# Books by Albert Bharucha-Reid

Albert Bharucha-Reid wrote a number of important books and also edited two series each of three books. Below we give some information about these books by giving extracts from reviews, extracts from Prefaces and, in a couple of cases, the Contents.

Elements of the Theory of Markov Processes and Their Applications (1960)

Probabilistic Methods in Applied Mathematics, Vol. I (1968)

Probabilistic Methods in Applied Mathematics, Vol. 2 (1970)

Probabilistic Methods in Applied Mathematics, Vol. 3 (1973)

Probabilistic methods in applied mathematics. Vol. 1-3 (1968, 1970, 1973)

Random Integral Equations (1972)

Probabilistic Analysis and Related Topics, Vol. 1. (1978)

Probabilistic Analysis and Related Topics, Vol. 2 (1979)

Probabilistic Analysis and Related Topics, Vol. 3 (1983)

Random Polynomials (1986) with M Sambandham

**Click on a link below to go to information on that book**Elements of the Theory of Markov Processes and Their Applications (1960)

Probabilistic Methods in Applied Mathematics, Vol. I (1968)

Probabilistic Methods in Applied Mathematics, Vol. 2 (1970)

Probabilistic Methods in Applied Mathematics, Vol. 3 (1973)

Probabilistic methods in applied mathematics. Vol. 1-3 (1968, 1970, 1973)

Random Integral Equations (1972)

Probabilistic Analysis and Related Topics, Vol. 1. (1978)

Probabilistic Analysis and Related Topics, Vol. 2 (1979)

Probabilistic Analysis and Related Topics, Vol. 3 (1983)

Random Polynomials (1986) with M Sambandham

**1. Elements of the Theory of Markov Processes and Their Applications (1960), by A T Bharucha-Reid.**

**1.1. From the Preface:**

My purpose in this book is twofold: first, to present a nonmeasure-theoretic introduction to Markov processes, and second, to give a formal treatment of mathematical models based on this theory which have been employed in various fields. Since the main emphasis is on applications, this book is intended as a text and reference in applied probability theory. The book is divided into three parts: Part I, Theory; Part II, Applications; and Appendixes. Part I consists of three chapters which are respectively devoted to processes discrete in space and time, processes discrete in space and continuous in time, and processes continuous in space and time (diffusion processes). In the first two chapters we restrict our attention to Markov chains with a denumerable number of states, with particular reference to branching stochastic processes. The reader interested in chains with a finite number of states should consult, for example, the books of Doob, Feller, Fréchet, Kemeny and Snell, and Romonovskii. While the main purpose of Part I is to present the elements of the theory required for the applications given in Part II, we hope that this material can also serve as a text for an introductory course (senior or first-year graduate level) on Markov processes for students of probability and mathematical statistics and research workers in applied fields. Since applications are given only in Part II, the instructor can select examples from the chapters devoted to applications. The prerequisites for Part I, and the rest of the book, are a knowledge of elementary probability theory (the first nine chapters of Feller, say), mathematical statistics (Mood level), and analysis (Rudin level). Some knowledge of matrices and differential equations is also required. Part II consists of six chapters devoted to applications in biology, physics, astronomy and astrophysics, chemistry, and operations research. An attempt has been made to consider in detail representative applications of the theory of Markov processes in the above areas, with particular emphasis on the assumptions on which the stochastic models are based and the properties of these models. We have restricted our attention to a formal treatment of the models involved and have, therefore, refrained from presenting any numerical results. The fields we consider in Part II are not the only ones where Markov processes have found applications. We refer in particular to important studies in economics, psychology, sociology, and actuarial science. It is possible that applications in these areas will be treated in a subsequent volume, but for the present we have decided to restrict our attention to the biological and physical sciences and operations research. The three appendixes are concerned with generating functions, integral transforms, and Monte Carlo methods. In the first two appendixes we have listed some properties of generating functions and Laplace and Mellin transforms that are required in the text. The third appendix is devoted, in the main, to references dealing with the use of Monte Carlo methods in the study of stochastic processes occurring in different applied fields. A bibliography is given at the end of each chapter and appendix. In addition, a general bibliography of texts and monographs on stochastic processes is given at the end of the Introduction. With particular regard to Parts I and II, an attempt has been made to present bibliographies that are relatively complete and up to date. Reference in the text to items in the bibliography is denoted by the number of the item enclosed in brackets.

