# Paul Lévy's books

Paul Lévy published ten books which we list below. We give separate volumes and second editions of books as separate entries. We give some information about some of these books, such as extracts from some reviews.

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Leçons d'analyse fonctionelle

Calcul des probabilités

Analyse fonctionnelle

Cours de Mécanique

Cours d'analyse Vol 1

Cours d'analyse Vol 2

Théorie de l'addition des variables aléatoires

Processus stochastique et mouvement brownien

Problemes concrets d'analyse fontionelle

Théorie de l'addition des variables aléatoires (2nd edition)

Le mouvement brownien

Processus stochastique et mouvement brownien

Quelques aspects de la pensée d'un mathématicien

Click on a link below to go to information about that book

Leçons d'analyse fonctionelle

Calcul des probabilités

Analyse fonctionnelle

Cours de Mécanique

Cours d'analyse Vol 1

Cours d'analyse Vol 2

Théorie de l'addition des variables aléatoires

Processus stochastique et mouvement brownien

Problemes concrets d'analyse fontionelle

Théorie de l'addition des variables aléatoires (2nd edition)

Le mouvement brownien

Processus stochastique et mouvement brownien

Quelques aspects de la pensée d'un mathématicien

**1. Leçons d'analyse fonctionelle (Gauthier-Villars, Paris, 1922), by Paul Lévy.**

**1.1. Preface by: Jacques Hadamard.**

The task of bringing Functional Calculus to the reader's attention might have seemed important a quarter of a century ago; it is much simplified today.

Logically speaking, the functional calculus should have been constituted from the very birth of the notion of definite integral or, more exactly, when with this notion was elaborated the very notion of function in the sense of Dirichlet, the function being considered as defined not by such or such a series of analytical operations, but through the knowledge of all its values; the definite integral precisely brings into play the set of these values and thus constituted a first fact of functional calculus. It was not noticed, however, that there was here, and in the Calculus of Variations which appeared soon after, a new branch of Science. It was reserved for M Pincherle and above all for M Volterra to identify its individuality and show its importance. This importance, which it is no longer possible for a mathematician to ignore, after the Memoirs of M Volterra on the functions of lines and the Lessons that the illustrious geometer taught at the University of Paris on the same subject.

But another fact has particularly contributed to making the discipline of which we speak natural and familiar to all geometers: I mean the theory of the Dirichlet problem and the new view which has opened up this subject to us by the discovery of M Fredholm. This taught us that even in the study of a partial differential equation it can be essential to consider the relations of one of the values of the unknown function not only with the infinitely close values, but also with all those that this unknown can take in its entire domain of existence. In short, today it is impossible to treat the theory of partial differential equations without relating it to functional calculus.

If it has become unnecessary to dwell on the subject of the Volume we are about to read, is it necessary to present the author to the public? Hardly more necessary. We know by what remarkable beginnings M Paul Lévy stood out in the scientific world. We know how, next to the generalisation of the notion of the total differential, as it results from the research of M Volterra, he similarly extended to the new field the notion of the completely integrable total differential equation and how this new generalisation has cast over all the most vivid and fruitful light.

For five years, M Lévy gave La Patrie an activity that would have been precious for Science. To this, he again devoted himself entirely. At the same time as the work of M Volterra and his own, he continues another which promised to be admirable: that of R Gateaux, killed by the enemy in September 1914, and whose work, so quickly and so brutally interrupted, had opened up the new path of integration to functional calculus.

The reader will see to what degree of clarity and harmony this lofty theory has been brought about by the efforts of such scholars. There is no doubt, and this is the main thing, that it also finds an opportunity for important applications and new progress.

**1.2. Review by: T H Hildebrandt.**

*Amer, Math. Monthly*

**32**(6) (1925), 309-311.

No doubt one of the largest fields in analysis to be opened up during the last three decades has come through the discovery that many of the results which can be generalised from functions of a single variable, i.e., a linear space, to functions of n variables, or n-space, can also be extended to the function space, i.e., the space in which the variable is a function of a continuous variable. The impetus for this development came from the study of the linear integral equation, with the deduction of its solution from the solution of algebraic equations in n variables, the work of Volterra and Fredholm. The influence of the former has been potent for further development, and one feels something of his influence in the work under discussion, a work which is calculated to open up new vistas and possibilities in the field of functionals, i.e., functions of functions.

The purpose of the volume as stated in the introduction is to extend to function space (1) the elementary results of the differential calculus, (2) the theory of partial differential equations, and (3) the theory of multiple integrals. A closer examination of the work reveals the fact that the author's primary interest is in the second of these two topics, the theory of partial differential equations in function space, and this has influenced to a large extent the choice of material. As a consequence, one finds that many matters which one might expect to be treated in one of the first two books on this subject are touched upon only lightly, and other matters which may not seem to be of major importance in the field of functionals are treated extensively. Such however is the privilege of authorship.

