1.1. Review by: A C Aitken.
The Mathematical Gazette 22 (252) (1938), 518-519.

English authors do not seem to have engaged explicitly in the study of Markov chains, though problems of the "random walk", of the mixing of liquids, of molecules under random impacts and the like are instances. ... The tract before us gives a good introduction to Markov chains and a survey of their principal properties. Families of chains, reversibility, stable chains, periodic chains; a secular equation, like that for the modes of a vibrating system (to which the subject has a certain analogy); asymptotic properties; an ergodic principle, to the effect roughly that the probability that a sufficient remote member of the sequence takes the value $x$ depends less and less on a finite number of initial values taken; the tendency to normal distribution of sums of consecutive values; model urn-schemes and other examples; these are some of the topics studied. A very interesting point is that the limiting properties of relative frequency in a chain are often enunciable in terms of the limits as defined in classical analysis, not the restricted limits "en probabilité" of Fréchet. Thus in the very simple example of a bag containing two balls, one white, one black, subject to random drawings one at a time with replacement, the relative frequency of white balls in the total of balls drawn tends to $\large\frac{1}{2}\normalsize$ only in the sense that the probability of its differing from $\large\frac{1}{2}\normalsize$ by less than tends to 1 as $n$ tends to infinity. If, however, whenever a white ball is drawn we put a black ball in the bag, and whenever a black is drawn we put in a white, the relative frequency of white balls drawn tends to $\large\frac{1}{2}\normalsize$ in the true classical sense.

This book would make a good introduction to the second volume of Professor Frechet's Recherches Theoriques Modernes sur le Calcul des Probabilités, which incidentally gives valuable observations on the relation of Markov's chains to Poincaré's method of arbitrary functions.

2.1. Review by: Editors.
Mathematical Reviews MR0085637 (19,69h).

There are three parts. The first part is classical. The second, entitled "Stochastic processes", deals with Markov chains, ergodic problems, distributions, limit theorems, time series, aleatory mechanics, and mixing processes. The third part, entitled "Applications", includes statistical mechanics, mathematical statistics, demography and actuarial theory (13 pages are devoted to insurance other than life insurance), and finally there are 6 pages on stellar statistics. There is a bibliography of 221 titles.

3.1. Review by: Jim Douglas Jr.
Mathematical Reviews MR0180400 (31 #4635).

This textbook serves as an introduction to game theory. In the first four chapters the author treats the definitions of games, strategies, convex sets and convex polyhedra, and mixed strategies. In the next three he discusses coalitions, composition and decomposition of games, and excesses. In the final chapter the minimax theorem is applied to linear programming.

4.1. Review by: I J Good.
Journal of the Royal Statistical Society. Series A (General) 129 (2) (1966), 291.

A man's knowledge of a subject can be measured as a weighted geometric mean of his knowledge of its mutually orthogonal aspects. The main aspects of the subject of stochastic processes are pure mathematics and applications to the real world, the philosophical aspects being covered by probability and statistics at large. By this measure Onicescu has an impressive knowledge of stochastic processes, and for this reason it is a pleasure to browse in this book, although the author is difficult to understand when he writes like a physicist.

The book is a translation into French from the original Romanian of 1962, with little change of substance. The titles of Russian references are unfortunately left untranslated. There is a small minority of people, among whom was G H Hardy, who think that a mathematical book, being a work of art, should be unsullied by an index. The author of the present book, or perhaps the translator, is a Menshevik in this respect; we Bolsheviks think that a good index doubles the value of a book. Fortunately, the table of contents of this book is very detailed.

Among other things, the book discusses collectives, or random sequences of digits, information theory and entropy, time series and statistical mechanics, limits of sequences of random variables, estimation problems for Markov chains, random walks and models for polymers, relation of random walks to elliptic difference equations, and the transition of this theory to differential equations, multiplicative processes and nuclear reactors, statistical metric spaces including Wiener spaces, Brownian motion and stationary Markov processes and the relationship with potential theory. In all, a book of some individuality.

