[Math.PR] 3 Sep 2002 Metric N N a Rce Oakqaiaieqetosaottena the About Questions Qualitative Ask to Convergence
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Weak Convergence of Probability Measures Revisited
WEAK CONVERGENCE OF PROBABILITY MEASURES REVISITED Cabriella saltnetti' Roger J-B wetse April 1987 W P-87-30 This research was supported in part by MPI, Projects: "Calcolo Stocastico e Sistemi Dinamici Statistici" and "Modelli Probabilistic" 1984, and by the National Science Foundation. Working Papers are interim reports on work of the lnternational Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute or of its National Member Organizations. INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSlS A-2361 Laxenburg, Austria FOREWORD The modeling of stochastic processes is a fundamental tool in the study of models in- volving uncertainty, a major topic at SDS. A number of classical convergence results (and extensions) for probability measures are derived by relying on new tools that are particularly useful in stochastic optimization and extremal statistics. Alexander B. Kurzhanski Chairman System and Decision Sciences Program ABSTRACT The hypo-convergence of upper semicontinuous functions provides a natural frame- work for the study of the convergence of probability measures. This approach also yields some further characterizations of weak convergence and tightness. CONTENTS 1 About Continuity and Measurability 2 Convergence of Sets and Semicontinuous Functions 3 SC-Measures and SGPrerneasures on 7 (E) 4 Hypo-Limits of SGMeasures and Tightness 5 Tightness and Equi-Semicontinuity Acknowledgment Appendix A Appendix B References - vii - WEAK CONVERGENCE OF PROBABILITY MEASURES REVISITED Gabriella salinettil and Roger J-B wetsL luniversiti "La Sapienza", 00185 Roma, Italy. 2~niversityof California, Davis, CA 95616. 1. ABOUT CONTINUITY AND MEASURABILITY A probabilistic structure - a space of possible events, a sigma-field of (observable) subcollections of events, and a probability measure defined on this sigma-field - does not have a built-in topological structure. -
Wasserstein Distance W2, Or to the Kantorovich-Wasserstein Distance W1, with Explicit Constants of Equivalence
THE EQUIVALENCE OF FOURIER-BASED AND WASSERSTEIN METRICS ON IMAGING PROBLEMS G. AURICCHIO, A. CODEGONI, S. GUALANDI, G. TOSCANI, M. VENERONI Abstract. We investigate properties of some extensions of a class of Fourier-based proba- bility metrics, originally introduced to study convergence to equilibrium for the solution to the spatially homogeneous Boltzmann equation. At difference with the original one, the new Fourier-based metrics are well-defined also for probability distributions with different centers of mass, and for discrete probability measures supported over a regular grid. Among other properties, it is shown that, in the discrete setting, these new Fourier-based metrics are equiv- alent either to the Euclidean-Wasserstein distance W2, or to the Kantorovich-Wasserstein distance W1, with explicit constants of equivalence. Numerical results then show that in benchmark problems of image processing, Fourier metrics provide a better runtime with respect to Wasserstein ones. Fourier-based Metrics; Wasserstein Distance; Fourier Transform; Image analysis. AMS Subject Classification: 60A10, 60E15, 42A38, 68U10. 1. Introduction In computational applied mathematics, numerical methods based on Wasserstein distances achieved a leading role over the last years. Examples include the comparison of histograms in higher dimensions [6,9,22], image retrieval [21], image registration [4,11], or, more recently, the computations of barycenters among images [7, 15]. Surprisingly, the possibility to identify the cost function in a Wasserstein distance, together with the possibility of representing images as histograms, led to the definition of classifiers able to mimic the human eye [16, 21, 24]. More recently, metrics which are able to compare at best probability distributions were introduced and studied in connection with machine learning, where testing the efficiency of new classes of loss functions for neural networks training has become increasingly important. -
Arxiv:1009.3824V2 [Math.OC] 7 Feb 2012 E Fpstv Nees Let Integers
OPTIMIZATION AND CONVERGENCE OF OBSERVATION CHANNELS IN STOCHASTIC CONTROL SERDAR YUKSEL¨ AND TAMAS´ LINDER Abstract. This paper studies the optimization of observation channels (stochastic kernels) in partially observed stochastic control problems. In particular, existence and continuity properties are investigated mostly (but not exclusively) concentrating on the single-stage case. Continuity properties of the optimal cost in channels are explored under total variation, setwise convergence, and weak convergence. Sufficient conditions for compactness of a class of channels under total variation and setwise convergence are presented and applications to quantization are explored. Key words. Stochastic control, information theory, observation channels, optimization, quan- tization AMS subject classifications. 