1st Edition

Patterned Random Matrices

By Arup Bose Copyright 2018
    291 Pages
    by Chapman & Hall

    292 Pages
    by Chapman & Hall

    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.



     



     



     

    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

    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.

    ". . . 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