...

There is an enormous amount of material in this volume, and most of it is not of an elementary character, although the author has tried to give it an elementary tone by introducing a chapter on Lebesgue integration and linear integral equations in the first part. In order to follow the author easily in his deliberations, it is desirable to have a considerable acquaintance with the literature on functionals, and a thorough knowledge of partial differential equations of the first order in n-space, as well as a thorough acquaintance with the solution of Laplace differential equations. From the point of view of the reviewer the author's style is not lucid; it is not what one usually expects to find in the French mathematical presentations, and particularly in most of the books of the Borel series of monographs. And there are misprints, some of them rather puzzling. But much of the contents of the book is worth making the effort to understand, and it is true that it points the way towards much that is of value in the direction of new research in the functional field.

**2. Calcul des probabilités (Gauthier-Villars, Paris, 1925), by Paul Lévy.**

**2.1. Contents.**

**I - THE PRINCIPLES OF CALCULATION OF PROBABILITY**

**1 - Subjective probability and the principles of the mathematical theory.**

Definition of probability from the subjective point of view.

Measure of probability.

Rational and irrational probabilities.

First principle: principle of addition of probabilities or principle of total probabilities.

Definition of a probability law.

Experiments involving an infinity of cases.

Second principle: principle of compound probabilities.

Bayes principle.

A posteriori probabilities and probability of the causes.

Relationship between Bayes' principle and the principle of compound probabilities.

**2 - The verifiable consequences of the theory. Probability and frequency.**

The notion of a very unlikely event.

First remarks on the repetition of experiments.

Case of the coin toss.

Application of the principle of compound probabilities.

Generalisation of previous results.

The law of large numbers.

The probability measured by the frequency.

**3 - Objective value of the probability.**

Chance, according to Poincaré.

Example taken from mathematics.

Objective value of the notion of very unlikely event.

Objective value of probability.

Study of roulette.

Study of the shuffling of cards.

Conclusion on probabilities in games of chance.

Miscellaneous remarks.

**4 - Various notions relating to the laws of probability. Gauss's law and its relation to the law of large numbers.**

The notion of probable value.

Relationship between the notion of probable value and that of centre of gravity.

Averages of the various orders.

The order of magnitude of an error and that of a possible variable.

First applications of the previous concepts.

Demonstration of the law of large numbers by Chebyshev's method.

The notion of reduced law.

Gauss's law and the coin toss game.

Gauss's law and games of chance in general.

Gauss's law and the theory of errors.

The notion of precision of a measurement.

Other applications of Gauss's law.

**5 - Probability deduced from experiment and the statistical sciences.**

The probability deduced from experiment.

Discussion of a problem introducing subjective probability.

Need to supplement the experimental results by a theoretical study of the conditions of the experiment.

Special case of the previous problem: recognise if a given sequence of digits has been chosen at random.

General information on statistical problems.

First example: the size of the population of a country.

Second example: study of mortality.

General observations on insurance operations.

**6 - Critique of the theory of probable gain.**

The notion of probable value and games of chance.

Fundamental principle according to which one should, in principle, seek to make the probable gain as large as possible.

First restriction to the fundamental principle, relating to the case where the number of experiments is not sufficient to allow the application of the law of large numbers.

Case of the lottery ticket.

The Saint Petersburg paradox.

Second restriction to the fundamental principle.

Distinction between the value of a gain and the interest there is in achieving it, or moral gain.

The advantage of the richer of the two players in a fair game.

The theory of moral hope.

Conclusions and remarks.

**II - MATHEMATICAL THEORY OF PROBABILITY**

**1 - General notions on the laws of probability and on the theory of sets.**

Definition of a one-variable probability law.

Countable sets.

Countable sets of points located on a line.

Zero measure sets.

First class of masses; finite masses concentrated at certain points.

Second class; masses of summable density.

Third class; masses distributed in a set of zero measure, without any point containing finite mass.

General case.

Notions on the two-variable probability laws.

**2 - Probable values, characteristic coefficients and the characteristic function.**

Analytical expression of the probable value.

The integral of Stieltjes.

Properties of the means of the various orders.

Characteristic coefficients.

Characteristic function.

Integral of Dirichlet.

Determination of a probability law by its characteristic function.

Case of absolutely continuous laws. Fourier formulas.

Determination of the masses of the first class. Relation with the theory of Fourier series. Miscellaneous remarks.

Conditions for a function to be characteristic. Properties of a probability law, deduced from its characteristic function.

Example of continuous law. Gauss's law.

