Chance and Choice

The following sections are included:

• Foreword

• Contents

If the Sun has risen on n successive days, what is the PROBABILITY that it will rise on the next day?

Laplace gave the answer: For n = 1, this is . Since he assumes that it is “equally likely” that the Sun rises or not on each day, how come the answer is not ½?…

下述之一串定理(實係一定理及其數推論), 常見引於論科學方法之著作中其證明用積分法為之甚易, 下述之證明則僅用或然率原理與數數求和法, 而無損於理論之嚴密.按此證係亡友劉炳震君與余所合作, 成於去歲, 嗣因悉前人已有類似之證 (Prevost 與 Lhuilier 之證, 見 Todhunter: History of Probability, 但其立論及方法與此頗有不同, 竊謂本文較為直捷), 無意發表, 今劉君不幸自盡, 其家屬命余整理其遺作, 乃敢試將此文公之於世, 以資紀念劉君, 并非掠人之美也.

Dans cette Note je me propose de donner une démonstration très simple du théorème suivant de M. Gumbel et une généralisation de ce théorème…

The following sections are included:

• Introduction

• Generalization of Poincaré's formula; Generalization and sharpening of Boole's inequality

• Generalization of Fréchet's inequalities and related inequalities

• The p_{α1 … αi}'s in terms of the p_m(v₁, …, v_k)'s and the p[_{α1 … αi]}'s in terms of the p₁(v₁, …, v_k)'s

• A condition for existence of systems of events associated with the probabilities p₁(v₁, ⋯ , v_k).

The following sections are included:

• Introduction

• Postscript

Soit M un module, c'est-à-dire un ensemble de nombres tels que a ∈ M et (a + b) ∈ M soient équivalentes à a ∈ M et b ∈ M.

Let X_n, n = 1, 2,… be independent random variables with moments

The reduction of the variance to 1 is not necessary, but is made here solely for the sake of simplicity…

In the classical coin-tossing game we have a sequence of independent random variables X_v, v = 1, 2,… each taking the values ± 1 with probability 1/2. We are interested in the signs of the partial sums . To eliminate the zeros of S_n we make the following convention: S_n is “positive” if S_n > 0 or if S_n = 0 but S_n−1 > 0; otherwise S_n is “negative.” The elegance of the results to be announced depends on this convention. Let N_n denote the number of “positive” terms among S₁, …, S_n. We shall confine ourselves to an even n, noting only that 0 ≤ N_n+1 − N_n ≤ 1. In the following r and m are positive integers and P(B∣A) is the conditional probability of B under the hypothesis A…

The following sections are included:

• Historical origin

• An integer representation theorem

• The probability setting

• Laplace's asymptotic and a lower estimate

• The main argument

• Fourier analysis in the symmetric case

• References

• PROBLEM

Renewal theory has been treated by many pure and applied mathematicians. Among the former we may mention Feller, Täcklind and Doob. The principal limit theorem (for one-dimensional, positive, lattice random variables) was however proved earlier by Kolmogorov in 1936 as the ergodic theorem for denumerable Markov chains. A partial result for the non-lattice case was first proved by Doob using the theory of Markov processes, and the complete result by Blackwell. The extension of the renewal theorem to random variables taking both positive and negative values was first given by Wolfowitz and the author [1], for the lattice case. A partial result for the non-lattice case, using a purely analytical approach, was obtained by Pollard and the author [3].² For the literature see [1]…

The following sections are included:

• Summary

• Introduction

• Lemmas

• First case

• Second case

• Linear case

• Consequences

• REFERENCES

• Historical Note

The fundamentals of the theory of denumerable Markov chains with stationary transition probabilities were laid down by Kolmogorov, and further work was done by Doblin. The theory of recurrent events of Feller is closely related, if not coextensive. Some new results obtained by T. E. Harris turn out to tie up very nicely with some amplifications of Doblin's work. Harris was led to consider the probabilities of hitting one state before another, starting from a third one. This idea of considering three states, one initial, one “taboo”, and one final, is more fully developed in the present work. The notion of first passage time to the “union” or “intersection” of two states is also introduced here. The interplay between these notions is illustrated.