4.2. Review by: Joseph Leo Doob.
Mathematical Reviews MR0172314 (30 #2533).

There are eight chapters: (I) Sequences of random variables. This chapter includes a discussion of collectives, as defined by von Mises and his successors. (II) Random structures, irregular structures. Systems having a double structure. This chapter includes material on information theory. (III) Random sequences in time. In this chapter the author discusses Markov chains and prediction theory and suggests a new treatment of prediction, but the details of the method and its results are not given. (IV) Random walk. The corresponding finite difference equations. (V) Procedures for the integration of finite difference equations of elliptic type. (VI) Multiplicative processes. It is not made clear how these are related to the matrices with positive elements which are discussed. (VII) Statistical metric spaces. Random metric spaces. (VIII) Brownian motion and numerical solution of certain boundary value problems. For the most part the discussion is carried on in general terms without formal definitions and theorems, and the logic is not always easy to follow for a reader not at least as familiar with the subject as the author. ... The author's writing has a strongly individual flavour, and the book will be stimulating to those who can understand it.

5.1. Review by: Harry H Cohn.
Mathematical Reviews MR0259963 (41 #4592).

This book consists of three parts. In the first the author outlines some classical and contemporary models of probability theory (such as those of Bernoulli, Gauss, Poisson, Renyi, de Finetti, etc.) pointing out their common traits and flaws (from the author's point of view). In the second part a probabilistic theory for a system of ordered events is given. The third part of the book deals with a new theory initiated by the author. This theory is constructed directly on the set of events and its elements are, instead of random variables as in the classical theory, the integrals taken over the set of events and called sum-functions. Some aspect of probability such as convergence in various senses, the central limit problem, martingales, etc., are treated in accordance with this new theory. At the end, a paper of I Cuculescu on supermartingales from the point of view of sum-functions, is added to the book.

6.1. Review by: Hans-Otto Georgii.
Mathematical Reviews MR0418792 (54 #6828).

The book gives a detailed, easy-to-read presentation of the fundamental, meanwhile mostly classical, concepts when investigating the temporal evolution of mechanical systems with finite degrees of freedom. Apparently intended for readers who are familiar with analytical mechanics but not with probability theory, it aims to explain and justify the probabilistic concepts in statistical mechanics from a mechanical point of view. At the same time, it conveys to the reader the historical discussion of the fundamental ergodic problems. The book is divided into two parts: conservative and non-conservative statistical mechanics.

7.1. From the Preface.

The undertaking of constructing a theory of probability on a lattice of events without set support, on the one hand, and the difficulties encountered in the analysis of certain problems of mathematical logic, on the other hand, were the motive for most of the studies I present here.

I had first been led to the attempt to constitute a logic built on the propositional material of the established sciences, of mathematics in particular, and then to the systematic resumption of the study of the 'object', a study which must also lead to the various forms of mathematical creation and to which I had already devoted a good part of my former little book entitled: Principii de cunoastere stiintifica (Principles of scientific knowledge (Romanian), as well as an essay on determinism.

Linking mathematical logic to all the propositions of an established mathematical doctrine, as it is presented in a specific book, obliges us to limit the field of this logic to a single value, that of truth. There is no room for the false. I have tried to construct such a logic. Two things have resulted: the epistemological category and the ontological category.

These logics, the last of which leads directly to arithmetic, already affirm the fundamental function of the 'object' in each science. Most of the following chapters are devoted to it. I used there my former research, published in the book already quoted, especially concerning probability and statistics. But also, on the occasion of the study of some of the important objects of the mathematical sciences such as number, space, the principles and objects of invariant mechanics or those of Einstein's relativity, we were able to highlight all the unforeseen that intervenes in the formation of objects, and how much it goes beyond the limits that one believes are set for it by its axiomatic definition, which only gives a criterion of recognition and identification.

The point of view that we have adopted in this research has enabled us to elucidate, at least within the framework of mathematics, a certain number of difficulties, such as, for example, those concerning the paradoxes of set theory. Because we can consider as belonging to mathematics only the well-defined objects of this theory.

But also we are obliged to recognise that, as soon as we go beyond the framework of established mathematical doctrines and enter the philosophical domain concerning them, the propositions with 'I', which have no place in these doctrines, are in common and even necessary use. These propositions are not easily objectifiable and it is not even possible for us for the moment to affirm that they will be so one day according to the mathematical mode. And yet these propositions are not for that less necessarily linked to the construction of our science. It is for this reason that we gave in the last chapter of the book an opening towards this universe of propositions to which the 'Essai de philosophie de l'information' is devoted.

8.1. From the Preface.

Mechanics is the science of motion of the bodies of the material universe. For centuries, this frame of our experience has been conceived in various manners.

Sometimes in a very complete and precise manner, in the sense that the universe includes both stars, the solar system and the bodies of the earthly experiments. It was the case of Aristotle's mechanics and also that of Ptolemy's; but each of these three worlds conserved its special motion laws. A proper and universal mechanical principle, unique for the whole universe, was formulated for the first time by Archimedes, who did not try to build a corresponding theory. I am thinking of the principle of the lever that was, however, a universal principle of equilibrium between action and reaction.