**1.2. Review by: M. S.**

*Population (French Edition)*

**17**(1) (1962), 181.

Bharucha-Reid brings together most of the basic elements of the theory of Markov processes that now make up a significant part of probability calculations. The first part is devoted to the exposition of fundamental theorems and properties of discrete or continuous processes in space and time. The general case is studied under the title of "Diffusion Processes" (Theories of Kolmogorov and Feller). The second part is devoted to applications of Markov processes in different domains such as biology, physics, astrophysics, and queuing theory.

**1.3. Review by: W M Gilbert.**

*Econometrica*

**32**(3) (1964), 457.

The author states that his purpose is to give a non-measure theoretic introduction to Markov processes, and to illustrate applications of Markov processes to biology, physics, chemistry, and operations research. The book is divided into two parts, approximately 150 pages of "theory" and some 250 pages of "applications." Prerequisites for reading the book are described as equivalent to the first nine chapters of Feller, Rudin's 'Principles of Mathematical Analysis', and Mood's 'Introduction to the Theory of Statistics'. Each of the three chapters on theory is followed by a rather short but interesting set of problems, and each of the nine chapters of the book is followed by a remarkably extensive bibliography. The book is written in a pleasant style. If occasionally lucidity seems to suffer from over simplification, it is still true that a reader with only the stated prerequisites can learn a good bit about the modern theory of Markov processes. This is a considerable accomplishment, for probability theory done with precision requires an imposing amount of analysis.

**1.4. Review by: George Weiss.**

*Science, New Series*

**132**(3435) (1960), 1244.

The staggering range of applications for Markov processes covers almost every subject from astronomy to zoology. Hence, the appearance of a book which brings together between its covers an introduction to most of these applications is indeed welcome. Although the choice of topics in this book is fascinating, it is probably a necessary corollary that not all topics are treated equally well. Some, such as the discussion of stochastic models in biology, fare quite well; while others, such as applications in astronomy and chemistry, are discussed only superficially and seem to have been transported unchanged from the original papers to the book.

**1.5. Review by: R Syski.**

*SIAM Review*

**5**(2) (1963), 173-174.

Markov processes can be studied either from the point of view of Measure Theory or in a less formal, but not necessarily less rigorous, manner by purely analytical methods. This book is written entirely from the latter point of view and it gives a very comprehensive survey of basic theory and numerous applications. A useful feature is the exposition of several results not available in English, and the enormous list of references (about 650 items) arranged according to chapters. The book is divided into two parts, one dealing with the theory and the other with applications. Theoretical chapters contain small selections of problems supplementing the main text.

**1.6. Review by: P A Moran.**

*Journal of the Royal Statistical Society. Series A (General)*

**124**(1) (1961), 94-95.

This book gives a detailed account of Markov processes in which the variate, which may be one or many dimensional, can only take a discrete set of values whilst the time variable is discrete or continuous. As the main aim of the book is to describe the many applications of this subject, which for the most part require the variate to take an enumerably infinite set of values, the theory of Markov chains with only a finite number of states is not discussed in any detail. Thus the formal object of this book is the same as that of the recent book by K L Chung but covers hardly any of the same ground since the latter is concerned with the deep general theorems of the subject. This means that the author can develop all the basic theory required without an elaborate foundation of basic measure and operator theory. ... Such a wide coverage means that each problem cannot be dealt with completely, but nevertheless the main facts are described and, since extensive references are given and a large number of special points are considered in the lengthy collections of examples at the end of each chapter, the coverage of the book is surprisingly complete. The particular value of the book is therefore that it gives a clear introduction to such a large number of important fields of research and summarizes the known results in such a way that the reader can easily follow up any particular line. This is a good and valuable book.