Another example. Cauchy's law.

**3 - Composition of the laws of probability.**

General notions.

Addition of probable values.

Composition of characteristic coefficients.

Composition of characteristic functions.

Application to the laws of Gauss and Cauchy.

Direct formulas for the composition of probabilities.

Application of the direct method to the laws of Gauss and Cauchy.

**4 - Variable probability laws. The notion of reduced law.**

Limit of a probability law.

Limit of the characteristic function.

Reciprocal of the previous theorem.

Laws of probability tending to a given type.

The notion of reduced law.

First reduction process: probable value and mean squared error.

Application to the probability laws defined by their characteristic functions.

Generalisation of the first reduction process.

Second reduction process.

**5 - The law of large numbers.**

Stirling's formula.

Various statements of the law of large numbers.

The Chebyshev and Poisson theorems.

Study of the coin toss by the direct method.

First generalisation of the results obtained by the direct method.

Second generalisation.

Third generalisation.

Application of the concept of characteristic function.

Fundamental theorem establishing the role of Gauss's law.

Remarks on the previous theorem.

Representation of the laws of probability in functional space.

The example of M Lindeberg.

M Lindeberg's point of view.

M Lindeberg's method.

Comparison of the different methods used to demonstrate the law of large numbers.

**6 - Exceptional laws.**

General notions.

Search for stable laws.

Area of attraction of the $L_{\{\alpha,\beta\}}$ laws.

Existence of the laws $L_{\{\alpha,\beta\}}$.

Relations between the probabilities of the large values of the variable and the shape of the characteristic function at the origin.

Study of the composition of a large number of errors in the case of laws which are not $L_{\{\alpha,\beta\}}$ laws.

Composition of laws belonging to different areas of attraction.

Semi-stable laws.

**7 - Notions on the theory of errors.**

General notions.

Compensation for accidental errors.

Importance of the fact that the accidental error obeys Gauss's law.

Determination of precision parameters.

Application of the principle of probabilities of causes.

The method of least squares.

Application to a levelling problem.

**8 - Notions on the kinetic theory of gases.**

General notions.

Maxwell's law. Methods of Maxwell and M Borel to obtain this law.

Justification of Maxwell's law.

Reversibility and irreversibility. Conclusion relating to Maxwell's law.

Average free course.

Calculation of the pressure on the walls.

Various generalisations.

Determination of molecular constants.

**Note.**

The laws of probability in abstract sets.

**2.2. From the Preface.**

We know that the calculus of probabilities is based essentially on a single theorem, the law of large numbers. We can say that the sole object of the theory is to prove this theorem, and some others which are connected with it. The problems relating to card games, which often occupy a large place in treatises on the calculus of probability, are only of interest as simple examples which will help to understand the scope of the principles. But as soon as they get more complicated, they are exercises in combinatorial analysis rather than in the calculus of probabilities.

To establish the law of large numbers, or rather to establish the few results which complete and clarify it, and highlight the role of Gauss's law in the theory of errors, one can not seek to make a precise mathematical theory and be satisfied with reasoning of common sense. This is what MM Borel and Deltheil said in their little book published in the Armand Colin collection, and there is nothing to add to what they said. But for the mathematician, this is not enough. It is necessary to justify the fundamental principles of the theory of errors by adequately specifying the intuitive notion of accidental error and by deducing from it by a rigorous reasoning that the accidental error obeys Gauss's law, M Borel considers that this result does not justify the mathematical apparatus necessary to achieve it.

The reader will see that this mathematical apparatus is not as large as it is generally believed. We arrive very simply at the result, by using the notion of a characteristic function. I indicated this method as early as 1920 in my course at the École Polytechnique. Noting that it seemed little known, and that its systematic use led to new results, I, in 1922 and 1923, presented to the Academy of Sciences some notes on this subject. Great was my surprise on learning from a letter from G Polya, written following the first of these notes, that this method and some of the results which I believed to be new had been developed in notes presented by Cauchy to the Academy of Sciences in 1853; it is even possible that not all of Cauchy's results were published; The Academy of Sciences indeed found that the illustrious scientist put too little discretion in filling in the

*Comptes rendus*of his discoveries. However that may be, it is singular that a few notes by the greatest mathematician of the time, relating to a problem which has been the subject of so much research, did not attract attention. Poincaré, in the second edition of his

*Calcul des Probabilités*, indicates in a few lines the principle of Cauchy's method; it seems, moreover, that he did not know that it was from Cauchy and it is almost certain that he did not see the full significance of it. None of the works published since indicates this method, which seems to have been ignored even by the editors of the

*Encyclopédie des Sciences Mathématiques*. So, I thought that a new book, devoted largely to the systematic development of this method, would not duplicate the previous ones.