The Martin boundary for a discrete parameter Markov chain was first considered by Doob,² using the theory of R. S. Martin after whom the boundary is named. A direct and ingenious method was later found by Hunt,³ who also strengthened Doob's results in several points. In this note, I shall sketch a natural approach to the theory in the continuous parameter case. There, upon the introduction of certain intrinsic quantities (probabilities) which have no obvious discrete analogues, it is possible to derive the main results by familiar methods developed in reference…

1. L'objet de cette Note est l'étude de l'équation

que nous écrirons symboliquement σ ⋆ E = İ. La fonction , donnée dans [o, ∞), est non croissante et vérifie les conditions

Doob's version of the fundamental convergence theorem of potential theory asserts that if (f_n) is a decreasing sequence of excessive functions and f is the supermedian function inf f_n, then the set where f differs from (its regularized function) is semi-polar. Many beautiful proofs of this result are available in the literature. Here is a trivial one…

1. To M. Brelot we owe various basic principles and methods of general potential theory (see e.g. [1]). The relationship between them constitutes an important part of the development. In a recent paper [2] I established an equilibrium principle for a broad class of Markov processes by a simple probabilistic method which may be succinctly described as that of ≪ last exit ≫ In contrast, the probabilistic method of solving the Dirichlet problem, due largely to Doob, may be described as that of the ≪ first exit ≫. Now the classical method of solving the equilibrium problem, introduced by Gauss and perfected by Frostman, accrues from the minimization of a quadratic functional involving the ≪ energy ≫. It is natural to ask whether the method of last exit has anything to do with that of energy. Indeed, the first question that arises is whether the equilibrium measure obtained in [2] does minimize some kind of energy. The hypotheses made there are free from the usual duality assumptions and certainly do not require the symmetry of the (potential density) kernel, on which the classical method of energy relies heavily. Although the concept of energy has been extended to the nonsymmetric case, its utilization in a general probabilistic context appears to be still a distant goal.In this note we shall show that for a symmetric kernel, the equilibrium measure obtained by the last exit method does in fact minimize the energy. Even this little step requires much strengthening of the hypotheses in [2] to be specified below.

Let {X(t), t ≥ 0} be the standard one-dimensional Brownian motion starting at 0. For t > 0 define

…

It happens sometimes that a great thing in one branch of mathematics is equally great in another. Consider the following function:

In a fundamental paper by KOLMOGOROV in 1931 [4], this was shown to be the distribution of the maximum deviation of the empiric distribution from the theoretical distribution (apart from an unessential change of variables). Kolmogorov did not say, but I am almost sure that he knew the series in (1) is a case of the celebrated Jacobi identity (see, e.g., [5; p. 8] and [2; p. 284]). Namely we have the infinite product representation: …

Schrödinger's equation is put in this reduced form:

where D is a bounded domain in R^d, d ≥ 1; Δ is the Laplacian; q is a function in D satisfying some conditions to be specified later. The Dirichlet boundary condition is …

The proposition quoted in the title above is a folklore. It can be argued as follows. If a Markov chain has run an infinitely long time, then every “state” will have become a “steady state”,and nothing will happen any more. This is physicists' talk, which sounds apt for the occasion. The formal definitions are given below…

On the return to equilibrium. We consider, on the group of integers, a random walk starting from the origin and whose steps admit as possible values exactly two integers, a and b, with a < 0 < b. In the particular case a = −1, we give an explicite expression for the law of the first return time to the origin.