**1.7. Review by: R P Eddy.**

*Mathematics of Computation*

**15**(75) (1961), 304-306.

In recent years the notions of probability have become increasingly important in the building of models of the world around us. This is true, for example, in certain of the physical sciences, social sciences, and in the simulation of military and other operations. Sometimes probabilistic notions appear directly as basic ingredients of the model, sometimes indirectly as the result of applying Monte Carlo methods to the solution of certain types of functional equations. The mathematical abstraction of an empirical process whose development is governed by probabilistic laws is known as a stochastic process. A special class of these processes are Markov processes in which the development subsequent to a time t depends (probabilistically) only upon the state of the process at t and not upon its previous history. Their theory has been developed extensively in the past three decades and they have enjoyed increasingly wide application. Although the subject has been masterly treated from an advanced standpoint in Doob's classical 'Stochastic Processes', there remains a dearth of textbooks in English which are accessible to the beginning graduate student who has not yet mastered the subtleties of modern analysis, especially measure theory. Notable exceptions, for the case of a finite number of possible states, are Feller's highly regarded 'Introduction to Probability Theory and its Applications' and the recent book on 'Finite Markov Chains' by Kemeny and Snell. The present work is intended as a graduate-level text and reference in applied probability theory. According to the author's preface, its purpose is twofold: "first, to present a nonmeasure-theoretic introduction to Markov processes, and second, to give a formal treatment of mathematical models based on this theory which have been employed in various fields." Further, he states that the prerequisites are "a knowledge of elementary probability theory (the first nine chapters of Feller, say), mathematical statistics (Mood level), and analysis (Rudin level). Some knowledge of matrices and differential equations is also required."

**1.8. Review by: J L Doob.**

*Mathematical Reviews*MR0112177

**(22 #3032)**.

The book is written as a compendium rather than as a complete treatment of the topics covered. The stress is on manipulation rather than ideas.

**1.9. Review by: G. J.**

*The Incorporated Statistician*

**11**(2) (1961), 126-128.

Without any doubt, this is one of the most interesting books which has been written about probability theory since the pioneer works of Feller Bartlett. The work is divided into two parts, the first comprising three chapters and dealing with the basic mathematical theory of Markov processes and the second containing six chapters devoted respectively to applications in biology, physics (two chapters), astronomy and astrophysics, chemistry and finally queueing theory and operations research. There are also three appendices dealing with generating functions, integral transforms and Monte Carlo methods. Inevitably, it will be bought by a wide range of people with a wide spectrum of interests. In the writer's opinion, the type of person who is likely to benefit most from its reading is the specialist in probability theory whose interest lies with mathematical techniques rather than the individual who is genuinely interested in applications. By the latter is meant the practising scientist, statistician or operations research worker who measures his success by the extent to which the models constructed are physically interesting, are not over simplified, lead to predictions which are compatible with experimental or empirical data and consequently may be used with some confidence in further predictions or decisions. This is not to say that the author has ignored this aspect; on the contrary, he has displayed in the last six chapters a very sound grasp of an amazingly wide range of practical problems. However, after some highly interesting introductory material in these chapters, one is eventually led to the consideration of mathematical models usually consisting of differential-difference-integral equations for certain probabilities associated with the problem. From then onwards the author directs all his enthusiasm towards the application of as many techniques as are available to the "solutions" of these equations. There is also a battery of theorems relating to the uniqueness and existence of these "solutions." But what, in fact, does a "solution" consist of? The author remarks in a number of places that progress is being held up in many applications owing to the difficulties of obtaining explicit solutions to the non-linear equations which arise so often. In the writer's experience, if an equation is really worth solving, then with the aid of an electronic computer it should be possible to adapt numerical or sampling techniques so as to provide answers which may be compared with actual data. It is unfortunate, however, that in many applications, the models are so over simplified that they have no real practical interest even if they were capable of solution. With the advent of the computer, a new and exciting era is beginning in applied mathematics in which the field is wide open for obtaining approximate answers to real problems as opposed to exact answers to very approximate problems.

**1.10. Review by: John B Lathrop.**

*Operations Research*

**11**(2) (1963), 290-293.

This book, one of the McGraw-Hill Series in Probability and Statistics, was brought to our attention some time after publication. It is included in this review because of its fairly bulky chapter (65 pages) on "Applications in Operations Research: The Theory of Queues." Most of the book is concerned with stochastic processes and their uses in biology, physics, astronomy, and chemistry. The chapter of interest is devoted not so much to the theory of queues, as to the adaptability of the theory of Markov processes to queues. It opens with the observation that "... the stochastic processes occurring in the theory of queues are in general non-Markovian ...," shows how imbedded Markov chains can throw light on the associated non-Markovian processes, then turns briefly to other approaches to queues. Applications to telephone traffic and machine servicing are discussed. The several formulations of the machine-interference problem and the variety of measures developed make this a worthwhile operations-research reference when considering this type of problem or its analogues. Short discussions of balking and bulk service are intended only to illustrate the effects of variations in queueing situations.

