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 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.
". . . 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
Empirical and limiting spectral distribution
A metric for probability measures
Patterned matrices: A unified approach
Toeplitz and Hankel matrices
Reverse Circulant matrix
Symmetric Circulant and related matrices
Additional properties of the LSDs
A unified setup
Aspect ratio y = 0
Aspect ratio y = 0
LSD for band matrices
Triangular Wigner matrix
Non-commutative probability space
Nature of the limit
Definitions and notation
Sum of independent patterned matrices