The name given in the title is an abbreviation of ‘Markov processes with continuous time parameter, denumerable state space and stationary transition probabilities’. This theory is to the discrete parameter theory as functions of a real variable are to infinite sequences. New concepts and problems arise which have no counterpart in the latter theory. Owing to the sharply denned nature of the process, these problems are capable of precise and definitive solutions, and the methodology used well illustrates the general notions of stochastic processes. It is possible that the results obtained in this case will serve as a guide in the study of more general processes. The theory has contacts with that of martingales and of semi-groups which have been encouraging and may become flourishing. For lack of space the developments from the standpoint of semi-groups or systems of differential equations cannot be discussed here…

The following sections are included:

• Introduction

• Comments on the conditions (1.3) and (1.4)

• Two analytical lemmas

• Review of the strong Markov property

• Transition from and to a stable state

• First entrance and last exit

• The minimal chain

• Beyond the first infinity

• REFERENCES

Markov processes have thrived under a set of cozy yet cogent hypotheses for a decade now. The prevailing conditions imply that almost every sample function is right continuous and has finite left limits except perhaps at infinity. Hence in particular the discontinuities are ordinary jumps, of which the number is finite in every time interval. If one replaces the word “infinity” above by “terminal time”, one gets a somewhat more general situation called a “standard process”. Between the two there is also something called “special standard process”, in which the behavior of the path near the terminal time is suitably restricted (see [7] for terminology)…

It is well known that a Markov process reversed in time remains Markovian. Indeed, the most elegant formulation of the Markov property is symmetric with respect to the direction of time, as follows. Let {x_t} be the process and let ℱ_t σ(x_s, S < t), be respectively the Borel fields generated by the process before and after t; then for any Λ ∈ ℱ_t and we have

…

In contemporary studies of homogeneous Markov processes on a topological space, under the name of Hunt or standard process, it is assumed that the only discontinuities of (almost all) sample functions are jumps, for all time or up to the lifetime of the process, respectively. If the same assumption is made on a Markov chain, where the state space is discrete and may be labeled by the integers, this results in a rather trivial situation long since “solved”. If other types of discontinuity are allowed, then the typical sample function will have infinity as a limiting value when such a discontinuity is approached, from one or both directions of time. Various ways of reaching and returning from infinity should then be distinguished, and the consequent ramification has been called a boundary in analogy with classical potential theory. The problem is then to set up a suitable boundary and investigate the behavior of the sample functions relative to it. For the proper object of study of stochastic processes is the collection of sample functions or paths – it is through the underpinning a groundwork of paths that modern probability theory makes its most original contribution to mathematics(²)…

Let (Ω, ℱ, P) be a probability space and X = {X_t, t > 0} a homogeneous Markov process adapted to the Borel subfields {ℱ_t} of ℱ. The process X is supposed to be measurable and it takes values in (E, ε), where E is a locally compact space with countable base and ε its Borel field. The transition function P_t(x, A), t > 0, x ε E, A ∈ ε, is Borelian in (t, x). The field family {ℱ_t} is augmented by all P-null sets but not necessarily right continuous. A stopping time relative to {ℱ_t+} will be called « optional » here. The augmented Borel field generated by (…) will be denoted by σ(…). C_K denotes the class of functions continuous on E and having compact supports. « Almost surely » (« a.s. ») refers to P and is often omitted in an obvious context. Any statement below regarding X_T is automatically understood to be under the proviso « a.s. on {T < ∞} » since X_∞ is not defined. « The path » is a circumlocution for « almost all paths ». The symbol « ↑ » means « increasing » but not necessarily strictly. Other terminology, notation and conventions follow generally those in [1] and [5]…

The following sections are included:

