G. H. HARDY AND PROBABILITY???

Stochastic Integrals and Abelian Processes

These notes are an elementary introduction to classical potential theory and to its connection with probabilistic tools such as stochastic calculus and the Markov property. In particular we review the probabilistic interpretations of harmonicity, of the Dirichlet problem, and of the Poisson equation using Brownian motion and stochastic calculus.

A general theory which provides asymptotic tail expansions for density, survival, and hazard rate functions is developed for both absolutely continuous and integer-valued distributions. The expansions make use of Tauberian theorems which apply to moment generating functions (MGFs) with boundary singularities that are of gamma-type or log-type. Standard Tauberian theorems from Feller [An Introduction to Probability Theory and Its Applications II (1971) Wiley] can provide a limited theory but these theorems do not suffice in providing a complete theory as they are not capable of explaining tail behaviour for compound distributions and other complicated distributions which arise in stochastic modelling settings. Obtaining such a complete theory for absolutely continuous distributions requires introducing new “Ikehara” conditions based upon Tauberian theorems whose development and application have been largely confined to analytic number theory. For integer-valued distributions, a complete theory is developed by applying Darboux’s theorem used in analytic combinatorics. Characterizations of asymptotic hazard rates for both absolutely continuous and integer-valued distributions are developed in conjunction with these expansions. The main applications include the ruin distribution in the Cramer–Lundberg and Sparre Andersen models, more general classes of compound distributions, and first-passage distributions in finite-state semi-Markov processes. Such first-passage distributions are shown to have exponential-like/geometric-like tails which mimic the behaviour of first-passage distributions in Markov processes even though the holding-time MGFs involved with such semi-Markov processes are typically not rational.

If you check the weather prediction on your phone, you might read that there is a 30 percent chance of rain at 4:00 in the afternoon. What does this mean? More precisely, what is the fraction 30/100 a measure of? Probability is a concept that is widespread both in everyday life and in science. Ordinary speakers of English utter and have some understanding of sentences such as “I will probably be late for the meeting,” or “it’s unlikely that Luxembourg will win the next World Cup.” Various sciences make explicit probabilistic claims: “the probability that a radium atom will decay in 1620 years is 0.5”; “the probability that a house mouse whose father is heterozygous for the t haplotype will inherit that trait is 0.9.” Other claims implicitly invoke probability: “the life expectancy of a child born in Japan today is 85.3 years.” Probability theory is also a major branch of mathematics, and it was given its modern formulation by Kolmogorov in 1933. Kolmogorov’s formalism presents a function P that satisfies a set of axioms: it is non-negative, normalized, and additive. These axioms entail a rich set of theorems concerning the behavior of P; together they make up the probability calculus. While the resulting theory is a formal theory in its own right, it is also natural to interpret P—to attach meanings, or truth conditions to claims involving it. “What is P?”, one may ask. This may be understood as a metaphysical question about what kinds of things are probabilities, or more generally as a question about what makes probability statements true or false. The various interpretations of probability attempt to answer this question, one way or another. This article surveys the literature on the interpretations of probability, due to mathematicians and especially philosophers. It divides the interpretations into two broad categories. Epistemological interpretations understand probability in terms of an agent’s beliefs, the strength of evidence in support of a statement, or other epistemological categories. Physical interpretations view probability as a feature of the world that would exist regardless of what evidence exists or what agents believe. This is a natural taxonomy, but others could be adopted, and its sub-categories are also somewhat pliable. The authors would like to thank Kim Border, Chris Bottomley, Kenny Easwaran, Hanti Lin, Michael Nielsen, Charles Sebens, Glenn Shafer, Julia Staffel, Jeremy Strasser, Michael Titelbaum, and an anonymous referee for many helpful suggestions.

A retrospective on 60 years of approximation theory and associated fields

The Nyman-Beurling criterion, equivalent to the Riemann hypothesis, is an approximation problem in the space of square integrable functions on $(0,\infty)$, involving dilations of the fractional part function by factors $\theta_k\in(0,1)$, $k\ge1$. Randomizing the $\theta_k$ generates new structures and criteria. One of them is a sufficient condition that splits into (i) showing that the indicator function can be approximated by convolution with the fractional part, (ii) a control on the coefficients of the approximation. This self-contained paper aims at identifying functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener's Tauberian theorem. In order to tackle (ii) in the future, we give some expressions of the scalar products. New and remarkable structures arise for the Gram matrix, in particular moment matrices for a suitable weight that may be the squared $\Xi$-function for instance.

