高維隨機(jī)矩陣的譜理論及其在無線通信和金融統(tǒng)計(jì)中的應(yīng)用

前言

大學(xué)最重要的功能是向社會(huì)輸送人才．大學(xué)對于一個(gè)國家、民族乃至世界的重要性和貢獻(xiàn)度，很大程度上是通過畢業(yè)生在社會(huì)各領(lǐng)域所取得的成就來體現(xiàn)的．中國科學(xué)技術(shù)大學(xué)建校只有短短的50年，之所以迅速成為享有較高國際聲譽(yù)的著名大學(xué)之一，主要就是因?yàn)樗囵B(yǎng)出了一大批德才兼?zhèn)涞膬?yōu)秀畢業(yè)生．他們志向高遠(yuǎn)、基礎(chǔ)扎實(shí)、綜合素質(zhì)高、創(chuàng)新能力強(qiáng)，在國內(nèi)外科技、經(jīng)濟(jì)、教育等領(lǐng)域做出了杰出的貢獻(xiàn)，為中國科大贏得了“科技英才的搖籃”的美譽(yù)．2008年9月，胡錦濤總書記為中國科大建校五十周年發(fā)來賀信，信中稱贊說：半個(gè)世紀(jì)以來，中國科學(xué)技術(shù)大學(xué)依托中國科學(xué)院，按照全院辦校、所系結(jié)合的方針，弘揚(yáng)紅專并進(jìn)、理實(shí)交融的校風(fēng)，努力推進(jìn)教學(xué)和科研工作的改革創(chuàng)新，為黨和國家培養(yǎng)了一大批科技人才，取得了一系列具有世界先進(jìn)水平的原創(chuàng)性科技成果，為推動(dòng)我國科教事業(yè)發(fā)展和社會(huì)主義現(xiàn)代化建設(shè)做出了重要貢獻(xiàn)．據(jù)統(tǒng)計(jì)，中國科大迄今已畢業(yè)的5萬人中，已有42人當(dāng)選中國科學(xué)院和中國工程院院士，是同期（自1963年以來）畢業(yè)生中當(dāng)選院士數(shù)最多的高校之一．其中，本科畢業(yè)生中平均每1，000人就產(chǎn)生1名院士和。700多名碩士、博士，比例位居全國高校之首．還有眾多的中青年才俊成為我國科技、企業(yè)、教育等領(lǐng)域的領(lǐng)軍人物和骨干．在歷年評選的“中國青年五四獎(jiǎng)?wù)隆鲍@得者中，作為科技界、科技創(chuàng)新型企業(yè)界青年才俊代表，科大畢業(yè)生已連續(xù)多年榜上有名，獲獎(jiǎng)總?cè)藬?shù)位居全國高校前列．鮮為人知的是，有數(shù)千名優(yōu)秀畢業(yè)生踏上國防戰(zhàn)線，為科技強(qiáng)軍做出了重要貢獻(xiàn)，涌現(xiàn)出20多名科技將軍和一大批國防科技中堅(jiān)．

內(nèi)容概要

本書講述了隨機(jī)矩陣譜理論的主要結(jié)果和前瞻研究，以及它在無線通信和現(xiàn)代金融風(fēng)險(xiǎn)理論中的應(yīng)用。書中前面講解基本知識(shí)，后面分析重要范例，全面介紹了隨機(jī)矩陣譜理論在這兩個(gè)領(lǐng)域中的成果。本書對其他需要高維數(shù)據(jù)分析的領(lǐng)域，能起到示范作用。本書可作為統(tǒng)計(jì)學(xué)、計(jì)算機(jī)科學(xué)、現(xiàn)代物理、量子力學(xué)、無線通信、金融工程、經(jīng)濟(jì)學(xué)等領(lǐng)域本科生、研究生和工程技術(shù)人員學(xué)習(xí)隨機(jī)矩陣?yán)碚摰闹匾獏⒖假Y料。