**2. Probabilistic Methods in Applied Mathematics, Vol. I (1968), by A T Bharucha-Reid (ed.).**

**2.1. From the Preface.**

An examination of the current literature of science, engineering, and technology shows that modern probability theory is exerting a profound influence on the formulation of mathematical models and on the development of theory in many applied fields; in turn, problems posed in applied fields are motivating research in probability theory. In addition, probability theory is finding increased applications in, and interaction with, other branches of mathematics. This interaction is very desirable, and of particular interest are the uses of probabilistic methods in various branches of analysis and those developments in abstract probability theory which are of value in the field of applied mathematics. As is well known, progress in any science is highly dependent upon developments in methodology. Applied mathematics is a good example of this phenomenon. Within recent years research in stochastic processes, functional analysis, and numerical analysis has led to the development of powerful methodological tools for the applied mathematician; and there is considerable evidence that applied mathematicians are indeed using the results of research in the above fields to formulate theories and study more realistic mathematical representations of concrete natural phenomena. This present serial publication, which will be published in several volumes at irregular intervals, is devoted to the role of modern probability theory, in particular the theory of stochastic processes, in the general field of applied mathematics. We will not attempt to define "applied mathematics," but will assume that its objective is the development and utilization of mathematical methods to understand natural phenomena and technological systems quantitatively. We propose to cover a rather wide range of general topics, special topics, and problem areas in the mathematical sciences, and each volume of this serial publication will contain several articles, each being written by an expert in the field. Although each article will be reasonably self-contained and fully referenced, the reader is assumed to be familiar with measure-theoretic probability and the basic classes of stochastic processes, and the elements of functional analysis The individual articles are not intended to be popular expositions of the survey type, but are to be regarded, m a sense, as brief monographs which can serve as introductions to specialized study and research In view of the above aims, the nature of the subject matter, and the manner in which the text is organized, these volumes will be addressed to a broad audience of mathematicians specializing in probability theory and its applications, applied mathematicians working in those areas in which probabilistic methods are being employed, physicists, engineers, and other scientists interested in probabilistic methodology and its potential applicability in their respective areas.

**2.2. Review by: David W Miller.**

*Management Science*

**16**(1), Theory Series (1969), 141.

This is Volume I of what is intended to be a "multivolume work which will be published at irregular intervals." ... There are some interesting results on the probability distribution of the number of particles in the nth generation conditional on the fact that there are particles present. These results are more complete for the critical case but some conclusions are reached for the subcritical and supercritical cases. This book will be of interest primarily to specialists.

**2.3. Review by: George H Weiss.**

*SIAM Review*

**11**(1) (1969), 96-98.

This volume contains three long articles, two on different aspects of the general problem of linear operator equations with random coefficients, and the third on the theory of branching processes in neutron transport theory. ... The idea of a series on probabilistic methods in applied mathematics seems a very worthwhile one as judged from this first volume. I look forward to the continuation of this series.

**3. Probabilistic Methods in Applied Mathematics, Vol. 2 (1970) by A T Bharucha-Reid (ed.).**

**3.1. Review by: B Nicolaneko, G Papanicolaou and D Stevens.**

*American Scientist*

**59**(1) (1971), 119.

The second volume of this series contains the following three articles: A T Bharucha-Reid's "Random Algebraic Equations," Stanley Gudder's "Axiomatic Quantum Mechanics," and W M Wonham's "Random Differential Equations in Control Theory;" These are, as the editor explains, "Brief monographs which can serve as introductions to specialized study and research." It is assumed that the reader is familiar with modern probability theory, stochastic processes, and elements of functional analysis. The first article is an up-to-date survey of the statistical properties of the roots of algebraic equations whose coefficients are random variables. Proofs are given for some results but omitted for others, and the reader is referred to the original works. The article ends with a brief survey of problems associated with the study of random matrices.

**4. Probabilistic Methods in Applied Mathematics, Vol. 3 (1973) by A T Bharucha-Reid (ed.).**

**4.1. Contents.**

G Birkhoff, J Bona and J Kampé de Fériet, Statistically well-set Cauchy problems; D Dence and J E Spence, Wave propagation in random anisotropic media; V A LoDato, Stochastic processes in heat and mass transport; R Syski, Potential theory for Markov chains; and Chris P Tsokos and William G Nichols, On some stochastic differential games.

