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 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.

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 XXmatrices

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

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.