高维随机矩阵的谱理论及其在无线通信和金融统计中的应用（全英文） pdf epub mobi txt 电子书 下载 2025
- Random Matrix Theory
- Spectral Theory
- High-Dimensional Statistics
- Wireless Communication
- Financial Statistics
- Asymptotic Analysis
- Probability
- Mathematical Finance
- Signal Processing
- Information Theory
具体描述
内容简介
本书讲述了随机矩阵谱理论的主要结果和前瞻研究，以及它在无线通信和现代金融风险理论中的应用。书中前面讲解基本知识，后面分析重要范例，全面介绍了随机矩阵谱理论在这两个领域中的成果。本书对其他需要高维数据分析的领域，能起到示范作用。本书可作为统计学、计算机科学、现代物理、量子力学、无线通信、金融工程、经济学等领域本科生、研究生和工程技术人员学习随机矩阵理论的重要参考资料。目录
Preface of Alumnis SerialsPreface
1 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 Statistics
2 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 2.33
3 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 Matrix
4 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 Xnf
5 Limiting Behavior of Eigenmatrix of Sample Covariance Matrix
5.1 Earlier Work by Silverstein
5.2 Further Extension of Silversteins 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 5.23
6 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 Receivers
8 Applications to Finace Statistics
References
Index
前言/序言
用户评价
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评分I've always been fascinated by the sheer potential of large, complex datasets, and this book, "Spectral Theory of High-Dimensional Random Matrices and its Applications in Wireless Communications and Financial Statistics," promised a deep dive into a fundamental mathematical framework. From what I’ve gathered, it seems to tackle the intricate world of random matrix theory, specifically focusing on scenarios where the dimensionality of the matrices becomes exceedingly large. This immediately piqued my interest, as it directly addresses the increasing scale of data we encounter in modern scientific and technological fields. The title suggests a rigorous exposition of the spectral properties – eigenvalues, eigenvectors, and their distributions – of these high-dimensional random matrices. I imagine the text would carefully build the theoretical foundations, perhaps starting with simpler models like the Wigner and GOE ensembles and then progressing to more complex ones relevant to practical applications. The mention of "spectral theory" implies a significant emphasis on analytical tools and mathematical rigor, which is exactly what I'm looking for to truly understand the underlying principles. It’s not just about the results, but the journey of deriving them. I anticipate sections dedicated to limit theorems, convergence properties, and the asymptotic behavior of spectral statistics, perhaps including discussions on universality phenomena. The comprehensive nature suggested by the title also leads me to believe that it would cover various types of random matrix ensembles, possibly including those with structured entries or non-Gaussian distributions, all crucial for modeling diverse real-world phenomena. The prospect of understanding how these abstract mathematical concepts translate into tangible insights is incredibly exciting.
评分From what I understand, this book aims to bridge the gap between the abstract mathematical framework of random matrix theory and its concrete manifestations in cutting-edge technological and economic sectors. The "spectral theory" aspect implies a deep dive into the mathematical underpinnings of how the eigenvalues and eigenvectors of large, randomly generated matrices behave. I expect a rigorous development of theorems and proofs, meticulously laid out to guide the reader through the complexities of high-dimensional probability and linear algebra. The fact that it specifically targets applications in wireless communications and financial statistics suggests that the theoretical discussions will be directly motivated by the challenges and opportunities within these domains. For instance, in wireless communications, one might expect to see how random matrix theory helps in understanding the capacity of wireless channels, the design of robust communication systems, or the analysis of interference in dense networks. In financial statistics, the applications could range from portfolio optimization and risk management to the detection of market patterns and the analysis of complex financial instruments. The book likely provides a detailed exploration of how spectral properties, such as the Marchenko-Pastur law or Tracy-Widom distributions, find practical utility in these application areas, offering quantitative insights and predictive capabilities.
评分The application-driven aspect of this book, specifically its focus on wireless communications and financial statistics, is what truly sets it apart for me. It’s one thing to grasp abstract mathematical theories, but it’s another entirely to see them directly inform and solve critical problems in these highly dynamic and data-intensive fields. For wireless communications, I can envision chapters delving into how random matrix theory can be used to analyze the performance of large-scale MIMO (Multiple-Input Multiple-Output) systems. This might involve understanding signal-to-noise ratios, channel capacity, and the impact of interference in scenarios with a vast number of antennas and users. The ability to model and predict the behavior of such complex communication channels using rigorous mathematical tools is invaluable. Similarly, in financial statistics, the book likely explores applications in risk management, portfolio optimization, and detecting market anomalies. The inherent randomness and high dimensionality of financial markets make them a prime candidate for random matrix theory. I’m particularly keen to learn how spectral properties can reveal underlying structures in correlation matrices, identify systemic risks, or even predict the emergence of financial crises. The transition from pure theory to practical problem-solving, as suggested by the title, is a strong draw, promising insights that are both intellectually stimulating and practically relevant.
评分The reputation of the authors and the potential for this book to become a definitive resource in its field is a significant factor in my interest. Spectral theory, especially when applied to high-dimensional random matrices, is a sophisticated area of mathematics with profound implications. This title suggests a comprehensive treatment that could potentially consolidate existing knowledge and introduce new perspectives. I envision the book starting with a historical overview of random matrix theory, perhaps tracing its origins in nuclear physics and its subsequent expansion into various scientific disciplines. The core of the book, I presume, will be a rigorous exposition of the spectral theory itself, covering topics such as the eigenvalue distributions for different types of random matrices (e.g., Gaussian, Wishart, non-Hermitian ensembles), the properties of eigenvectors, and the behavior of spectral statistics like extreme eigenvalues and level spacing. The emphasis on "high-dimensional" is key, indicating a focus on asymptotic regimes where traditional analytical methods may fail. I'm particularly interested in how the book addresses the universality of spectral distributions, a fascinating phenomenon where the statistical properties of eigenvalues become independent of the specific distribution of the matrix entries under certain conditions. The dual focus on applications in wireless communications and financial statistics suggests that the book will not only delve into the theoretical intricacies but also provide practical tools and insights for researchers and practitioners in these fields.
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