**4.2. Publisher's Description.**

Probabilistic Methods in Applied Mathematics, Volume 3 focuses on the influence of the probability theory on the formulation of mathematical models and development of theories in many applied fields. The selection first offers information on statistically well-set Cauchy problems and wave propagation in random anisotropic media. Discussions focus on extension to biaxial anisotropic random media; an effective medium description for a random uniaxial anisotropic medium and the resulting dyadic Green's function; evolution of the spectral matrix measure; and well-set Cauchy problems. The text then examines stochastic processes in heat and mass transport, including mass transport, velocity field, temperature transport, and coupling of mass and heat transport. The manuscript takes a look at the potential theory for Markov chains and stochastic differential games. Topics include formal solutions for some classes of stochastic linear pursuit-evasion games; solution of a stochastic linear pursuit-evasion game with nonrandom controls; problems of potential theory; and hitting distributions. The selection is a vital source of data for mathematicians and researchers interested in the probability theory.

**5. Probabilistic methods in applied mathematics. Vol. 1-3 (1968, 1970, 1973) by A. T.Bharucha-Reid (ed.).**

**5.1. Review by: Lorenzo Peccati.**

*Giornale degli Economisti e Annali di Economia, Nuova Serie*

**34**(11/12) (1975), 808-809.

These are the first three volumes of a series that aims to collect applied mathematical studies in which the use of probability calculations is significant. Generally speaking, it is not a question of opening new fields of investigation, but rather of the first attempts to settle matters already considered, but not in an organic way. There are many topics, and in fact, all at a good level.

**6. Random Integral Equations (1972), by A T Bharucha-Reid.**

**6.1. From the Preface.**

At the present time the theory of random equations is a very active area of mathematical research; and applications of the theory are of fundamental importance in the formulation and analysis of various classes of operator equations which arise in the physical, biological, social, engineering, and technological sciences. Of the several classes of random equations which have been studied, random integral equations (and random differential equations formulated as integral equations) have been studied rather extensively. This book is intended as an introductory survey of research on random integral equations and their applications. Research on random integral equations has, in the main, proceeded along two lines. There are the fundamental studies on random integral equations associated with Markov processes, these studies being initiated by Itô in 1951; and there are the studies on classical linear and nonlinear integral equations with random right-hand sides, random kernels, or defined on random domains, these studies being initiated by Spacek in 1955. In this book we attempt to present a complete account of the basic results that have been obtained in both of the above rather broad areas of research. The material in this book is presented in seven chapters. In Chapters I and 2 we present a survey of those basic concepts and theorems of probability theory in Banach spaces required for the formulation and study of random equations in Banach spaces. Chapter 3 is an introduction to the theory of random equations. In this chapter the material presented in Chapters I and 2 are utilized to discuss the basic concepts and to formulate various methods of solving random equations. Chapters 1-3 are intended as an introduction to probabilistic functional analysis, and they can be read independently of the remaining chapters. In Chapter 4 linear Fredholm and Volterra integral equations with random right-hand sides and/or random kernels are considered. Chapter 5 is devoted to the formulation and analysis of eigenvalue problems for some random Fredholm equations. In Chapter 6 we consider random nonlinear integral equations, in particular Hammerstein and Volterra equations. Finally, in Chapter 7 we study Itô random integral equations. This book is intended primarily for probabilists, applied mathematicians, and mathematical scientists interested in probabilistic functional analysis and the theory of random equations and its applications. Readers are assumed to have some knowledge of probabilistic measure theory, the basic classes of stochastic processes, and the elements of functional analysis in Banach spaces. Since there are many excellent texts and reference works available which cover the background material needed, no effort has been made to make this volume self-contained.

