Patterned Random Matrices

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 Sankyha 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