• Introduction

• The gauge

• Semigroup and potential

• Conditional gauge

• Footnotes

• References

Hsu returned from England to Kunming in the middle of the Sino-Japanese war, apparently in 1940. He gave a long course beginning with the theory of measure and integration, through probability theory, and leading to mathematical statistics. For the first part he based his lectures on Carathéodory's Vorlesungen #x00FC;ber Reelle Funktionen, for the second on Cramerés Random Variables and Probability Distributions. He was a polished and vivacious lecturer with complete notes written out beforehand in a notebook he carried. He enjoyed making a fine point in class. For instance, when he was doing the inversion formula for characteristic functions, he took delight in the fact that one could integrate over a single point by the Lebesgue-Stieltjes integral. He had a tattered copy of Cramér's book with many marginal notes and used to say that it was written in a more difficult way than necessary but contained all that was essential for probability. He was a true virtuoso in the method of characteristic functions. His papers [17], [23], [25], [26], [31] and [35] all showed his fascination for, as well as mastery of, this precious tool. Mathematical literature was hard to come by in Kunming during those years, and some of the old volumes shipped from Peking (then called Peiping) were kept in caves to preserve them from air raids. (We did not actually live in caves, but frequently had to run to them for hours at a stretch under raids or alerts.) Among the books so kept was, for example, Kolmogorov's Grundbegriffe der Wahrscheinlichkeitsrechnung. At my request Hsu had this book extracted from the caves, but I remember his saying of it: “This is another kind of mathematics”. He was not, of course, a probabilist per se at a time when such an appellation hardly existed anywhere in the world; by education and inclination he was more fond of problem-solving than generalities. However he never tried to dissuade others from following their own bent and getting interested in those other kinds of things. Indeed, if he could be drawn into a new topic, he would work at it in earnest and quickly produce something of value. For instance, when Borel's work on what he called “denumerable probabilities” attracted my attention, Hsu took an interest also, which led to the paper [21]. This was an early approach to what became known as “recurrent events” under Feller's development. Papers [23] and [30] were probably written under similar circumstances…

We begin with a resume. Let {P(t), t ≥ 0} be a semigroup of stochastic matrices with elements P_ij(t),(i,j) ∈ I × I where I is a countable set, satisfying the condition…

Pólya's first publication, his 1913 Dissertation [1] in Hungarian, treats a problem in probability. His lifelong interest in this field is evident from his Bibliography (see Collected Papers, Vol.4; numbers in [ ] below are those given there). Besides the twenty papers reprinted in that volume, some ten more titles indicate probability and statistics. Some of his contributions to probability are classified as analysis, others perhaps as combinatorics. The major items which will be discussed here have long since become textbook material, [R1]. (References for this article are indicated by the letter R and listed at the end.) Readers wishing to learn more details of the results summarized below should consult these texts…

In June of 1940, shortly after the fall of Prance in which Doeblin died, I received a package of reprints sent from France by Fréchet. It was probably among those reprints that I first saw Doeblin's paper [1] cited. I was then a student at Kunming, China and had read Fréchet's book [2]. Kolmogorov's paper [3] which laid the foundation of the theory of Markov chains with denumerable states was not accessible to me, although the journal containing it might have been under the caves of Yunnanfu to escape Japanese bombing, as I reminisced at the Paul Lévy memorial in 1987¹. I read it later after learning Russian, and translated it in an exercise book which I still have. Kolmogorov mentioned Doeblin's independent work but did not state the two problems treated in [1]. I think Feller first showed that paper to me. Later Mrs. Doeblin sent me several of her son's reprints, among which was this one, with a cover but without the year of publication nor the number of the volume, testifying to the wartime conditions in France. What I remembered distinctly is that I asked a visitor, M.L., to report its content to a small audience, including Doob and Snell. Afterwards we had a good chuckle over it because the oral presentation did not sound any different from the rather obscure original script. The following year I was living in New York City and giving a course on limit theorems for sums of independent random variables at the Department of Mathematical Statistics of Columbia University, and the Korean war broke out. I must have tired of those sums and been looking for something else to think about. It was in this way that I returned to the first problem in [1]…

Mathematical probability began with the letter exchanges between Fermat and Pascal, both older than Newton who together with Leibniz invented calculus. Despite such names as Jacques/Jakob Bernoulli, DeMoivre, Laplace, not to mention Gauss who with his “error function” might have edged the theory in the other direction, in my opinion, Probability somehow did not enter the main stream of Mathematics until the twentieth century…

Paul Lévy's books were not easy of access in Kunming, China, during the war years at the Southwest Associated Universities, owing to Japanese bombing of the city¹. I arrived at Princeton in December 1945 but it was later that I saw Captain Gil Hunt poring over l'Addition. He told me that von Neumann said: reading Lévy, one soon realizes that the author thinks mathematics in a different way from nous autres…

The following sections are included:

• Chapter 1. Introduction and probability background (45 pages).

