Strong Approximation of the Empirical Distribution Function for Absolutely Regular Sequences in Rd
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On the Mean Speed of Convergence of Empirical and Occupation Measures in Wasserstein Distance Emmanuel Boissard, Thibaut Le Gouic
On the mean speed of convergence of empirical and occupation measures in Wasserstein distance Emmanuel Boissard, Thibaut Le Gouic To cite this version: Emmanuel Boissard, Thibaut Le Gouic. On the mean speed of convergence of empirical and occupation measures in Wasserstein distance. Annales de l’Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2014, 10.1214/12-AIHP517. hal-01291298 HAL Id: hal-01291298 https://hal.archives-ouvertes.fr/hal-01291298 Submitted on 8 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ON THE MEAN SPEED OF CONVERGENCE OF EMPIRICAL AND OCCUPATION MEASURES IN WASSERSTEIN DISTANCE EMMANUEL BOISSARD AND THIBAUT LE GOUIC Abstract. In this work, we provide non-asymptotic bounds for the average speed of convergence of the empirical measure in the law of large numbers, in Wasserstein distance. We also consider occupation measures of ergodic Markov chains. One motivation is the approximation of a probability measure by finitely supported measures (the quantization problem). It is found that rates for empirical or occupation measures match or are close to previously known optimal quantization rates in several cases. This is notably highlighted in the example of infinite-dimensional Gaussian measures. -
Introduction to Empirical Processes and Semiparametric Inference1
This is page i Printer: Opaque this Introduction to Empirical Processes and Semiparametric Inference1 Michael R. Kosorok August 2006 1c 2006 SPRINGER SCIENCE+BUSINESS MEDIA, INC. All rights re- served. Permission is granted to print a copy of this preliminary version for non- commercial purposes but not to distribute it in printed or electronic form. The author would appreciate notification of typos and other suggestions for improve- ment. ii This is page iii Printer: Opaque this Preface The goal of this book is to introduce statisticians, and other researchers with a background in mathematical statistics, to empirical processes and semiparametric inference. These powerful research techniques are surpris- ingly useful for studying large sample properties of statistical estimates from realistically complex models as well as for developing new and im- proved approaches to statistical inference. This book is more a textbook than a research monograph, although some new results are presented in later chapters. The level of the book is more introductory than the seminal work of van der Vaart and Wellner (1996). In fact, another purpose of this work is to help readers prepare for the mathematically advanced van der Vaart and Wellner text, as well as for the semiparametric inference work of Bickel, Klaassen, Ritov and Wellner (1997). These two books, along with Pollard (1990) and chapters 19 and 25 of van der Vaart (1998), formulate a very complete and successful elucida- tion of modern empirical process methods. The present book owes much by the way of inspiration, concept, and notation to these previous works. What is perhaps new is the introductory, gradual and unified way this book introduces the reader to the field. -
Limit Theorems for Random Fields
Limit Theorems for Random Fields A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematical Sciences College of Arts and Sciences University of Cincinnati, July 2019 Author: Na Zhang Chair: Magda Peligrad, Ph.D. Degrees: B.S. Mathematics, 2011, Committee: Wlodzimierz Bryc, Ph.D. Huaibei Normal University Yizao Wang, Ph.D. M.S. Financial Mathematics, 2014, Soochow University Abstract The focus of this dissertation is on the dependent structure and limit theorems of high dimensional probability theory. In this dissertation, we investigate two related topics: the Central Limit Theorem (CLT) for stationary random fields (multi-indexed random variables) and for Fourier transform of stationary random fields. We show that the CLT for stationary random processes under a sharp projective condition introduced by Maxwell and Woodroofe in 2000 [MW00] can be extended to random fields. To prove this result, new theorems are established herein for triangular arrays of martingale differences which have interest in themselves. Later on, to exploit the richness of martingale techniques, we establish the necessary and sufficient conditions for martingale approximation of random fields, which extend to random fields many corresponding results for random sequences (e.g. [DMV07]). Besides, a stronger form of convergence, the quenched convergence, is investigated and a quenched CLT is obtained under some projective criteria. The discrete Fourier transform of random fields, (Xk)k2Zd (d ≥ 2), where Z is the set of integers, is defined as the rotated sum of the random fields. Being one of the important tools to prove the CLT for Fourier transform of random fields, the law of large numbers (LLN) is obtained for discrete Fourier transform of random sequences under a very mild regularity condition. -
