J. R. Norris

Bio: J. R. Norris is an academic researcher. The author has an hindex of 1, co-authored 1 publications receiving 269 citations.

Probability theory - an analytic view, by Daniel W. Stroock. Pp 512. £30. 1994. ISBN 0-521-43123-9 (Cambridge University Press)

Abstract: This second edition of Daniel W. Stroock's text is suitable for first-year graduate students with a good grasp of introductory, undergraduate probability theory and a sound grounding in analysis. It is intended to provide readers with an introduction to probability theory and the analytic ideas and tools on which the modern theory relies. It includes more than 750 exercises. Much of the content has undergone significant revision. In particular, the treatment of Levy processes has been rewritten, and a detailed account of Gaussian measures on a Banach space is given. The first part of the book deals with independent random variables, Central Limit phenomena, and the construction of Levy processes, including Brownian motion. Conditioning is developed and applied to discrete parameter martingales in Chapter 5, Chapter 6 contains the ergodic theorem and Burkholder's inequality, and continuous parameter martingales are discussed in Chapter 7. Chapter 8 is devoted to Gaussian measures on a Banach space, where they are treated from the abstract Wiener space perspective. The abstract theory of weak convergence is developed in Chapter 9, which ends with a proof of Donsker's Invariance Principle. The concluding two chapters contain applications of Brownian motion to the analysis of partial differential equations and potential theory.

Gradient Flows: In Metric Spaces and in the Space of Probability Measures

Abstract: Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence Theorems.- Uniqueness, Generation of Contraction Semigroups, Error Estimates.- Gradient Flow in the Space of Probability Measures.- Preliminary Results on Measure Theory.- The Optimal Transportation Problem.- The Wasserstein Distance and its Behaviour along Geodesics.- Absolutely Continuous Curves in p(X) and the Continuity Equation.- Convex Functionals in p(X).- Metric Slope and Subdifferential Calculus in (X).- Gradient Flows and Curves of Maximal Slope in p(X).

Elements of large-sample theory

Abstract: This introductory book on the most useful parts of large-sample theory is designed to be accessible to scientists outside statistics and certainly to master’s-level statistics students who ignore most of measure theory. According to the author, “the subject of this book, first-order large- sample theory, constitutes a coherent body of concepts and results that are central to both theoretical and applied statistics.” All of the other existing books published on the subject over the last 20 years, from Ibragimov and Has’minskii in 1979 to the most recent by Van der Waart in 1998 have a common prerequisite in mathematical sophistication (measure theory in particular) that do not make the concepts available to a wide audience.

Blow-up of semilinear PDE's at the critical dimension. A probabilistic approach

Abstract: We present a probabilistic approach which proves blow-up of solutions of the Fujita equation ∂w/∂t = -(-Δ) α/2 w + w 1+β in the critical dimension d = α/β. By using the Feynman-Kac representation twice, we construct a subsolution which locally grows to infinity as t → ∞. In this way, we cover results proved earlier by analytic methods. Our method also applies to extend a blow-up result for systems proved for the Laplacian case by Escobedo and Levine (1995) to the case of α-Laplacians with possibly different parameters a.

Game theory, maximum entropy, minimum discrepancy and robust Bayesian decision theory

Abstract: We describe and develop a close relationship between two problems that have customarily been regarded as distinct: that of maximizing entropy, and that of minimizing worst-case expected loss Using a formulation grounded in the equilibrium theory of zero-sum games between Decision Maker and Nature, these two problems are shown to be dual to each other, the solution to each providing that to the other Although Topsoe described this connection for the Shannon entropy over 20 years ago, it does not appear to be widely known even in that important special case We here generalize this theory to apply to arbitrary decision problems and loss functions We indicate how an appropriate generalized definition of entropy can be associated with such a problem, and we show that, subject to certain regularity conditions, the above-mentioned duality continues to apply in this extended context This simultaneously provides a possible rationale for maximizing entropy and a tool for finding robust Bayes acts We also describe the essential identity between the problem of maximizing entropy and that of minimizing a related discrepancy or divergence between distributions This leads to an extension, to arbitrary discrepancies, of a well-known minimax theorem for the case of Kullback–Leibler divergence (the “redundancy-capacity theorem” of information theory) For the important case of families of distributions having certain mean values specified, we develop simple sufficient conditions and methods for identifying the desired solutions We use this theory to introduce a new concept of “generalized exponential family” linked to the specific decision problem under consideration, and we demonstrate that this shares many of the properties of standard exponential families Finally, we show that the existence of an equilibrium in our game can be rephrased in terms of a “Pythagorean property” of the related divergence, thus generalizing previously announced results for Kullback–Leibler and Bregman divergences

Opinion Fluctuations and Disagreement in Social Networks

Abstract: We study a tractable opinion dynamics model that generates long-run disagreements and persistent opinion fluctuations. Our model involves an inhomogeneous stochastic gossip process of continuous opinion dynamics in a society consisting of two types of agents: 1 regular agents who update their beliefs according to information that they receive from their social neighbors and 2 stubborn agents who never update their opinions and might represent leaders, political parties, or media sources attempting to influence the beliefs in the rest of the society. When the society contains stubborn agents with different opinions, the belief dynamics never lead to a consensus among the regular agents. Instead, beliefs in the society fail to converge almost surely, the belief profile keeps on fluctuating in an ergodic fashion, and it converges in law to a nondegenerate random vector. The structure of the graph describing the social network and the location of the stubborn agents within it shape the opinion dynamics. The expected belief vector is proved to evolve according to an ordinary differential equation coinciding with the Kolmogorov backward equation of a continuous-time Markov chain on the graph with absorbing states corresponding to the stubborn agents, and hence to converge to a harmonic vector, with every regular agent's value being the weighted average of its neighbors' values, and boundary conditions corresponding to the stubborn agents' beliefs. Expected cross products of the agents' beliefs allow for a similar characterization in terms of coupled Markov chains on the graph describing the social network. We prove that, in large-scale societies, which are highly fluid, meaning that the product of the mixing time of the Markov chain on the graph describing the social network and the relative size of the linkages to stubborn agents vanishes as the population size grows large, a condition of homogeneous influence emerges, whereby the stationary beliefs' marginal distributions of most of the regular agents have approximately equal first and second moments.