15A15, 15A09, 15A23 1. Introduction. In stochastic control, one is often concerned with the following problem: Given a dynamical system, an observation channel (stochastic kernel), a cost function, and an action set, when does there exist an optimal policy, and what is an optimal control policy? The theory for such problems is advanced, and practically significant, spanning a wide variety of applications in engineering, economics, and natural sciences. In this paper, we are interested in a dual problem with the following questions to be explored: Given a dynamical system, a cost function, an action set, and a set of observation channels, does there exist an optimal observation channel? What is the right convergence notion for continuity in such observation channels for optimization purposes? The answers to these questions may provide useful tools for characterizing an optimal observation channel subject to constraints. We start with the probabilistic setup of the problem. Let X ⊂ Rn, be a Borel set in which elements of a controlled Markov process {Xt, t ∈ Z+} live. -
Arxiv:2102.05840V2 [Math.PR]
SEQUENTIAL CONVERGENCE ON THE SPACE OF BOREL MEASURES LIANGANG MA Abstract We study equivalent descriptions of the vague, weak, setwise and total- variation (TV) convergence of sequences of Borel measures on metrizable and non-metrizable topological spaces in this work. On metrizable spaces, we give some equivalent conditions on the vague convergence of sequences of measures following Kallenberg, and some equivalent conditions on the TV convergence of sequences of measures following Feinberg-Kasyanov-Zgurovsky. There is usually some hierarchy structure on the equivalent descriptions of convergence in different modes, but not always. On non-metrizable spaces, we give examples to show that these conditions are seldom enough to guarantee any convergence of sequences of measures. There are some remarks on the attainability of the TV distance and more modes of sequential convergence at the end of the work. 1. Introduction Let X be a topological space with its Borel σ-algebra B. Consider the collection M˜ (X) of all the Borel measures on (X, B). When we consider the regularity of some mapping f : M˜ (X) → Y with Y being a topological space, some topology or even metric is necessary on the space M˜ (X) of Borel measures. Various notions of topology and metric grow out of arXiv:2102.05840v2 [math.PR] 28 Apr 2021 different situations on the space M˜ (X) in due course to deal with the corresponding concerns of regularity. In those topology and metric endowed on M˜ (X), it has been recognized that the vague, weak, setwise topology as well as the total-variation (TV) metric are highlighted notions on the topological and metric description of M˜ (X) in various circumstances, refer to [Kal, GR, Wul]. -
Statistics and Probability Letters a Note on Vague Convergence Of
Statistics and Probability Letters 153 (2019) 180–186 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro A note on vague convergence of measures ∗ Bojan Basrak, Hrvoje Planini¢ Department of Mathematics, Faculty of Science, University of Zagreb, Bijeni£ka 30, Zagreb, Croatia article info a b s t r a c t Article history: We propose a new approach to vague convergence of measures based on the general Received 17 April 2019 theory of boundedness due to Hu (1966). The article explains how this connects and Accepted 6 June 2019 unifies several frequently used types of vague convergence from the literature. Such Available online 20 June 2019 an approach allows one to translate already developed results from one type of vague MSC: convergence to another. We further analyze the corresponding notion of vague topology primary 28A33 and give a new and useful characterization of convergence in distribution of random secondary 60G57 measures in this topology. 60G70 ' 2019 Elsevier B.V. All rights reserved. Keywords: Boundedly finite measures Vague convergence w#–convergence Random measures Convergence in distribution Lipschitz continuous functions 1. Introduction Let X be a Polish space, i.e. separable topological space which is metrizable by a complete metric. Denote by B(X) the corresponding Borel σ –field and choose a subfamily Bb(X) ⊆ B(X) of sets, called bounded (Borel) sets of X. When there is no fear of confusion, we will simply write B and Bb. A Borel measure µ on X is said to be locally (or boundedly) finite if µ(B) < 1 for all B 2 Bb. -
Weak Convergence of Measures
Mathematical Surveys and Monographs Volume 234 Weak Convergence of Measures Vladimir I. Bogachev Weak Convergence of Measures Mathematical Surveys and Monographs Volume 234 Weak Convergence of Measures Vladimir I. Bogachev EDITORIAL COMMITTEE Walter Craig Natasa Sesum Robert Guralnick, Chair Benjamin Sudakov Constantin Teleman 2010 Mathematics Subject Classification. Primary 60B10, 28C15, 46G12, 60B05, 60B11, 60B12, 60B15, 60E05, 60F05, 54A20. For additional information and updates on this book, visit www.ams.org/bookpages/surv-234 Library of Congress Cataloging-in-Publication Data Names: Bogachev, V. I. (Vladimir Igorevich), 1961- author. Title: Weak convergence of measures / Vladimir I. Bogachev. Description: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Mathe- matical surveys and monographs ; volume 234 | Includes bibliographical references and index. Identifiers: LCCN 2018024621 | ISBN 9781470447380 (alk. paper) Subjects: LCSH: Probabilities. | Measure theory. | Convergence. Classification: LCC QA273.43 .B64 2018 | DDC 519.2/3–dc23 LC record available at https://lccn.loc.gov/2018024621 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. -