Applications of Empirical Process Theory Contents
Applications of Empirical Process Theory Contents 1 Introduction 3 1.1 Historical Motivation . 3 1.2 Basic Definitions . 4 1.3 Organisation of Chapters . 5 1.4 Notations . 5 2 Tools for Studying Empirical Processes 6 2.1 Entropy . 6 2.2 Symmetrisation . 7 2.3 Maximal Inequalities . 9 3 Empirical Process Theory 13 3.1 Uniform Law of Large Numbers . 13 3.2 Uniform Central Limit Theorem . 15 3.3 Vapnik{Cervonenkisˇ Class . 17 4 Statistical Applications 21 4.1 Direct Applications . 21 4.2 Functional Delta Method . 24 4.3 Nonparametric Maximum Likelihood Estimators . 27 Bibliography 37 2 Chapter 1 Introduction 1.1 Historical Motivation Historically, empirical process theory has one of its roots in the study of goodness of fit statistics. The first goodness-of-fit statistic was Pearson's chi-square statistic. Recall that for Pearson's goodness-of-fit test, to test whether independent and identically distributed (i.i.d.) real random variables X1;:::;Xn come from the distribution F , we partition the real line into intervals (bins), indexed 1; : : : ; m, and compare Ei, the expected number of data points in the i-th interval under the null distribution F , with Oi, the observed number. The null hypothesis Pm 2 is rejected when the Pearson chi-square statistic i=1(Oi − Ei) =Ei is large compared with 2 a χm−1 distribution. Pearson's idea of binning discretises a continuous distribution into a more tractable multinomial distribution, making the chi-square statistics easy to understand and simple to implement. However, the downside of using bins is its arbitrariness and loss of information during discretisation, leading to a loss in statistical power. -
Mean Field Simulation for Monte Carlo Integration MONOGRAPHS on STATISTICS and APPLIED PROBABILITY
Mean Field Simulation for Monte Carlo Integration MONOGRAPHS ON STATISTICS AND APPLIED PROBABILITY General Editors F. Bunea, V. Isham, N. Keiding, T. Louis, R. L. Smith, and H. Tong 1. Stochastic Population Models in Ecology and Epidemiology M.S. Barlett (1960) 2. Queues D.R. Cox and W.L. Smith (1961) 3. Monte Carlo Methods J.M. Hammersley and D.C. Handscomb (1964) 4. The Statistical Analysis of Series of Events D.R. Cox and P.A.W. Lewis (1966) 5. Population Genetics W.J. Ewens (1969) 6. Probability, Statistics and Time M.S. Barlett (1975) 7. Statistical Inference S.D. Silvey (1975) 8. The Analysis of Contingency Tables B.S. Everitt (1977) 9. Multivariate Analysis in Behavioural Research A.E. Maxwell (1977) 10. Stochastic Abundance Models S. Engen (1978) 11. Some Basic Theory for Statistical Inference E.J.G. Pitman (1979) 12. Point Processes D.R. Cox and V. Isham (1980) 13. Identification of OutliersD.M. Hawkins (1980) 14. Optimal Design S.D. Silvey (1980) 15. Finite Mixture Distributions B.S. Everitt and D.J. Hand (1981) 16. ClassificationA.D. Gordon (1981) 17. Distribution-Free Statistical Methods, 2nd edition J.S. Maritz (1995) 18. Residuals and Influence in RegressionR.D. Cook and S. Weisberg (1982) 19. Applications of Queueing Theory, 2nd edition G.F. Newell (1982) 20. Risk Theory, 3rd edition R.E. Beard, T. Pentikäinen and E. Pesonen (1984) 21. Analysis of Survival Data D.R. Cox and D. Oakes (1984) 22. An Introduction to Latent Variable Models B.S. Everitt (1984) 23. Bandit Problems D.A. Berry and B. -
Chapter 6 Stationary Stochastic Processes
Chapter 6 Stationary Stochastic Processes. 6.1 Ergodic Theorems. A stationary stochastic process is a collection {ξn : n ∈ Z} of random vari- ables with values in some space (X, B) such that the joint distribution of ··· ··· (ξn1 , ,ξnk ) is the same as that of (ξn1+n, ,ξnk+n) for every choice of k ≥ 1, and n, n1, ··· ,nk ∈ Z. Assuming that the space (X, B) is reasonable and Kolmogorov’s consistency theorem applies, we can build a measure P on the countable product space Ω of sequences {xn : n ∈ Z} with values in X, defined for sets in the product σ-field F. On the space Ω there is the natural shift defined by (Tω)(n)=xn+1 for ω with ω(n)=xn. The random variables xn(ω)=ω(n) are essentially equivalent to {ξn}. The stationarity of the pro- cess is reflected in the invariance of P with respect to T i.e. PT−1 = P .We can without being specific consider a space Ω a σ-field F, a one to one invert- ible measurable map from Ω → Ω with a measurable inverse T −1 and finally a probability measure P on (Ω, F)thatisT -invariant i.e P (T −1A)=P (A) for every A ∈F. One says that P is an invariant measure for T or T is a measure preserving transformation for P . If we have a measurable map from n ξ :(Ω, F) → (X, B), then it is easily seen that ξn(ω)=ξ(T ω) defines a sta- tionary stochastic process. The study of stationary stochastic process is then more or less the same as the study of measure preserving (i.e. -
Recent Advances in Invariance Principles for Stationary Sequences 2