書籍目錄

Preface of Alumni's SerialsPreface1  Introduction　1.1 History of RMT and Current Development　　1.1.1 A Brief Review of RMT　　1.1.2 Spectral Analysis of Large Dimensional Random Matrices　　1.1.3 Limits of Extreme Eigenvalues　　1.1.4 Convergence Rate of ESD　　1.1.5 Circular Law　　1.1.6 Central Limit Theory (CLT) of Linear Spectral Statistics　　1.1.7 Limiting Distributions of Extreme Eigenvalues and Spacings  1.2 Applications to Wireless Communications  1.3 Applications to Finance Statistics2 Limiting Spectral Distributions  2.1  Semi-circular Law　　2.1.1 The lid Case　　2.1.2  Independent but not Identically Distributed   2.2  Marcenko-Pastur Law　　2.2.1 MP Law for lid Case　　2.2.2 Generalization to the Non-lid Case　　2.2.3 Proof of Theorem 2.11 by Stieltjes Transform   2.3 LSD of Products　　2.3.1 Existence of the ESD of SnTn　　2.3.2 Truncation of the ESD of Tn　　2.3.3 Truncation, Centralization and Rescaling of the  X-variables　　2.3.4 Sketch of the Proof of Theorem 2.12　　2.3.5 LSD of F Matrix　　2.3.6 Sketch of the Proof of Theorem 2.14    2.3.7 When T is a Wigner Matrix    2.4 Hadamard Product  4　  2.4.1 Truncation and Centralization 　 2.4.2 Outlines of Proof of the theorem   2.5 Circular Law　　2.5.1 Failure of Techniques Dealing with Hermitian Matrices 　　2.5.2 Revisit of Stieltjes Transformation　　2.5.3 A Partial Answer to the Circular Law　　2.5.4 Comments and Extensions of Theorem 2.333 Extreme Eigenvalues  3.1  Wigner Matrix  3.2 Sample Covariance Matrix 　 3.2.1 Spectral Radius  3.3  Spectrum Separation  3.4 Tracy-Widom Law  　3.4.1 TW Law for Wigner Matrix　　3.4.2 TW Law for Sample Covariance Matrix4 CLT of LSS  4.1 Motivation and Strategy  4.2 CLT of LSS for Wigner Matrix　　4.2.1 Outlines of the Proof  4.3 CLT of LSS for Sample Covariance Matrices    4.4 F Matrix　　4.4.1 Decomposition of Xnf　　4.4.2 Limiting Distribution of X+nf　　4.4.3 Limiting Distribution of Xnf5 Limiting Behavior of Eigenmatrix of Sample Covariance Matrix  5.1 Earlier Work by Silverstein  5.2 Further Extension of Silverstein's Work  5.3 Projecting the Eigenmatrix to a d-Dimensional Space　　5.3.1 Main Results　　5.3.2 Sketch of Proof of Theorem 5.19　　5.3.3 Proof of Corollary 5.236 Applications to Wireless Communications  6.1 Introduction  6.2 Channel Models.　　6.2.1 Basics of Wireless Communication Systems  ……7 Limiting Performances of Linear and Iterative Receivers8 Applications to Finace StatisticsReferencesIndex

章節(jié)摘錄

In applications of the asymptotic theorems of spectral analysis of large di-mensional random matrices, two important problems arose after the LSD wasfound. The first is the bound on extreme eigenvalues; the second is the con-vergence rate of the ESD, with respect to sample size. For the first problem,the literature is extensive. The first success was due to Geman （1980）, whoproved that the largest eigenvalue of a sample covariance matrix convergesalmost surely to a limit under a growth condition on all the moments of theunderlying distribution. Yin, Bai, and Krishnaiah （1988） proved the same re-sult under the existence of the 4th order moment, and Bai, Silverstein, andYin （1988） proved that the existence of the 4th order moment is also necessaryfor the existence of the limit. Bai and Yin （1988b） found the necessary andsufficient conditions for almost sure convergence of the largest eigenvalue of aWigner matrix. By the symmetry between the largest and smallest eigenval-ues of a Wigner matrix, the necessary and sufficient conditions for almost sureconvergence of the smallest eigenvalue of a Wigner matrix were also found. Comparing to almost sure convergence of the largest eigenvalue of a sam-ple covariance matrix, a relatively harder problem is to find the limit of thesmallest eigenvalue of a large dimensional sample covariance matrix. The firstattempt made in Yin, Bai, and Krishnaiah （1983） proved that the almost surelimit of the smallest eigenvalue of a Wishart matrix has a positive lower boundwhen the ratio of dimension to the degrees of freedom is less than 1/2.

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