In modern times the idea of material identity among all the bodies of the universe made its way; it had a first great victory with the Copernican theory. It also has been recorded as an indisputable truth in Leonardo's manuscripts, and it reached the final victory with Galilei's celestial discoveries.

The science of natural motion began as early as people became convinced of a substantial identity among the bodies of our universe. Limited, first, with Galilei, at the motion of the bodies under the strength of their weight here on the Earth, then with Newton for all motions on the Earth and in the planetary system, it has since aspired to include the whole universe. The latter being conceived as a unity realised by the motion of all its material bodies, in a system of interactions which maintain its stability.

The astronomic discoveries which happened in an accelerated rhythm since Newton to our days have strengthened this idea of unity of the universe which reached its culminating point in the discovery of the dilatation phenomenon governed by Hubble's law in its general sense.

To this success of the experimental knowledge, corresponds the creation of the theory of relativity, which gave us a handy geometrical image of this universe in its spatial wholeness as a representation of its material structure and of its general properties.

Coming back to the position of a science of motion in which the presence of the whole universe is to be found in each of its components, not by structural geometric ways, as in relativity, but by means of the analysis of the elementary processes of motion following the line of Newtonian thought, the Invariant Mechanics, without leaving the spatio-temporal frame of the old science, has found, together with gravity, a second inertial interaction, similar to an elastic repulsive force. Slightly sensitive to current distances, but very sensitive to intergalactic distances, this interaction is for a great part responsible for Hubble's expansion and at the same time for the stability of the universe, in its limits, necessarily finite.

The elaboration of this doctrine began with the studies "A New Mechanics of Material Systems" (Revista Universitatii si Politechnicii din Bucuresti, nr. 3, 1954), "Introduction à une mecanique invariante des systèmes (Revue de Math. pures et appliquees, Bucharest, t.V., 1957), "Une mécanique des systemes inertiaux. Une théorie de la gravitation. Une mécanique des petites distances" (Journal of Math. and Mech., t.5, no. 7, 1958).

These papers have been followed by others dealing with continuous systems as for example "Die Mechanik des starren Körpers", (Revue de mécanique appliquée, Bucharest, t.II, nr. 3, 1958}; "La mécanique de certaines particules stables" (Rediconti Sem. Math. Univ. Padova, t. 28, 1958), "On The Two Bodies Problem" (Revue de Math. Pures et Appliquées, Bucharest, t.V. nr. 1, 1960), "L'Univers antiminkowskien" (Revue de Physique Acad. R.P.R., Bucharest, t.V., nr. 3-4, 1961 ), as the Lectures at the Institute for Mechanics - University of Trieste - 1966, when I gave the name of "Invariantive Mechanics" to this new theory.

A synthesis of these theories was given in the volume "Mecanica invariantiva si Cosmologia" Editura Acadamiei R.S.R., 1974. On this occasion was established that the theoretical law of conservation of the impulse of the whole system of bodies of the Universe corresponds to Hubble's empirical law (C.R. Acad. Sci. Paris 1972). In the same volume was incorporated Mihaila's correction of the gravity law.

The full content of the present book was delivered as lectures at the "International Centre of Mechanical Sciences" Udine, Italy, to whose leadership I am honoured to appertain.

8.2. From the Introduction.

The principles of Newton's Natural Philosophy have a vast field of application which includes the motion of bodies at velocities extending from zero to the velocities of planetary motion, and involving distances which approach the large scale and the dimensions of our Galaxy and reach the threshold of the nuclear universe at the opposite end. Only second order effects of planetary motion are outside the realm of Newton's Natural Philosophy, which is, within bounds to be ascertained, a science of nature.

A science of motion, where velocities and distances may exceed the limits assigned above, requires a new structure which should be reduced to Newtonian one for velocities and distances consistent with it; it is only beyond these limits that quantitative differences should become apparent and significant thus allowing the new structure to be still considered, in a broader sense, a science of nature. In order to build this new Mechanics it is necessary to reconsider the simplest problems of motion, of inertial motion first, of a single material particle, then of two or more material particles, for systems such as the solid body of the Newtonian mechanics and ultimately of the motion in a field.

The inertial motion of a material particle reveals an unexpectedly vast content of the inertia of a material mass in motion including its Einsteinean characteristics and other features besides. The law of inertial motion of a system of material particles appears as a theory of gravitation and at the same time of the expansion of the universe.