**6.2. Review by: Donald A Dawson.**

*SIAM Review*

**16**(2) (1974), 266-268.

Since the early fifties, the theory of stochastic equations, and in particular stochastic integral equations, has been the stage for intense activity. This has been stimulated in part by an immense wealth of applications beginning with the Langevin equation in statistical mechanics and N Wiener's emphasis on the role of randomness in the problems of cybernetics. Applications of random equations are now found in many areas of engineering, the physical and biological sciences and systems theory. In spite of the extensive literature on the subject, there are many potentially important aspects of the theory which have not yet been developed. For example, the theory of stochastic partial differential equations is in its infancy. In this book the author presents a bird's-eye view of the present state of the subject. The first half of the book consists of an introduction to "probabilistic functional analysis" which serves as the analytic foundation for the theory of stochastic integral equations. The theory of stochastic integral equations has grown along two parallel branches. In one, initiated by K Ito in 1951, the source of randomness is a white noise term leading to an important class of stochastic integral equations which has developed hand in hand with the theory of Markov processes. In the other, classical linear and nonlinear integral equations with random right-hand sides, random kernels or defined on random domains are studied. In the former, the distribution of the solution can be studied via the Fokker-Planck equation, while in the latter, the statistical properties of the solution are usually studied by consideration of the moments or cumulants. The second half of the book forms an introduction to both of these lines of development. Throughout the book, the theory is illustrated by a judicious choice of examples of stochastic integral equations and indications of their role in applications. ... Although the book covers a large number of topics, it does not cover them all to the same depth and omits others. For example, there is no discussion of stability theory, and a little more discussion of "dishonest" approximation methods would have been useful to the reader who hopes to understand much of the applied literature. However, these omissions are offset by an extensive bibliography of 448 references which cover both the topics discussed and others which have been omitted. To conclude, Bharucha-Reid has succeeded in giving an introduction to basic probabilistic functional analysis and a global perspective of the present state of the theory of random integral equations.

**6.3. Review by: J S Milton.**

*Mathematical Reviews*MR0443086

**(56 #1459)**.

This book will be of interest to probabilists, applied mathematicians, and mathematical scientists interested in probabilistic functional analysis and the theory of random equations and its applications. ... The material in the text is thoroughly documented and provides the reader with a core bibliography of the most recent work done in the field of random integral equations.

**7. Probabilistic Analysis and Related Topics, Vol. 1. (1978), by A T Bharucha-Reid (ed.).**

**7.1. From the Preface.**

Probabilistic analysis and related topics, which will be published in several volumes at irregular intervals, is devoted to current research in probabilistic analysis and its applications in the mathematical sciences. We propose to cover a rather wide range of general and special topics. Each volume will contain several articles, and each article will be by an expert in the subject area. Although these articles are reasonably self-contained and fully referenced, it is assumed that the reader is familiar with measure-theoretic probability, the basic classes of stochastic processes, functional analysis, and various classes of operator equations. The individual articles are not intended to be popular expositions of the survey type, but are to be regarded, in a sense, as brief monographs that can serve as introductions to specialized study and research. In view of the above aims, the nature of the subject matter, and the manner in which the text is organized, these volumes will be addressed to a broad audience of mathematicians specializing in probability and stochastic processes, applied mathematical scientists working in those areas in which probabilistic methods are being employed, and other research workers interested in probabilistic analysis and its potential applicability in their respective fields.

**7.2. Review by: Rodney Coleman.**

*Journal of the Royal Statistical Society. Series A (General)*

**143**(1) (1980), 79-80.

This is volume one of an occasional series of long articles on recent research in probabilistic analysis. This is defined in the preface to be that branch of the theory of random functions (stochastic processes) concerned with their analytical properties. This has expanded in recent years from being a study of their continuity, differentiability measure theory and so on. The reader is expected to have some familiarity with these branches of mathematics. The editor writes that the articles are not popular surveys, but rather introductory research monographs.

**7.3. Review by: J Gani.**

*International Statistical Review / Revue Internationale de Statistique*

**48**(2) (1980), 231.

In the preface of this book, the aim of the projected series of which it is the first volume is stated. This is to publish current research in probabilistic analysis, and its applications in the mathematical sciences. By probabilistic analysis is understood that branch of the theory of stochastic processes primarily concerned with the analytical properties of random functions. The book consists of five papers. all written by specialists.

**8. Probabilistic Analysis and Related Topics, Vol. 2 (1979), by A T Bharucha-Reid (ed.).**

**8.1. Contents.**

N U Ahmed, Optimal control of stochastic systems; Ryszard Jajte, Gleason measures; Peter A Loeb, An introduction to nonstandard analysis and hyperfinite probability theory; and Arunava Mukherjea, Limit theorems: stochastic matrices, ergodic Markov chains, and measures on semigroups.

