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Patterned Random Matrices





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ISBN 9781138591462
Published May 17, 2018 by Chapman and Hall/CRC
291 Pages

 
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Book Description

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 Marchenko-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 Sankyhā 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.



 



 



 

Table of Contents

A unified framework



Empirical and limiting spectral distribution



Moment method



A metric for probability measures



Patterned matrices: a unified approach



Scaling



Reduction to bounded case



Trace formula and circuits



Words



Vertices



Pair-matched word



Sub-sequential limit



Exercises





Common symmetric patterned matrices



Wigner matrix



Semi-circle law, non-crossing partitions, Catalan words



LSD



Toeplitz and Hankel matrices



Toeplitz matrix



Hankel matrix



Reverse Circulant matrix



Symmetric Circulant and related matrices



Additional properties of the LSD



Moments of Toeplitz and Hankel LSD



Contribution of words and comparison of LSD



Unbounded support of Toeplitz and Hankel LSD



Non-unimodality of Hankel LSD



Density of Toeplitz LSD



Pyramidal multiplicativity



Exercises





Patterned XX matrices



A unified set up



Aspect ratio y =



Preliminaries



Sample variance-covariance matrix



Catalan words and Marˇcenko-Pastur law



LSD



Other XX matrices



Aspect ratio y =



Sample variance-covariance matrix



Other XX matrices



Exercises





k-Circulant matrices



Normal approximation



Circulant matrix



k-Circulant matrices



Eigenvalues



Eigenvalue partition



Lower order elements



Degenerate limit



Non-degenerate limit



Exercises





Wigner type matrices



Wigner-type matrix



Exercises





Balanced Toeplitz and Hankel matrices



Main results



Exercises





Patterned band matrices



LSD for band matrices



Proof



Reduction to uniformly bounded input



Trace formula, circuits, words and matches



Negligibility of higher order edges



(M) condition



Exercises





Triangular matrices



General pattern



Triangular Wigner matrix



LSD



Contribution of Catalan words



Exercises





Joint convergence of iid patterned matrices



Non-commutative probability space



Joint convergence



Nature of the limit



Exercises





Joint convergence of independent patterned matrices



Definitions and notation



Joint convergence



Freeness



Sum of independent patterned matrices



Proofs



Exercises





Autocovariance matrix



Preliminaries



Main results



Proofs



Exercises

...
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Author(s)

Biography

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 Sankyhā 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.

Reviews

". . . this book can be recommended for students and researchers interested in a broad overview of random matrix theory. Each chapter ends with plenty of problems useful for exercises and training." ~ Statistical Papers

 

" . . . the authors should be congratulated for producing two highly relevant and well-written books. Statisticians would probably gravitate to LCAM in the first instance and those working in linear algebra would probably gravitate to PRM." ~Jonathan Gillard, Cardiff University