Probabilistic Inequalities PDF
Preview Probabilistic Inequalities
PROBABILISTIC INEQUALITIES SERIES ON CONCRETE AND APPLICABLE MATHEMATICS Series Editor: Professor George A. Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, USA Published Vol. 1 Long Time Behaviour of Classical and Quantum Systems edited by S. Graffi & A. Martinez Vol. 2 Problems in Probability by T. M. Mills Vol. 3 Introduction to Matrix Theory: With Applications to Business and Economics by F. Szidarovszky & S. Molnár Vol. 4 Stochastic Models with Applications to Genetics, Cancers, Aids and Other Biomedical Systems by Tan Wai-Yuan Vol. 5 Defects of Properties in Mathematics: Quantitative Characterizations by Adrian I. Ban & Sorin G. Gal Vol. 6 Topics on Stability and Periodicity in Abstract Differential Equations by James H. Liu, Gaston M. N’Guérékata & Nguyen Van Minh Vol. 7 Probabilistic Inequalities by George A. Anastassiou ZhangJi - Probabilistic Inequalities.pmd 2 6/24/2009, 4:38 PM Series on Concrete and Applicable Mathematics – Vol. 7 P R O B A B I L I S T I C I N E Q U A L I T I E S George A Anastassiou University of Memphis, USA World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. PROBABILISTIC INEQUALITIES Series on Concrete and Applicable Mathematics — Vol. 7 Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-4280-78-5 ISBN-10 981-4280-78-X Printed in Singapore. ZhangJi - Probabilistic Inequalities.pmd 1 6/24/2009, 4:38 PM 22ndJune2009 18:37 WorldScienti(cid:12)cBook-9.75inx6.5in Book Dedicated to my wife Koula v 22ndJune2009 18:37 WorldScienti(cid:12)cBook-9.75inx6.5in Book TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk 22ndJune2009 18:37 WorldScienti(cid:12)cBook-9.75inx6.5in Book Preface In this monographwe present univariate and multivariate probabilistic inequalities regarding basic probabilistic entities like expectation, variance, moment generat- ing function and covariance. These are built on recent classical form real analy- sis inequalities also given here in full details. This treatise relies on author’s last twenty one years related research work, more precisely see [18]-[90], and it is a natural outgrowth of it. Chapters are self-contained and several advanced courses can be taught out of this book. Extensive background and motivations are given per chapter. A very extensive list of references is given at the end. The topics covered are very diverse. Initially we present probabilistic Ostrowski type inequal- ities, other various related ones, and Grothendieck type probabilistic inequalities. A great bulk of the book is about Information theory inequalities, regarding the Csiszar’s f-Divergence between probability measures. Another great bulk of the book is regarding applications in various directions of Geometry Moment Theory, inparticulartoestimatetherateofweakconvergenceofprobabilitymeasurestothe unit measure, to maximize pro(cid:12)ts in the stock market, to estimating the di(cid:11)erence of integral means, and applications to political science in electorial systems. Also wedevelopGru(cid:127)sstypeandChebyshev-Gru(cid:127)sstypeinequalitiesforStieltjes integrals and show their applications to probability. Our results are optimal or close to op- timal. We end with important real analysismethods with potential applicationsto stochastic. The exposed theory is destined to (cid:12)nd applications to all applied sci- ences and related subjects, furthermore has its own theoretical merit and interest from the point of view of Inequalities in Pure Mathematics. As such is suitable for researchers,graduatestudents, andseminarsof theabovesubjects, alsoto be in all science libraries. The (cid:12)nal preparation of book took place during 2008 in Memphis, Tennessee, USA. I would like to thank my family for their dedication and love to me, which was the strongest support during the writing of this monograph. vii 22ndJune2009 18:37 WorldScienti(cid:12)cBook-9.75inx6.5in Book viii PROBABILISTIC INEQUALITIES IamalsoindebtedandthankfultoRodicaGalandRazvanMezeifortheirheroic and great typing preparation of the manuscript in a very short time. George A. Anastassiou Department of Mathematical Sciences The University of Memphis, TN U.S.A. January 1, 2009 22ndJune2009 18:37 WorldScienti(cid:12)cBook-9.75inx6.5in Book Contents Preface vii 1. Introduction 1 2. Basic Stochastic Ostrowski Inequalities 3 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 One Dimensional Results . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Multidimensional Results . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3. Multidimensional Montgomery Identities and Ostrowski type Inequalities 15 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Application to Probability Theory . . . . . . . . . . . . . . . . . . 29 4. General Probabilistic Inequalities 31 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Make Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 Remarks on an Inequality . . . . . . . . . . . . . . . . . . . . . . 35 4.4 L ;q >1, Related Theory . . . . . . . . . . . . . . . . . . . . . . 36 q 5. About Grothendieck Inequalities 39 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.2 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6. Basic Optimal Estimation of Csiszar’s f-Divergence 45 6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.2 Main Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ix
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