Probability Surveys Vol. 3 (2006) 1–36 ISSN: 1549-5787 DOI: 10.1214/154957806100000202 Recent advances in invariance principles for stationary sequences∗ Florence Merlev`ede Universit´eParis 6, LPMA and C.N.R.S UMR 7599, 175 rue du Chevaleret, 75013 Paris, FRANCE e-mail: [email protected] Magda Peligrad† Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, Oh 45221-0025, USA e-mail: [email protected] Sergey Utev School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, ENGLAND e-mail: [email protected] Abstract: In this paper we survey some recent results on the central limit theorem and its weak invariance principle for stationary sequences. We also describe several maximal inequalities that are the main tool for obtaining the invariance principles, and also they have interest in themselves. The classes of dependent random variables considered will be martingale-like sequences, mixing sequences, linear processes, additive functionals of er- godic Markov chains. AMS 2000 subject classifications: 60G51, 60F05. Keywords and phrases: Brownian motion, weakly dependent sequences, martingale differences, linear process, central limit theorem, invariance prin- ciple. Received November 2004. Contents 1 Introduction 2 arXiv:math/0601315v2 [math.PR] 24 Feb 2006 2 Definitions and notations 3 3 Maximal inequalities 5 ∗This is an original survey paper †Supported in part by a Charles Phelps Taft Memorial Fund grant and NSA grant H98230- 05-1-0066. 1 F. Merlev`ede et al./Recent advances in invariance principles for stationary sequences 2 4 Invariance principles 8 4.1 The normalization √n ........................ 8 4.2 Examples.Part1.......................... -
The Convergence of Monte Carlo Estimates of Distributions
Noname manuscript No. (will be inserted by the editor) The convergence of Monte Carlo estimates of distributions Nicholas A. Heard and Melissa J. M. Turcotte Received: date / Accepted: date Abstract It is often necessary to make sampling-based sta- where δXi is a Dirac measure at Xi and X is the support tistical inference about many probability distributions in par- of π. The random measure (1) is a maximum likelihood es- allel. Given a finite computational resource, this article ad- timator of π and is consistent: for all π-measurable sets B, dresses how to optimally divide sampling effort between the limn→∞ Πˆn(B)= π(B). samplers of the different distributions. Formally approach- Sometimes estimating the entire distribution π is of in- ing this decision problem requires both the specification of trinsic inferential interest. In other cases, this may be de- an error criterion to assess how well each group of samples sirable if there are no limits on the functionals of π which represent their underlying distribution, and a loss function might be of future interest. Alternatively, the random sam- to combine the errors into an overall performance score. For pling might be an intermediary update of a sequential Monte the first part, a new Monte Carlo divergence error criterion Carlo sampler (Del Moral et al., 2006) for which it is desir- based on Jensen-Shannon divergence is proposed. Using re- able that the samples represent the current target distribution sults from information theory, approximations are derived well at each step. for estimating this criterion for each target based on a sin- Pointwise Monte Carlo errors are inadequate for captur- gle run, enabling adaptive sample size choices to be made ing the overall rate of convergence of the realised empirical during sampling. -
Arxiv:1606.07664V1 [Math.PR]
A GLIVENKO–CANTELLI THEOREM FOR ALMOST ADDITIVE FUNCTIONS ON LATTICES CHRISTOPH SCHUMACHER, FABIAN SCHWARZENBERGER, IVAN VESELIĆ Abstract. We develop a Glivenko–Cantelli theory for monotone, almost additive functions of i. i. d. sequences of random variables indexed by Zd. Under certain condi- tions on the random sequence, short range correlations are allowed as well. We have an explicit error estimate, consisting of a probabilistic and a geometric part. We ap- ply the results to yield uniform convergence for several quantities arising naturally in statistical physics. 1. Introduction The classical Glivenko–Cantelli theorem states that the empirical cumulative distribution functions of an increasing set of independent and identically distributed random variables converge uniformly to the cumulative population distribution function almost surely. Due to its importance to applications, e. g. statistical learning theory, the Glivenko– Cantelli theorem is also called the “fundamental theorem of statistics”. The theorem has initiated the study of so-called Glivenko–Cantelli classes as they feature, for instance, in the Vapnik–Chervonenkis theory [VC71]. Generalizations of the fundamental theorem rewrite the uniform convergence with respect to the real variable as convergence of a supremum over a family (of sets or functions) and widen the family over which the supremum is taken, making the statement “more uniform”. However, there are limits to this uniformization: For instance, if the original distribution is continuous, there is no convergence if the supremum is taken w. r. t. the family of finite subsets of the reals. Thus, a balance has to be found between the class over which the supremum is taken and the distribution of the random variables, the details of which are often dictated by the application in mind. -