**2.3. Review by: W D Cairns.**

*Amer, Math. Monthly*

**33**(6) (1926), 328-330.

Lévy consciously makes the law of large numbers (Bernoulli's) the central feature of his treatise and the foundation of the calculus of probability, problems in games of chance being regarded merely as simple problems which enable one to grasp the real meaning of the later principles. The second outstanding feature of the book is that Lévy bases his rigorous development of Gauss' law for accidental errors on the notion of characteristic function, a method which at least in considerable part goes back to Cauchy and 1853. Lévy justifies this, as against the direct derivation of Gauss' law from the binomial formula, on the ground that when once the fundamental properties of the characteristic function are found, a large number of important consequences are obtained, adopting thus a unifying principle similar, say, to that which Hilbert uses in his organisation of integral equations.

The book is divided into two main parts. The first part on the principles of the calculus of probability contains chapters on subjective probability, principles derived experimentally, objective value of probability, laws of probability (the law of Gauss and its relation to the law of large numbers), probability derived from experience (statistics), and a critique of the theory of expectation or risk. The second part on the mathematical theory of probability treats in successive chapters general notions about the laws of probability and the theory of sets, probable values, characteristic coefficients and functions (his basic chapter, including his proof of Gauss' law of error), the composition or combination of laws of probability, laws of variable probabilities, a more extended study of the law of large numbers, exceptional laws the theory of errors, and the kinetic theory of gases.

**2.4. Review by: F P W.**

*Science Progress in the Twentieth Century (1919-1933)*

**20**(80) (1926), 712-713.

The main part of M Lévy's book deals with the law of Gauss as the limit to which, in very general circumstances, any law of probability tends. The method consists in the use of the idea of "characteristic function." The author published several notes on the method in 1922-3 in the

*Comptes Rendus*, and was surprised to learn by a letter from M G Pólya, whose knowledge of all branches of mathematics is astounding, that it had already been developed to some extent by Cauchy in 1853. No subsequent writer had, however, referred to it, and M Lévy therefore judged that a systematic treatment would not be out of place. He further gives indications of a more recent method, due to M Lindeberg, and has chapters on exceptional laws, and on applications of the theory to kinetic theory and to the theory of errors. ... He has added a preliminary part, philosophical rather than mathematical, which strikes one as rather vague and unsatisfactory, but the main part of the book is well worth detailed study.

**2.5. Review by: J Marshall.**

*The Mathematical Gazette*

**13**(184) (1926), 214.

This book by M Lévy on the Theory of Probability is written in order to emphasise a point of view which seems to have been ignored by other writers on the subject. Its special feature is the prominent place given to the use of the characteristic function which Cauchy was the first to introduce. M Lévy attempts to justify the fundamental principles of the theory of errors by giving a suitable precision to the intuitive idea of accidental error, and by deducing in a rigorous manner that the accidental error obeys the law of Gauss. Several writers think that this result does not justify the mathematical apparatus necessary to establish it, and the author attempts to combat this idea by showing that the mathematical methods required are not so formidable as is generally supposed.

...

On the whole M Lévy's work will be considered by many to be rather prolix. Readers who find the first part new and exciting are likely to be overwhelmed by the analysis in the second part, and those who can follow the second part with ease will find the first part rather dull. But the book contains a fund of valuable information, and the author has certainly succeeded in showing that the use of the characteristic function has a synthetic value which deserves a prominent place in the development of the subject.

**3. Analyse fonctionnelle (Gauthier-Villars, Paris, 1925), by Paul Lévy.**

**3.1. Review by: G C Evans.**

*Bull. Amer. Math. Soc.*

**33**(3) (1927), 372.

The author gives in this brief discussion an exposition of what is necessarily only a part of the subject of functional analysis, but in such a way as to impress the reader with the existence of new ideas under the sun. The condensation naturally enforces slow reading.

Chapter I gives the fundamental notions of function space and the generalised definition of distance ...

Chapter II gives an introduction to equations in functional derivatives. The reader will probably turn to references where he can find more detail, but he will be able here to get a notion of complete integrability and of characteristics.

In Chapter III is developed the idea of integration in function space. On account of the fact that the ratio of the volumes of two similar figures becomes infinite or zero as the number of dimensions becomes infinite, the idea of volume must be replaced by that of average. Here again, however, we meet the same difficulty, in that the neighbourhood of one position is apt to predominate to an infinite extent in the determination of the average. (This aspect of infinity will be more or less familiar to the reader as the characteristic of one of the well known methods of arriving at Maxwell's law of distribution for the velocities in a gas of a large number of molecules.) It becomes desirable therefore to consider other kinds of weighted means. ...