Lecture 20: Weak Convergence (PDF)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 20 11/25/2013 Introduction to the theory of weak convergence Content. 1. σ-fields on metric spaces. 2. Kolmogorov σ-field on C[0;T ]. 3. Weak convergence. In the first two sections we review some concepts from measure theory on metric spaces. Then in the last section we begin the discussion of the theory of weak convergence, by stating and proving important Portmentau Theorem, which gives four equivalent definitions of weak convergence. 1 Borel σ-fields on metric space We consider a metric space (S; ρ). To discuss probability measures on metric spaces we first need to introduce σ-fields. Definition 1. Borel σ-field B on S is the field generated by the set of all open sets U ⊂ S. Lemma 1. Suppose S is Polish. Then every open set U ⊂ S can be represented as a countable union of balls B(x; r). Proof. Since S is Polish, we can identify a countable set x1; : : : ; xn;::: which is dense. For each xi 2 U, since U is open we can find a radius ri such that ∗ B(xi; ri) ⊂ U. Fix a constant M > 0. Consider ri = min(arg supfr : B(x; r) ⊂ Ug;M) > 0 and set r = r∗=2. Then [ B(x ; r ) ⊂ U. In i i i:xi2U i i order to finish the proof we need to show that equality [i:xi2U B(xi; ri) = U holds. Consider any x 2 U. There exists 0 < r ≤ M such that B(x; r) ⊂ U. -
Probability Measures on Metric Spaces
Probability measures on metric spaces Onno van Gaans These are some loose notes supporting the first sessions of the seminar Stochastic Evolution Equations organized by Dr. Jan van Neerven at the Delft University of Technology during Winter 2002/2003. They contain less information than the common textbooks on the topic of the title. Their purpose is to present a brief selection of the theory that provides a basis for later study of stochastic evolution equations in Banach spaces. The notes aim at an audience that feels more at ease in analysis than in probability theory. The main focus is on Prokhorov's theorem, which serves both as an important tool for future use and as an illustration of techniques that play a role in the theory. The field of measures on topological spaces has the luxury of several excellent textbooks. The main source that has been used to prepare these notes is the book by Parthasarathy [6]. A clear exposition is also available in one of Bour- baki's volumes [2] and in [9, Section 3.2]. The theory on the Prokhorov metric is taken from Billingsley [1]. The additional references for standard facts on general measure theory and general topology have been Halmos [4] and Kelley [5]. Contents 1 Borel sets 2 2 Borel probability measures 3 3 Weak convergence of measures 6 4 The Prokhorov metric 9 5 Prokhorov's theorem 13 6 Riesz representation theorem 18 7 Riesz representation for non-compact spaces 21 8 Integrable functions on metric spaces 24 9 More properties of the space of probability measures 26 1 The distribution of a random variable in a Banach space X will be a probability measure on X. -
Free Complete Wasserstein Algebras
FREE COMPLETE WASSERSTEIN ALGEBRAS RADU MARDARE, PRAKASH PANANGADEN, AND GORDON D. PLOTKIN Abstract. We present an algebraic account of the Wasserstein distances Wp on complete metric spaces, for p ≥ 1. This is part of a program of a quantitative algebraic theory of effects in programming languages. In particular, we give axioms, parametric in p, for algebras over metric spaces equipped with probabilistic choice operations. The axioms say that the operations form a barycentric algebra and that the metric satisfies a property typical of the Wasserstein distance Wp. We show that the free complete such algebra over a complete metric space is that of the Radon probability measures with finite moments of order p, equipped with the Wasserstein distance as metric and with the usual binary convex sums as operations. 1. Introduction The denotational semantics of probabilistic programs generally makes use of one or another monad of probability measures. In [18, 10], Lawvere and Giry provided the first example of such a monad, over the category of measurable spaces; Giry also gave a second example, over Polish spaces. Later, in [15, 14], Jones and Plotkin provided a monad of evaluations over directed complete posets. More recently, in [12], Heunen, Kammar, et al provided such a monad over quasi-Borel spaces. Metric spaces (and nonexpansive maps) form another natural category to consider. The question then arises as to what metric to take over probability measures, and the Kantorovich-Wasserstein, or, more generally, the Wasserstein distances of order p > 1 (generalising from p = 1), provide natural choices. A first result of this kind was given in [2], where it was shown that the Kantorovich distance provides a monad over the sub- category of 1-bounded complete separable metric spaces. -
