Spectral Theory of Large Dimensional Random Matrices and Its Applications to Wireless Communications and Finance Statistics : Random Matrix Theory and Its Applications PDF
Preview Spectral Theory of Large Dimensional Random Matrices and Its Applications to Wireless Communications and Finance Statistics : Random Matrix Theory and Its Applications
Spectral Theory of Large Dimensional Random Matrices and its Applications to Wireless Communications and Finance Statistics Random Matrix Theory and its Applications 9049_9789814579056_tp.indd 1 13/12/13 11:48 am TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Spectral Theory of Large Dimensional Random Matrices and its Applications to Wireless Communications and Finance Statistics Random Matrix Theory and its Applications Zhidong Bai Northeast Normal University, China & National University of Singapore, Singapore Zhaoben Fang University of Science and Technology of China, China Ying-Chang Liang The Singapore Infocomm Research Institute, Singapore University of Science and World Scientific Technology of China Press 9049_9789814579056_tp.indd 2 13/12/13 11:48 am 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. Spectral Theory of Large Dimensional Random Matrices and Its Applications to Wireless Communications and Finance Statistics © Bai Zhidong, Fang Zhaoben, Liang Yingchang The work is originally published by University of Science and Technology of China Press in 2010. This edition is published by World Scientific Company Pte Ltd by arrangement with University of Science and Technology of China Press, Anhui, China. All rights reserved. No reproduction and distribution without permission. SPECTRAL THEORY OF LARGE DIMENSIONAL RANDOM MATRICES AND ITS APPLICATIONS TO WIRELESS COMMUNICATIONS AND FINANCE STATISTICS Random Matrix Theory and Its Applications Copyright © 2014 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 978-981-4579-05-6 Printed in Singapore Dedicated to Fiftieth Anniversary of Alma Mater University of Science and Technology of China To my great teachers Professors Yongquan Yin and Xiru Chen my wife, Xicun Dan and my sons, Li, and Steve Gang and grandsons Yongji, Yonglin, and Yongbin — Zhidong Bai To my wife Guo Fujia and my daughter Fang Min — Zhaoben Fang To To my wife Shuo Gao my son, Paul, and daughter, Wendy — Liang Ying-Chang TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Preface Sincethelastthreeorfourdecades,computerscienceandcomputingfacilities have been developed and adopted to almost every discipline. In consequence, datasetscollectedfromexperimentsand/ornaturalphenomenahavebecome largeinboththeirsizesaswellasdimensions.Atthesametime,thecomputing ability has become capable of dealing with extremely huge data sets. For example,in the biologicalsciences,a DNA sequence canbe as long as several billions. In finance research, the number of different stocks can be as large as tens of thousands. In wireless communications, the number of users can be several millions. In image processing, the pixels of a picture may be several thousands. On the other hand, however, in statistics, classical limit theorems have been found to be seriously inadequate in aiding in the analysis of large di- mensional data. All these are challenging classical statistics. Nowadays, an urgent need to statistics is to create new limiting theories that are applicable to large dimensional data analysis. Therefore, since last decade, the large di- mensional data analysis has become a very hot topic in statistics and various disciplines where statistics is applicable. Currently,spectralanalysisof largedimensionalrandommatrices(simply Random Matrix Theory (RMT)) is the only systematic theory that can be appliedto someproblemsoflargedimensionaldata analysis.The RMT dates back to the early development of Quantum Mechanics in the 1940’s and 50’s. Inanattempttoexplainthecomplexorganizationalstructureofheavynuclei, E.Wigner,ProfessorofMathematicalPhysicsatPrincetonUniversity,argued that one should not compute energy levels from Schr¨odinger’s equation. In- stead,oneshouldimaginethecomplexnucleisystemasablackboxdescribed by n n Hamiltonian matrices with elements drawn from a probability dis- × tribution with only mild constraints dictated by symmetry considerations. Under these assumptions and a mild conditions imposed on the probability measureinthespaceofmatrices,onefindsthejointprobabilitydensityofthe neigenvalues.Basedonthisconsideration,Wignerestablishedthewellknown vii viii Preface semi-circular law. Since then, RMT has been developed into a big research area in mathematical physics and probability. Duetoneedsoflargedimensionaldataanalysis,thenumberofresearchers and publications on RMT has been rapidly increasing.As an evidence, it can beseenfromthefollowingstatisticsfromMathscinetdatabaseunderkeyword Random Matrix on 11 April 2008: 1955—1964 1965—1974 1975—1984 1985—1994 1995—2004 2005–04.2008 23 138 249 635 1205 493 Table 0.1. Publication numberson RMT in 10 year periods since 1955 The purpose of this monograph is to introduce the basic concepts and results of RMT and some applications to Wireless Communications and Fi- nance Statistics. The readers of this book would be graduate students and beginning researchers who are interested in RMT and/or its applications to their own research areas. As for the theorems in RMT, we only provide an outline of their proofs. The detailed proofs are referred to the book Spectral Analysis of Large Dimensional Random Matrices by Bai, Z. D. and Silver- stein, J. W. (2006). A for the applications to Wireless communications and Finance Statistics, we are more emphasizing its formulation to illustrate how the RMT is applied to, rather than detailed mathematical proofs. Special thanks go to Mr. Liuzhi Yin who contributed to the book by pro- viding editing and extensive literature review. Changchun, China Bai Zhidong Hefei, China Fang Zhaoben Singapore Liang Ying-Chang April 2008 Contents 1 Introduction............................................... 1 1.1 History of RMT and Current Development ................. 1 1.1.1 A brief review of RMT............................. 2 1.1.2 Spectral Analysis of Large Dimensional Random Matrices ......................................... 3 1.1.3 Limits of Extreme Eigenvalues ...................... 4 1.1.4 Convergence Rate of ESD .......................... 4 1.1.5 Circular Law ..................................... 5 1.1.6 CLT of Linear Spectral Statistics .................... 5 1.1.7 Limiting Distributions of Extreme Eigenvalues and Spacings ......................................... 6 1.2 Applications to Wireless Communications .................. 6 1.3 Applications to Finance Statistics ......................... 7 2 Limiting Spectral Distributions ............................ 11 2.1 Semicircular Law ........................................ 11 2.1.1 The iid Case...................................... 12 2.1.2 Independent but not Identically Distributed .......... 18 2.2 Marˇcenko-PasturLaw.................................... 22 2.2.1 MP Law for iid Case............................... 22 2.2.2 Generalization to the Non-iid Case .................. 25 2.2.3 Proof of Theorem 2.11 by Stieltjes Transform ......... 26 2.3 LSD of Products ........................................ 27 2.3.1 Existence of the ESD of S T ...................... 28 n n 2.3.2 Truncation of the ESD of T ....................... 29 n 2.3.3 Truncation, Centralization and Rescaling of the X-variables....................................... 30 2.3.4 Sketch of the Proof of Theorem 2.12 ................. 31 2.3.5 LSD of F Matrix.................................. 32 2.3.6 Sketch of the Proof of Theorem 2.14 ................. 36 2.3.7 When T is a Wigner Matrix ........................ 42 ix
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