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Book Cover
Author Bose, Arup, author.

Title Patterned Random Matrices / Arup Bose
Edition First edition
Published Boca Raton, FL : CRC Press, 2018
Online access available from:
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Description 1 online resource : text file, PDF
Contents 2.2 Toeplitz and Hankel matrices2.2.1 Toeplitz matrix; 2.2.2 Hankel matrix; 2.3 Reverse Circulant matrix; 2.4 Symmetric Circulant and related matrices; 2.5 Additional properties of the LSDs; 2.5.1 Moments of the Toeplitz and Hankel LSDs; 2.5.2 Contribution of words and comparison of LSDs; 2.5.3 Unbounded support of the Toeplitz and Hankel LSDs; 2.5.4 Non-unimodality of the Hankel LSD; 2.5.5 Density of the Toeplitz LSD; 2.5.6 Pyramidal multiplicativity; 2.6 Exercises; Chapter 3 Patterned XX' matrices; 3.1 A unified setup; 3.2 Aspect ratio y = 0; 3.2.1 Preliminaries
3.2.2 Sample variance-covariance matrix3.2.2.1 Catalan words and the MarĖ‡cenko-Pastur law; LSD; 3.2.3 Other XX0 matrices; 3.3 Aspect ratio y = 0; 3.3.1 Sample variance-covariance matrix; 3.3.2 Other XX0 matrices; 3.4 Exercises; Chapter 4 k-Circulant matrices; 4.1 Normal approximation; 4.2 Circulant matrix; 4.3 k-Circulant matrices; 4.3.1 Eigenvalues; 4.3.2 Eigenvalue partition; 4.3.3 Lower-order elements; 4.3.4 Degenerate limit; 4.3.5 Non-degenerate limit; 4.4 Exercises; Chapter 5 Wigner-type matrices; 5.1 Wigner-type matrix; 5.2 Exercises
9.3 Nature of the limit9.4 Exercises; Chapter 10 Joint convergence of independent patternedmatrices; 10.1 Definitions and notation; 10.2 Joint convergence; 10.3 Freeness; 10.4 Sum of independent patterned matrices; 10.5 Proofs; 10.6 Exercises; Chapter 11 Autocovariance matrix; 11.1 Preliminaries; 11.2 Main results; 11.3 Proofs; 11.4 Exercises; Bibliography; Index
Chapter 6 Balanced Toeplitz and Hankel matrices6.1 Main results; 6.2 Exercises; Chapter 7 Patterned band matrices; 7.1 LSD for band matrices; 7.2 Proof; 7.2.1 Reduction to uniformly bounded input; 7.2.2 Trace formula, circuits, words and matches; 7.2.3 Negligibility of higher-order edges; 7.2.4 (M1) condition; 7.3 Exercises; Chapter 8 Triangular matrices; 8.1 General pattern; 8.2 Triangular Wigner matrix; 8.2.1 LSD; 8.2.2 Contribution of Catalan words; 8.3 Exercises; Chapter 9 Joint convergence of i.i.d. patterned matrices; 9.1 Non-commutative probability space; 9.2 Joint convergence
Cover; Half Title; Title; Copyright; Contents; Preface; Introduction; About the Author; Chapter 1 A unified framework; 1.1 Empirical and limiting spectral distribution; 1.2 Moment method; 1.3 A metric for probability measures; 1.4 Patterned matrices: A unified approach; 1.4.1 Scaling; 1.4.2 Reduction to bounded case; 1.4.3 Trace formula and circuits; 1.4.4 Words; 1.4.5 Vertices; 1.4.6 Pair-matched word; 1.4.7 Sub-sequential limit; 1.5 Exercises; Chapter 2 Common symmetric patterned matrices; 2.1 Wigner matrix; 2.1.1 Semi-circle law, non-crossing partitions, Catalan words; 2.1.2 LSD
Summary "Large dimensional random matrices (LDRM) with specific patterns arise in econometrics, computer science, mathematics, physics, and statistics. This book provides an easy initiation to LDRM. Through a unified approach, we investigate the existence and properties of the limiting spectral distribution (LSD) of different patterned random matrices as the dimension grows. The main ingredients are the method of moments and normal approximation with rudimentary combinatorics for support. Some elementary results from matrix theory are also used. By stretching the moment arguments, we also have a brush with the intriguing but difficult concepts of joint convergence of sequences of random matrices and its ramifications. This book covers the Wigner matrix, the sample covariance matrix, the Toeplitz matrix, the Hankel matrix, the sample autocovariance matrix and the k-Circulant matrices. Quick and simple proofs of their LSDs are provided and it is shown how the semi-circle law and the March?enko-Pastur law arise as the LSDs of the first two matrices. Extending the basic approach, we also establish interesting limits for some triangular matrices, band matrices, balanced matrices, and the sample autocovariance matrix. We also study the joint convergence of several patterned matrices, and show that independent Wigner matrices converge jointly and are asymptotically free of other patterned matrices. Arup Bose is a Professor at the Indian Statistical Institute, Kolkata, India. He is a distinguished researcher in Mathematical Statistics and has been working in high-dimensional random matrices for the last fifteen years. He has been the Editor of Sankyh? for several years and has been on the editorial board of several other journals. He is a Fellow of the Institute of Mathematical Statistics, USA and all three national science academies of India, as well as the recipient of the S.S. Bhatnagar Award and the C.R. Rao Award. His forthcoming books are the monograph, Large Covariance and Autocovariance Matrices (with Monika Bhattacharjee), to be published by Chapman & Hall/CRC Press, and a graduate text, U-statistics, M-estimates and Resampling (with Snigdhansu Chatterjee), to be published by Hindustan Book Agency."--Provided by publisher
Bibliography Includes bibliographical references and index
Subject Algebras, Linear.
Multilinear algebra.
Random matrices.
Random variables.
Algebras, Linear.
MATHEMATICS -- Probability & Statistics -- General.
Multilinear algebra.
Random matrices.
Random variables.
Form Electronic book
ISBN 0429488432