Keynes begun his scientifïc career with probability theory. But, he had not the idea, as far as we can know, to give a probabilistic interpretation of his famous multiplier. This article is aimed at showing that probability theory, and especially finite Markov chains theory, gives an easier and even more natural interpretation of the keynesian multiplier than the traditional methods. Multiplier theory may be looked on as old-fashioned today, but it is still at the heart of most of macroeconometric models. So, we define first the relative position of the multiplier, which is linear and actually static, inside these models which are non-linear and dynamic. Secondly, we give a markovian interpretation of the income multiplier in both cases of the simple multiplier and the matrix multiplier. We compare it with the traditional interpretation: in the probabilistic interpretation every kind of economic agents (banks and firms, and not only households) take a part in the process of incomes which leads to the multiplier. Finally, we enlarge our method to the neighbouring analysis of the money multiplier and of the velocity of money. Our conclusion is that the markovian method could also be used for a keynesian crisis analysis.

Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not related to the non-standard analysis) is used to work with finite, infinite, and infinitesimal numbers numerically. This can be done on a new kind of a computer—the Infinity Computer—able to work with all these types of numbers. The new computational tools both give possibilities to execute computations of a new type and open new horizons for creating new mathematical models where a computational usage of infinite and/or infinitesimal numbers can be useful. A number of numerical examples showing the potential of the new approach and dealing with divergent series, limits, probability theory, linear algebra, and calculation of volumes of objects consisting of parts of different dimensions are given.

Population-size-dependent and age-dependent branching processes

K. Ito invented his famous stochastic calculus on Brownian motion in the 1940s. In the 1960s and 1970s, the “Strasbourg school,” headed by P.A. Meyer, developed a modern theory of martingales, the general theory of stochastic processes, and stochastic calculus on semimartingales. It turned out soon that semimartingales constitute the largest class of right continuous adapted integrators with respect to which stochastic integrals of simple predictable integrands satisfy the theorem of dominated convergence in probability. Stochastic calculus on semimartingales not only became an important tool for modern probability theory and stochastic processes but also has broad applications to many branches of mathematics, physics, engineering, and mathematical finance.

Fractional Brownian Motions, Fractional Noises and Applications

Stochastic integrals, random sums, random contractions, and selfdecomposable random variables constitute fundamental concepts of probability theory with significant applications in several areas of systemics. The main results of the paper are a characterization of a selfdecomposable distribution and a formulation of a Poisson random sum of power contractions. These results are established by incorporating a type of stochastic integral for a continuous in probability, homogeneous stochastic process with independent increments, and the same type of stochastic integral for a compound Poisson stochastic process with positive jumps. Interpretations of the results in treatment of risks threatening various systems are also provided.

Viacheslav P. Belavkin was born in Lvov on 30 May 1946. A 1970 graduate of Physics Department of Moscow State University, he received his PhD in 1973. His thesis, entitled Optimal Estimation and Measurements in Quantum Systems, written under the supervision of Professor Ruslan Stratonovich, was a pioneering work that would lay the foundations for the new field of quantum optimal filtering and control. As a student and then a collaborator of Stratonovich, Slava learnt very well stochastic calculus along with the theory of nonlinear filtering. His scientific programme included an extension of the ideas of filtering and control from classical to the quantum domain. In this area of research Slava worked on the development of quantum Markov models and quantum stochastic processes, dynamical nondemolition principle for quantum continuous measurements and, finally, on the formulation of quantum filtering theory. During his life Slava obtained a number of remarkable and highly original results in mathematical physics and quantum probability, and there are theorems and equations named after him. One of such results is the Belavkin quantum filtering equation — the evolution equation (an analog of the Schrodinger equation) for observed quantum systems. Overall Professor Belavkin published more than 200 papers. His doctoral dissertation (habilitation) was presented and defended at the Steklov Institute of Mathematical Science in Moscow in 1991. In 1996, Belavkin shared the Main State Prize of the Russian Federation with R. Stratonovich. In 1978/79 Slava spent one year visiting quantum probability group of Professor Roman S. Ingarden in Torun. It was the origin of his long standing

This is the first fundamental book devoted to non-Kolmogorov probability models. It provides a mathematical theory of negative probabilities, with numerous applications to quantum physics, information theory, complexity, biology and psychology. The book also presents an interesting model of cognitive information reality with flows of information probabilities, describing the process of thinking, social, and psychological phenomena.

The Riesz means, or sometimes typical means, were introduced by M. Riesz and have been studied in connection with summability of Fourier series and of Dirichlet series [8] and [11]. In number-theoretic context, it is the Riesz sum rather than the Riesz mean that has been extensively studied. The Riesz sums appear as long as there appears the G-function. Cf. Remark 1 and [14]. As is shown below, the Riesz sum corresponds to integration while Landau's differencing is an analogue of differentiation. This integration-differentiation aspect has been the driving force of many researches on number-theoretic asymptotic formulas. Ingham's decent treatment [13] of the prime number theorem is one typical example. We state some efficient theorems that give asymptotic formulas for the sums of coefficients of the generating Dirichlet series not necessarily satisfying the functional equation.