The theories expounded owe, of course, to the founders of the subject, and much also to Lévy himself and to Gateaux, Wiener, Fréchet and Daniell.

**4. Cours de Mécanique (Gauthier-Villars, Paris, 1928), by Paul Lévy.**

**4.1. Note.**

This book contains lectures delivered at the École supérieure des mines.

**5. Cours d'analyse Vol 1 (Gauthier-Villars, Paris, 1929), by Paul Lévy.**

**5.1. Note.**

There are many versions of his course in the form of handouts to students which changed little from the 1920s to the 1950s. The course published by Gauthier-Villars in 1929 is an intermediate course, where the proof of Picard's theorem is missing but where there is not yet an introduction to the Lebesgue integral.

**6. Cours d'analyse Vol 2 (Gauthier-Villars, Paris, 1930), by Paul Lévy.**

**6.1. Note.**

There are many versions of his course in the form of handouts to students which changed little from the 1920s to the 1950s. The course published by Gauthier-Villars in 1929 is an intermediate course, where the proof of Picard's theorem is missing but where there is not yet an introduction to the Lebesgue integral.

**7. Théorie de l'addition des variables aléatoires (Gauthier-Villars, Paris, 1937), by Paul Lévy.**

**7.1. Note.**

For the Contents of the 2nd edition, and a review of the 2nd edition, see 10. below.

**7.2. From the Preface.**

The calculation of probabilities has made immense progress over the past fifteen years. Classic problems have been given a more complete solution than it seemed at the beginning of this century possible to hope. New problems, arising from the theory of countable probabilities, have been posed and often solved. So there could be no question of giving in one volume an exposition of the whole of the calculus of probabilities, in its current state. My goal is more restricted. My personal research having mainly been aimed, for several years, at the study of asymptotic problems relating to probabilities, it seemed to me that the moment had come to give a general account of the current state of the questions which I have thus studied, and which during the same period were the subject of numerous works, among which it is appropriate to mention in particular those of A Khinchin and A Kolmogorov.

I only thought about choosing a title for this book after I had finished writing it, and it was difficult to find one that exactly matched the questions presented. So the reader should not be surprised if he finds that my subject has not been treated in a complete way, or on the contrary that I left it until the last chapter; the title is wrong; my intention was to talk about the issues on which I had something to say.

**8. Processus stochastique et mouvement brownien (Gauthier-Villars, Paris, 1948), by Paul Lévy.**

**8.1. Note.**

The text by Lévy is followed by a 50-page note by M Loève on the general theory of stationary and related processes.

**9. Problemes concrets d'analyse fontionelle (Gauthier-Villars, Paris, 1951), by Paul Lévy.**

**9.1. Note.**

This is a second edition of

*Leçons d'analyse fontionnelle*(1922), with a supplement by F Pellegrino.

**10. Théorie de l'addition des variables aléatoires (2nd edition) (Gauthier-Villars, Paris, 1954), by Paul Lévy.**

**10.1. Note.**

The Preface to this work is by Émile Borel.

**10.2. Contents.**

**1 - The foundations of the notion of probability.**

Subjective probability and the law of probability.

Principle of compound probabilities.

The verifiable consequences of the theory.

Objective value of probability.

Probability and frequency; statistical determinations.

Critique of empirical definitions of probability.

**2 - Laws of probability and partitions.**

Case of a countable set. Principle of total probabilities.

Case of an uncountable set.

The measure of a linear set.

Definition of a probability law in an abstract set. The notion of partition.

Additional remarks.

**3 - Laws with one or more variables.**

One-variable laws. Distribution functions.

Probable value and the Stieltjes integral.

The characteristic function.

Gauss's law.

Dispersion of a random variable.

Variable probability laws.

Notions on sequences of random variables.

The notion of a compact set.

Application of the preceding notion to the laws of probability.

Types of laws and reduced laws.

Two-variable laws.

The notion of conditional probability.

**4 - The composition of probabilities and Bernoulli's theore**m.

Fundamental formulas.

Composition of characteristic moments and functions.

Proof and generalisation of Bernoulli's theorem by Chebyshev's method.

The case of Bernoulli and the method of de Moivre.

Poisson's law, or small probabilities.

The increase in dispersion.

**5 - Theorems relating to Gauss's law.**

First notions of stable laws.

The theorem of H Cramér.

A corollary of the theorem of H Cramér.

Lyapunov's theorem, or second limit theorem of the calculus of probabilities.

Extension of Lyapunov's theorem.

The inverse theorem.

The domain of attraction of Gauss's law.

Remarks and special cases.

Conclusion relating to the general case.

**6 - Countable probabilities and random series with independent terms.**

First notions on countable probabilities.