**9. Probabilistic Analysis and Related Topics, Vol. 3 (1983), by A T Bharucha-Reid (ed.).**

**9.1. Review by: Richard Durrett.**

*American Scientist*

**75**(3) (1987), 322.

This book continues the tradition of its two predecessors. The volume brings us articles on the qualitative theory of stochastic systems, Langevin equations with "multiplicative noise," stability theory of stochastic difference systems, Markov properties of random fields, and the method of random contractors. Of the five articles the first one is my favourite. It is clearly written and deals with a subject of significant interest to probabilists. The other four articles are somewhat lacking in one of these aspects but overall of high quality. My main complaint about the volume is that most of the articles mention all sorts of applications in the introduction which they do not deliver, and after that usually all-too short section is over they plunge head first into unmotivated technical details.

**10. Random Polynomials (1986), by A T Bharucha-Reid and M Sambandham**

**10.1. Publisher's Description.**

Probability and Mathematical Statistics: A Series of Monographs and Textbooks: Random Polynomials focuses on a comprehensive treatment of random algebraic, orthogonal, and trigonometric polynomials. The publication first offers information on the basic definitions and properties of random algebraic polynomials and random matrices. Discussions focus on Newton's formula for random algebraic polynomials, random characteristic polynomials, measurability of the zeros of a random algebraic polynomial, and random power series and random algebraic polynomials. The text then elaborates on the number and expected number of real zeros of random algebraic polynomials; number and expected number of real zeros of other random polynomials; and variance of the number of real zeros of random algebraic polynomials. Topics include the expected number of real zeros of random orthogonal polynomials and the number and expected number of real zeros of trigonometric polynomials. The book takes a look at convergence and limit theorems for random polynomials and distribution of the zeros of random algebraic polynomials, including limit theorems for random algebraic polynomials and random companion matrices and distribution of the zeros of random algebraic polynomials. The publication is a dependable reference for probabilists, statisticians, physicists, engineers, and economists.

**10.2. Review by: Richard H Glendinning.**

*Journal of the Royal Statistical Society. Series A (General)*

**150**(3) (1987), 281.

Over the last ten years there has been a resurgence of interest in the application of probability theory to problems in analysis. Interest has centred on three topics: polynomials and power series with random coefficients, random Fourier series and the eigenvalues of random matrices. This book is concerned with the study of polynomials whose coefficients are real or complex-valued random variables. In most cases these random variables have zero mean. Interest centres on the study of their zeros. Random polynomials arise in many branches of statistics and applied mathematics where there are numerous potential applications. However the current level of knowledge is largely restricted to problems of theoretical interest. The author of this timely book are to be congratulated for gathering together the large volume of published work scattered among many journals. Each chapter describes a group of broadly related problems and ends with a comprehensive list of references. Most chapters include results obtained by simulation. Program listings are included in an appendix. Unfortunately there is no author index. The mathematical background required varies between chapters and can be quite demanding for a statistical audience. ... This book will be an important tool to applied mathematicians and probability theorists interested in the study of random polynomials.

**10.3. Review by: Jürgen vom Scheidt.**

*Mathematical Reviews*MR0856019

**(87m:60118)**.

his monograph is concerned with zeros of various types of random polynomials. It summarizes many results on this subject and connects several basic probabilistic properties of algebraic polynomials with numerical applications. This book is of interest to probabilists, statisticians, physicists, engineers and economists. ... This book can be recommended to everyone who wishes to enter this interesting subject of random polynomials or to obtain a survey on it.

**10.4. Review by G Samal.**

*Bull. Amer. Math. Soc.*

**21**(1) (1989), 182-183.

In describing a physical phenomenon, mathematical equations come into the picture whose coefficients carry some physical significance. These coefficients are random variables following some probability distributions, since they are computed from experimental data or from natural observations. Thus random equations arise from many applied problems in mathematical physics, engineering and statistics. A polynomial whose coefficients are random variables is called a random polynomial. Then the coefficients are subject to random error. Although there are a lot of applications of random polynomials in various branches of science and technology, it is only recently that attempts have been made to develop the theory of random equations. The study of random algebraic polynomials was initiated by Bloch and Pólya (1932). Motivated by this work, the systematic study of random algebraic polynomials was initiated by Littlewood and Offord in 1938. At present active research is being carried out in several countries including United States, Great Britain and India. Yet no comprehensive treatment of this subject was so far available in book form. ... Almost all available results on random polynomials are presented in this book. The book is the first of its kind in presenting a rigorous treatment on the subject. For everybody who works in the field the book provides all information.

Last Updated November 2019