Statistical Aspects of Wasserstein Distances
Statistical Aspects of Wasserstein Distances Victor M. Panaretos∗and Yoav Zemely June 8, 2018 Abstract Wasserstein distances are metrics on probability distributions inspired by the problem of optimal mass transportation. Roughly speaking, they measure the minimal effort required to reconfigure the probability mass of one distribution in order to recover the other distribution. They are ubiquitous in mathematics, with a long history that has seen them catal- yse core developments in analysis, optimization, and probability. Beyond their intrinsic mathematical richness, they possess attractive features that make them a versatile tool for the statistician: they can be used to de- rive weak convergence and convergence of moments, and can be easily bounded; they are well-adapted to quantify a natural notion of pertur- bation of a probability distribution; and they seamlessly incorporate the geometry of the domain of the distributions in question, thus being useful for contrasting complex objects. Consequently, they frequently appear in the development of statistical theory and inferential methodology, and have recently become an object of inference in themselves. In this review, we provide a snapshot of the main concepts involved in Wasserstein dis- tances and optimal transportation, and a succinct overview of some of their many statistical aspects. Keywords: deformation map, empirical optimal transport, Fr´echet mean, goodness-of-fit, inference, Monge{Kantorovich problem, optimal coupling, prob- ability metric, transportation of measure, warping and registration, Wasserstein space AMS subject classification: 62-00 (primary); 62G99, 62M99 (secondary) arXiv:1806.05500v3 [stat.ME] 9 Apr 2019 1 Introduction Wasserstein distances are metrics between probability distributions that are inspired by the problem of optimal transportation. -
The Linear Topology Associated with Weak Convergence of Probability
The linear topology associated with weak convergence of probability measures Liang Hong 1 Abstract This expository note aims at illustrating weak convergence of probability measures from a broader view than a previously published paper. Though the results are standard for functional analysts, this approach is rarely known by statisticians and our presentation gives an alternative view than most standard probability textbooks. In particular, this functional approach clarifies the underlying topological structure of weak convergence. We hope this short note is helpful for those who are interested in weak convergence as well as instructors of measure theoretic probability. MSC 2010 Classification: Primary 00-01; Secondary 60F05, 60A10. Keywords: Weak convergence; probability measure; topological vector space; dual pair arXiv:1312.6589v3 [math.PR] 2 Oct 2014 1Liang Hong is an Assistant Professor in the Department of Mathematics, Robert Morris University, 6001 University Boulevard, Moon Township, PA 15108, USA. Tel.: (412) 397-4024. Email address: [email protected]. 1 1 Introduction Weak convergence of probability measures is often defined as follows (Billingsley 1999 or Parthasarathy 1967). Definition 1.1. Let X be a metrizable space. A sequence {Pn} of probability measures on X is said to converge weakly to a probability measure P if lim f(x)dPn(dx)= f(x)P (dx) n→∞ ZX ZX for every bounded continuous function f on X. It is natural to ask the following questions: 1. Why the type of convergence defined in Definition 1.1 is called “weak convergence”? 2. How weak it is? 3. Is there any connection between weak convergence of probability measures and weak convergence in functional analysis? 4. -
BILLINGSLEY Tee University of Chicago Chicago, Illinois
Convergence of Probability Measures WILEY SERIES IN PROBABILITY AND STATISTICS PROBABILITY AND STATISTICS SECTION Established by WALTER A. SHEWHART and SAMUEL S. WILKS Editors: Vic Barnett, Noel A. C. Cressie, Nicholas I. Fisher, Iain M, Johnstone, J. B. Kadane, David G. Kendall, David W. Scott, Bernard W. Silverman, Adrian F. M. Smith, Jozef L. Teugels; Ralph A. Bradley, Emeritus, J. Stuart Hunter, Emeritus A complete list of the titles in this series appears at the end of this volume. Convergence of Probability Measures Second Edition PATRICK BILLINGSLEY TEe University of Chicago Chicago, Illinois A Wiley-Interscience Publication JOHN WILEY & SONS,INC. NewYork Chichester Weinheim Brisbane Singapore Toronto This text is printed on acid-free paper. 8 Copyright 0 1999 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. No patt of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-601 1, fax (212) 850-6008, E-Mail: PERMREQ @, WILEY.COM. For ordering and customer service, call 1-800-CALL-WILEY.