The lemmas of Borel and Cantelli.

Probabilities equal to zero or one.

The limiting dispersion for a series with independent random terms.

Convergence from Bernoulli's point of view and convergence in probability.

The oscillations of the $S_{n}$ sequence and the probability of convergence.

The average convergence condition.

The condition of Khinchin and Kolmogorov for almost sure convergence.

A theorem on the case of divergence.

Sums of terms with dispersions bounded below.

**7 - Integrals with independent random elements.**

The position of the problem.

The condition of convergence and the reduction to the problem of indefinitely divisible laws.

Determination of $X(t)$.

Case where the function $X(t)$ is almost surely continuous; role of Gauss's law.

The role of mobile discontinuities and Poisson's law.

General determination of indefinitely divisible laws.

The arithmetic of indefinitely divisible laws.

The general arithmetic of the laws of probability.

A problem of A Khinchin.

Application to the study of stable laws.

Other methods for the formation of stable laws.

Semi-stable laws.

Quasi-stable laws.

Conclusion relating to the group of a given law.

Multiple integrals with independent random elements.

Laws indefinitely divisible by several variables.

Stable laws with several variables.

**8 - Miscellaneous questions relating to the sums of linked variables**.

The general problem of chain probabilities.

The simple chains of Markov.

The extension of Bernoulli's theorem and of Chebyshev's method to the sums of chained variables.

Lindeberg's method for Lyapunov's theorem.

Convergence of series with non-independent terms.

The strong law of large numbers.

The law of the iterated logarithm.

Remarks and additions.

The case of Bernoulli and the strong law of large numbers.

**9 - Application of the calculus of probabilities to the theory of continued fractions.**

Reminder of classical notions.

Borel's inequalities and the problems of the first group.

Gauss's problem. First recurrence formulas.

New recurrence formulas and proof of convergence.

Extension of the Gauss formula to certain cases where the distribution of the probability is not uniform.

Application of the strong law of large numbers.

Application of the preceding results to the study of the sums of Khinchin.

**Note I: Weak law and strong law of large numbers.**

**Note II: Basic notions on stochastic processes.**

**10.3. Publisher's description.**

After a short presentation concerning the issue of the foundations of the calculus of probabilities, a study of the notions of random variable and the law of probability and a reminder of the most classical theorems of the calculus of probabilities, this book deals with new problems, arising from the theory countable probabilities, of various questions relating to the sums of chained variables.

**10.4. Review by: B C Brookes.**

*The Mathematical Gazette*

**39**(330) (1955), 344.

This is a revised edition of the well-known book first published in 1937. The work of revision has been made difficult for the author because of the recent rapid development of the theory of random variables (a subject to which he has made important contributions) and because he has already expounded it more fully in his book

*Processus stochastiques et Mouvement brownien*, published in 1948. Though some amendments have been made to the original text the most important change in the new edition is the inclusion of two new appendixes.

The first appendix is a paper published in 1953 in the

*Bulletin des Sciences mathématiques*. It discusses the weak and strong laws of large numbers applied to sums of independent random variables, and introduces the concept of the "Laplacian sum", a concept due to Lévy. The second appendix is a commentary on stochastic processes inspired by Doob's treatise published in 1953 (reviewed in

*The Mathematical Gazette*). It is both a critical review of Doob's work and an introduction to the mathematical theory of stochastic processes. It is refreshing to find even a French mathematician expressing admiring astonishment at the "impitoyable rigueur" of Doob's logical development and demurring at the high level of abstraction which he maintains.

The first four of the nine chapters of the main book give an introduction to modern probability theory based on measure theory and the Stieltjes integral. In the next four chapters the author expounds those aspects of the theory of random variables in which he is particularly interested and to which he has contributed. The last chapter, on the application of probability theory to the theory of continued fractions, is interesting in itself but is irrelevant to the main purpose of the book.

The changes and additions in the revised edition do not help to unify a book which even in its first edition seemed to be more of a personal note-book than an objective treatise. Yet because of increasing interest in stochastic processes the publication of this revised edition is well justified.

**11. Le mouvement brownien (Gauthier-Villars, Paris, 1954), by Paul Lévy.**

**11.1. Review by: B C Brookes.**

*The Mathematical Gazette*

**39**(330) (1955), 344.

The title might suggest that the booklet describes physical processes, but though Lévy indicates analogies between some of the equations he derives and those of diffusion and conductivity processes, his interest in the Brownian motion is almost wholly mathematical. The greater part of the booklet is devoted to a logically developed theory the linear random function defined above'; the remainder, less coherent, consists of an account of additive random functions which have "Brownian motion" in the plane and in n-dimensional Euclidean space. Lévy provides an up-to-date summary of a subject to which he has made important contributions. But as the treatment is necessarily condensed this booklet be recommended only to those already acquainted with the theory of functions.

