Applications of Measure Theory to Statistics
This book aims to put strong reasonable mathematical senses in notions of objectivity and subjectivity for consistent estimations in a Polish group by using the concept of Haar null sets in the corresponding group. This new approach – naturally dividing t
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Applications of Measure Theory to Statistics
Applications of Measure Theory to Statistics
Gogi Pantsulaia
Applications of Measure Theory to Statistics
123
Gogi Pantsulaia Department of Mathematics Georgian Technical University Tbilisi Georgia
ISBN 978-3-319-45577-8 DOI 10.1007/978-3-319-45578-5
ISBN 978-3-319-45578-5
(eBook)
Library of Congress Control Number: 2016955334 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Contents
1 Calculation of Improper Integrals by Using Uniformly Distributed Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Some Auxiliary Notions and Facts from the Theory of Uniform Distribution of Sequences . . . . . . . . . . . . . . . . . . . . . . 1.3 Kolmogorov Strong Law of Large Numbers and Glivenko–Cantelli Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Calculation of a Certain Improper One-Dimensional Riemann Integral by Using Uniformly Distributed Sequences . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Infinite-Dimensional Monte Carlo Integration. . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Uniformly Distributed Sequences of an Increasing Family of Finite Sets in Infinite-Dimensional Rectangles . . . . . . . . . . 2.3 Monte Carlo Algorithm for Estimating the Value of Infinite-Dimensional Riemann Integrals . . . . . . . . . . . . . . . 2.4 Applications to Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . .
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