Likewise, the introduction of the field discloses a wealth of possible forms which are at the disposal of the physicist and require only to be recognised and eventually classified according to specific features which elude, at least on a first examination, the criteria of mechanics.

9.1. Review by: Alan J Mayne.
Operational Research Quarterly (1970-1977) 28 (4, 2) (1977), 1030.

This book applies mathematical concepts, such as lattice theory and Boolean algebras, to the construction of a more general formulation of probability theory. Most of the chapters develop the mathematical framework of this theory, but the final chapters discuss its application to random processes, and especially to Markov processes. The presentation is very condensed and very abstract, though clearly expressed, so that this book is most likely to interest mathematicians and probability theorists. Although its publishers claim that it is "also interesting by the vast perspective which it opens for the applications of the new theory", it is disappointing that no applications to specific practical problems are mentioned. There may well be results in the book that could be applied to queueing theory and other topics of interest to operational researchers, but it seems likely that a considerable effort will be needed to extract them!

9.2. Review by: Demetrios A Kappos.
Mathematical Reviews MR0448469 (56 #6775).

A probability theory on Boolean algebras can be introduced if the (random) events are regarded as abstract propositions of a logic. Since 1948 it has been shown by the reviewer in a series of papers and in his book [Probability algebras and stochastic spaces, Academic Press, New York, 1969] that the algebraic measure theory on Boolean algebras of Carathéodory is well suited to introduce all the basic notions of such a theory. The first author, in 1958, initiated a probability theory on Boolean algebras based on the concept of sum functions. Later he and other Romanian mathematicians worked in this direction, and many interesting results were obtained concerning the treatment of topics met in probability theory. In 1969, the first author [Principles of probability theory (Romanian), Bucharest, 1969] set the historical and logical framework of the new theory. Now in the present book this theory is explained systematically on a new basis and applied in the following problems: independence, strong law of large numbers (proof using martingales), martingales (discrete time), Markov processes, processes with independent increments, the strong Markov property for denumerable stopping times, Doeblin's theorem and the ergodic theorem for measure-preserving transformations. The authors give a description of the contents of the book and emphasise that the book is intended for students interested in a new probability theory and in the enlargement of their mathematical knowledge.

9.3. Review by: Sergiu Rudeanu.
Bulletin mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, Nouvelle Série 21 (69) (1/2) (1977), 220-222.

The book is self-contained; an impressive deal of work has been done in order to construct all Boolean prerequisites. The proofs are often highly technical, but always carefully written; the only difficulty which the reader might encounter is the lack od a list of symbols and terms.

This book is doubtless a welcome addition to the literature and it should stimulate further research.

10.1. Review by: Anca-Maria Precupanu.
Mathematical Reviews MR0725331 (86a:60080).

The aim of this book is to present the main results on random functions that are almost periodic in probability; these functions were considered, for the first time, by Onicescu and V I Istratescu in 1975.

The book is divided into eight chapters. Chapter I contains various theorems of Weierstrass type which are used in Chapter III. Chapter II contains a brief presentation of the basic results on classical almost periodic (a.p.) functions and asymptotically almost periodic (a.a.p.) functions. Chapter III is devoted to the study of some theorems of Weierstrass type for random functions. To this end the authors introduce some metrics with respect to which they define the notions of continuity and differentiability of random functions, and the notions of convergence and uniform convergence of sequences of random functions. Also, they consider the concept of integration in probability which is then related to the mean convergence and the mean continuity of random functions. In Chapter IV the authors establish some elementary properties of random functions that are a.p. or a.a.p. in the mean of order 2, define the mean and associate a Fourier series to every random function a.p. in the mean of order 2.

The most important chapter is Chapter V which contains the basic results concerning the random functions that are a.p. in probability. Here the authors give the definitions of the random functions that are a.p. in probability by a Bohr-type, a Bochner-type, or an approximation property and establish some relations between them. They also point out sufficient conditions for a random function obtained by means of an invariant transformation to be a.p. in probability, properties of P-summation functions generated by a random function that is a.p. in probability and some ergodic properties of functions that are a.p. in probability. The results of Chapter V are extended in Chapter VI to the case of random functions with values in the space of m-dimensional random vectors. Following the same scheme as in Chapter V, the authors study, in Chapter VI, some properties of functions that are weakly a.p. in probability. The last chapter contains certain applications of the a.p. functions in probability to differential equations and to systems of differential equations with random parameters.