• Chapter 2. Definition of a stochastic process—Principal classes (56 pages).

• Chapter 3. Processes with mutually independent random variables (46 pages).

• Chapter 4. Processes with mutually uncorrelated or orthogonal random variables (22 pages).

• Chapter 5. Markov processes—Discrete parameter (65 pages).

• Chapter 6. Markov processes—Continuous parameter (57 pages).

• Chapter 7. Martingales (99 pages).

• Chapter 8. Processes with independent increments (34 pages).

• Chapter 9. Processes with orthogonal increments (27 pages).

• Chapter 10. Stationary processes—discrete parameter (55 pages).

• Chapter 11. Stationary processes—continuous parameter (53 pages).

• Chapter 12. Linear least squares prediction—stationary (wide sense) processes (39 pages).

Until recently, probability theory has been called “calculus of probabilities” in the French and German languages. This book is so called and deserved the name in the best sense of the word. It is essentially a textbook of mathematical analysis as applied to the field of probability. By this it is not implied that the measure-theoretic foundations are not given adequately and rigorously. Indeed, the book begins with axiomatic Boolean algebra including a proof of M. H. Stone's isomorphism theorem, albeit in fine print. Kolmogorov's extension theorem is also given its full treatment while the Radon-Nikodym theorem, though not proved, is discussed in some detail with examples—which is probably more helpful than reproducing a standard proof. However, the unmistakable flavor of this book is the abundance of classical analytic techniques vigorously and interestingly employed to calculate the probabilities…

Volume 1 of this work has long been established as a classic in mathematical treatises and has gone through three quite different editions. It is elementary in the sense that it begins at the beginning and presupposes little background. [When it first came out it was much used as a textbook for a first course in probability; however, it is becoming increasingly difficult to do so as the audience has widened but its mathematical preparedness has not.] In fact, it deals only with a countable sample space so that all random variables and distributions are necessarily of the discrete type. This of course does not prevent Feller from going off at various deep ends and discussing some of the most up-to-date topics of interest. The restriction was deliberately imposed in order to waste no time on dull generalities (often indiscriminately referred to as “measure theory”) and to proceed at once to significant results. It is clear that many basic concepts cannot even be defined in the restricted context, and any serious exposition of stochastic processes in continuous time would be out of the question. But the latter is precisely where the action is at these days. Feller himself made fundamental contributions in the nineteen-fifties to this area called by him “generalized diffusions” and by others “Feller processes” (precursor of Doob-Hunt-Meyer processes); albeit largely in an analytic form, without “paths,” or rather with these only lurking in the background. It was therefore the general expectation that he would take this up in Volume 2 as a main theme. This would be a fitting sequel to his extremely successful popularization of Markov chain theory in Volume 1, acknowledged by himself in the Preface to the first edition. It must have come as a surprise to many that this is not what he did in Volume 2. Although there are recurrent and persistent allusions to bigger and better things to come (witness on page 333 “we are not at this juncture interested in developing a systematic theory”), general discussion of such processes is mostly relegated to heuristic descriptions, preparatory material and footnotes—not too copious. When asked why he chose not to expound his own theory of diffusion, Feller had responded by invoking the future promise of Volume 3, 4, ⋯. Alas, time has stopped too soon for him and his readers…

Is mathematics useful? We who are engaged in the profession must have had occasions to wonder about this question. It is often said that mathematics is not really a science; it is an art. It is an art in the sense that in its pursuit we strive for beauty, not utility. If we study mathematics as an art, perhaps we can justify it for its own sake, although there are some people who would disparage “art for art's sake”. When it comes to science, there is more reason to ask whether it is useful. Does it apply to the practical needs of our daily life? Does it contribute to the general well-being of society and mankind?…