarks should be made. At first: not a single reference is included in the book. Citing original papers is surely unnecessary in such an introductory book. Still, a beginner also should appreciate references to textbooks on background material in linear algebra, probability theory, or complex analysis, or to previous introductory textbooks of similar character. Secondly: the nonrigorous style makes the book inappropriate for undergraduate students of mathematics as a first book. Thirdly: it might have been proper to include several examples taken from the field of chemical and biological applications. The authors have also attempted to preserve Marc Kac’s lecture style manifesting in shortest and most elegant proofs. To sum up: for all its merits the book is highly recommended to students of physics and engineering, and to teachers of stochastic processes. Nevertheless, I would not propose it to undergraduate students of mathematics.

A reader who finished this book may consult the book by M. Iosifescu, Finite Markov processes and their applications. (1980; Zbl 0436.60001), for exact mathematical results, and the books by N. G. van Kampen [Stochastic processes in physics and chemistry. (1981; Zbl 0511.60038)] or by C. W. Gardiner [Handbook of stochastic methods for physics, chemistry and the natural sciences. (1983; Zbl 0515.60002)] for further research problems.

The main body of the text covers Poisson processes, shot noise, Gaussian processes including stationary ones, Ornstein-Uhlenbeck processes and Brownian motion and Markov chains related to Brownian motion. Included are descriptions of Bochner’s theorem, linear systems and the spectral density of stationary processes. Birth and death processes, martingales, branching processes, stochastic differential equations, and renewal processes are briefly mentioned.

There is an abundant set of problems (or rather, exercises) attached to each chapter including the zeroth one and naturally excluding the Appendix. The semi-intuitive method of the authors makes it possible to cover a surprisingly large material (e.g. even stochastic differential equations are mentioned).

Although the book is a real gain of the literature on applied mathematics, several critical remarks should be made. At first: not a single reference is included in the book. Citing original papers is surely unnecessary in such an introductory book. Still, a beginner also should appreciate references to textbooks on background material in linear algebra, probability theory, or complex analysis, or to previous introductory textbooks of similar character. Secondly: the nonrigorous style makes the book inappropriate for undergraduate students of mathematics as a first book. Thirdly: it might have been proper to include several examples taken from the field of chemical and biological applications. The authors have also attempted to preserve Marc Kac’s lecture style manifesting in shortest and most elegant proofs. To sum up: for all its merits the book is highly recommended to students of physics and engineering, and to teachers of stochastic processes. Nevertheless, I would not propose it to undergraduate students of mathematics.

A reader who finished this book may consult the book by M. Iosifescu, Finite Markov processes and their applications. (1980; Zbl 0436.60001), for exact mathematical results, and the books by N. G. van Kampen [Stochastic processes in physics and chemistry. (1981; Zbl 0511.60038)] or by C. W. Gardiner [Handbook of stochastic methods for physics, chemistry and the natural sciences. (1983; Zbl 0515.60002)] for further research problems.

Reviewer: J.Tóth

##### MSC:

60Gxx | Stochastic processes |

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

60Jxx | Markov processes |