**12. Processus stochastique et mouvement brownien (2nd edition) (Gauthier-Villars, Paris, 1965), by Paul Lévy.**

**12.1. Contents.**

**1 - Two simple examples of stochastic processes.**

The function $X(t)$ of linear Brownian motion.

The reduced function. A projective invariance theorem.

Remarks and corollaries.

Examples of discontinuous random functions related to Poisson's law.

**2 - General notions on stochastic processes.**

Stochastic processes and random functions.

Direct definition of stochastic processes.

Continuity of stochastic processes.

First notions on Markov processes.

Stochastic differential equations.

Cauchy's problem for stochastic differential equations.

Effective formation of $X(t)$.

Special cases and examples.

The derivation and integration of random functions.

Conditions of existence of the random derivative $m. q$.

**3 - Markoff processes and the diffusion of probability.**

The Chapman and Kolmogorov equations.

The probability diffusion equation.

Special cases.

Strongly continuous processes and Laplace's law.

Brownian motion and the heat equation. Cauchy problem, mixed type problems.

Applications.

Asymptotic theorems relating to Brownian motion.

Case of vector random functions.

**4 - Stationary processes.**

General notions.

Examples.

General theorems on covariance.

Derivatives of stationary random functions.

Harmonic analysis of stationary random functions.

The case of random functions of several variables.

**5 - Additive processes.**

Lemmas relating to series with independent random terms. Their application to additive processes.

Notes on some discontinuous functions.

The three types of additive processes.

The inverse theorem. Weakly continuous additive processes and indefinitely divisible laws.

The uniqueness of representation, and the arithmetic of indefinitely divisible laws.

Theorems relating to Laplace's law.

More and more divisible laws.

The method of B de Finetti and A Kolmogorov.

Types of stable laws.

A group of Pearson's laws.

Additive processes on the circumference.

Additive processes in multi-dimensional spaces.

**6 - In-depth study of linear Brownian motion.**

The concept of Brownian oscillation.

The functions $M(t)$ and $Y(t)$.

The lengths of the intervals $e'$

Formulas relating to the interval $e'$ containing a given value of $t$

Interpolation formulas. Application of Theorem 2.

The inversion of $M(t)$. Properties of the inverse function $T(x)$.

Direct construction of assembly $E_{1}$. Stochastic equivalence of $E_{0}$ and $E_{1}$. Second method for the direct construction of $E$.

Reconstruction of $X(t)$ via $E$.

The summation function of $(T)$. The law of the iterated logarithm.

**7 - The plane Brownian motion.**

The general shape of curve $C$.

Surface measurement and Brownian oscillation.

The closing of curve $C$.

The stochastic area of the curve $C$.

Intrinsic properties of the $C$ curve. Conformal representation and boundary problems.

**8 - Brownian motion with several parameters.**

Preliminary remarks.

The lemma of I J Schönberg and L Schwartz.

Definition of Brownian motion with p parameters.

Geometric representation of the random function of Brownian motion.

Study of some simple figures.

The line and an outside point.

The sphere and regular polyhedra. Asymptotic properties.

Supplement drafted for the second edition

**RECENT PROGRESS IN THE THEORY OF LAPLACIAN RANDOM FUNCTIONS**

**1 - General notions on Laplacian random functions.**

Definitions and preliminary remarks.

Conditional probabilities in a Laplacian system.

The integral of G Maruyama.

The integral with variable upper limit and the canonical representations.

A remarkable curve of Hilbert space.

Characterisation of canonical nuclei.

The infinitesimal variation of $X(t)$. The nuclei of Goursat.

Search for nuclei corresponding to a given Goursat covariance.

Remarks on the set of representations of a Laplacian function defined by its covariance.

Determination of the canonical representation as a function of the covariance.

**2 - Complementary notions on classical Brownian motion.**

Quadratic functionals of $X(t)$.

The Fourier-Wiener series.

Remarks on a general class of random Fourier series.

The $C$ curve of plane Brownian motion and the theorems of A Dvoretsky, P Erdős and S Kakutani.

The area between an arc of curve $C$ and its chord.

Brownian motion in Hilbert space.

**3 - The Brownian function of several parameters.**

The Brownian function on the Riemann sphere.

Return to the Euclidean case.

The Brownian function in Hilbert space.

The average of $X(A)$ over a sphere.

The determinism of $X(A)$ in Hilbert space.

The sets $K(E)$.

Minimising subsets and conjugate elements.

**Note from Michel LOÈVE**

**SECOND ORDER RANDOM FUNCTIONS**

**1 - Covariances.**

General.