Monte Carlo Methods
Adam M. Johansen and Ludger Evers Monte Carlo Methods Lecture Notes November 15, 2007 Department of Mathematics 2 Table of Contents Table of Contents ................................................... ................... 3 1. Introduction ................................................... .................... 5 1.1 What are Monte Carlo Methods?........................... ....................... 5 1.2 Introductory examples . ......................... 5 1.3 A Brief History of Monte Carlo Methods .................... ....................... 9 1.4 Pseudo-random numbers . ..................... 10 2. Fundamental Concepts: Transformation, Rejection, and Reweighting .............. 15 2.1 Transformation Methods ............................... .......................... 15 2.2 Rejection Sampling . ......................... 16 2.3 Importance Sampling . ......................... 19 3. Markov Chains ................................................... ................. 25 3.1 Stochastic Processes . ........................... 25 3.2 Discrete State Space Markov Chains . ..................... 26 3.3 General State Space Markov Chains . ................... 33 3.4 Selected Theoretical Results . ..................... 37 3.5 Further Reading . ......................... 38 4. The Gibbs Sampler ................................................... ............. 39 4.1 Introduction ....................................... ............................. 39 4.2 Algorithm ........................................ .............................. 40 4.3 The Hammersley-Clifford -
ONE-DIMENSIONAL EMPIRICAL MEASURES, ORDER STATISTICS, and KANTOROVICH TRANSPORT DISTANCES Sergey Bobkov and Michel Ledoux Univer
ONE-DIMENSIONAL EMPIRICAL MEASURES, ORDER STATISTICS, AND KANTOROVICH TRANSPORT DISTANCES Sergey Bobkov and Michel Ledoux University of Minnesota∗ and University of Toulousey December 19, 2016 Abstract. This work is devoted to the study of rates of convergence of the empiri- 1 Pn cal measures µn = n k=1 δXk , n ≥ 1, over a sample (Xk)k≥1 of independent identically distributed real-valued random variables towards the common distribution µ in Kan- torovich transport distances Wp. The focus is on finite range bounds on the expected p 1=p Kantorovich distances E(Wp(µn; µ)) or E(Wp (µn; µ)) in terms of moments and an- alytic conditions on the measure µ and its distribution function. The study describes a p1 variety of rates, from the standard one n to slower rates, and both lower and upper- bounds on E(Wp(µn; µ)) for fixed n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, log- concavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation. Keywords. Empirical measure, Kantorovich distance, rate of convergence, finite rate bound, order statistic, inverse distribution function, beta distribution, log-concave measure. Mathematics Subject Classification (2010). Primary 60B10, 60F99, 60G57, 62G30, 60B12; Secondary 62G20. ∗School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA, [email protected] yInstitut de Math´ematiquesde Toulouse, Universit´ede Toulouse, F-31062 Toulouse, France, and Institut Universitaire de France, [email protected] 1 2 Contents 1 Introduction 4 2 Generalities on Kantorovich transport distances 10 2.1 Kantorovich transport distance Wp .................... -
Donsker and Glivenko-Cantelli Theorems for a Class of Processes Generalizing the Empirical Process Davit Varron
Donsker and Glivenko-Cantelli theorems for a class of processes generalizing the empirical process Davit Varron To cite this version: Davit Varron. Donsker and Glivenko-Cantelli theorems for a class of processes generalizing the empiri- cal process. Electronic Journal of Statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2014, pp.2301-2325. hal-01158050 HAL Id: hal-01158050 https://hal.archives-ouvertes.fr/hal-01158050 Submitted on 31 Aug 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Electronic Journal of Statistics ISSN: 1935-7524 arXiv: math.PR/0000000 Donsker and Glivenko-Cantelli theorems for a class of processes generalizing the empirical process Davit Varron Laboratoire de mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté e-mail: [email protected] Abstract: We establish a Glivenko-Cantelli and a Donsker theorem for a class of random discrete measures which generalize the empirical measure, under conditions on uniform entropy numbers that are common in empirical pro- cesses theory. Some illustrative applications in nonparametric Bayesian theory and randomly sized sampling are provided. Primary 60F17, 60G57, 62F15. Keywords and phrases: Empirical processes, Random measures, Bayesian non- parametrics.