Characteristic properties of covariances.

Operations on covariances.

Partial covariances.

Orthogonal decompositions.

**2 - Properties of second order random functions.**

Root mean square differential properties.

Root mean square integral properties.

Integral properties: harmonic and spectral analysis.

Almost safe properties; differential properties.

Almost safe full properties.

Normal case.

Random functions and some theories of analysis.

**3 - Random functions of an exponential nature.**

Stationary random functions.

Exponentially convex random functions and generalisation.

**12.2. From the Introduction.**

The greater part of the present work is an overview of the results obtained by the author, from 1934 to 1939, on additive processes and on Brownian motion, and of some more recent results. But it seemed useful to us to precede this presentation with a general summary of stochastic processes, which, thanks to the work of A Khinchin, J Kampé de Fériet, H Cramér, M Loève, A Blanc-Lapierre and R Fortet, has made great progress in recent years.

**12.3. Review by: D G Kendall.**

*Biometrika*

**53**(1/2) (1966), 293-294.

It is convenient to review these three books together. That by Lévy is a new edition of a classic work; the format and binding are now quite attractive, and some 70 pages of new material have been added. As the author remarks, the edition of 1948 contained very little about general Markov processes, and the general theory of stationary and related processes was represented there by a 50-page note by M Loève. In the new edition there is hardly any difference in these respects; the note of Loève is reproduced without change, and there is no mention of Lévy's own profound contributions to the theory of denumerable Markov chains with a continuous parameter which have transformed that subject almost beyond recognition. The appearance since 1948 of the two great treatises by Loève and Chung make it quite reasonable for the author to maintain the emphasis of the original edition, but still the reviewer feels that the incorporation of Lévy's famous

*École Normale*papers on Markov chains, perhaps in an Appendix, without modification, would have been greatly welcomed by most readers of this book.

The new material consists of three additional chapters. The first of these is concerned with integral representations of Gaussian processes, while the second deals with Brownian motion in any number of dimensions (with a single time-variable t). The third new chapter gives an account of the author's work on one of his most characteristic inventions: Brownian motion in one dimension with a multi dimensional time-variable. The most exciting case is that in which the time-variable ranges through Hilbert space. Here the sample 'paths' have a curious quasi-analytic property: a 'path' is almost surely fully determined as soon as it is given on an open set of 'time' values. It will be recalled that the friends and admirers of M Lévy have instituted in his honour a prize to be awarded to the author of the best memoir on multi-dimensional time-parameter Brownian motion (last date of entry 1 January 1966).

**13. Quelques aspects de la pensée d'un mathématicien (A Blanchard, Paris, 1970), by Paul Lévy.**

**13.1. Review by: Kenneth O May.**

*Isis*

**62**(3) (1971), 415-416.

An autobiography by a distinguished mathematician is always a welcome document. Paul Lévy, member of the Academie des Sciences and professor emeritus at the École Polytechnique, presents at age eighty-three a very substantial footnote to more than sixty years of highly competent mathematical activity. The first and larger part of the book (entitled

*Souvenirs mathématique*on the cover,

*Autobiographie mathématique*inside) describes the expected precocity and then his achievements one by one in chronological order within major subdivisions (functional analysis and integral equations, probability, stochastic processes and Brownian motion, miscellaneous). The second part (

*Considération philosophiques, L'évolution de mes idées sur la philosophie*) deals with his religious and philosophical opinions in general and in relation to probability and the foundations of mathematics. Except for a brief footnote on his father added in proof, there is no information about family, personal life, or career. It is intellectual autobiography in the extreme with only incidental reference to the context.

The writing is factual, simple, and, above all, candid. Lévy makes it clear that he wishes to clarify and complete, to set the record straight, to improve faulty expositions, and to express long-held opinions that did not find a place in his specialised research publications. In the seven-page introduction and in later side remarks, he describes himself as a mathematician unable, even when young, to pay much attention to the ideas of others, not having attempted "to follow the main stream of modern mathematics," concentrating on developing his own ideas, usually unaware of how his work fitted into the collective effort (often failing to publish new ideas or duplicating previous work), and unwilling to read relevant earlier papers even when called to his attention. But he maintains that in following his natural bent he achieved the best possible results. His weakness in giving and receiving information undoubtedly accounts in part for slow recognition and for his complex of being misunderstood (

*méconnu*). On page 119 he gives this as one motive for writing the book. It is clear that he was one of the old guard from whom Nikolas Bourbaki had to emancipate himself in the thirties, and who regards the "new math" with grave misgivings ...

Lévy is to be thanked for offering such an honest and informative document which will be very useful to anyone assessing mathematics and mathematicians of the first half of this century.

Last Updated September 2020