Patterned random matrices
Bose, Arup
creator
text
Florida
CRC Press
2018
monographic
| 0
xxi, 267p. With index
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.
https://www.crcpress.com/Patterned-Random-Matrices/Bose/p/book/9781138591462
Table of Contents
1 A unified framework
2 Common symmetric patterned matrices
3 Patterned XX matrices
4 k-Circulant matrices
5 Wigner-type matrices
6 Balanced Toeplitz and Hankel matrices
7 Patterned band matrices
8 Triangular matrices
9 Joint convergence of i.i.d. patterned matrices
10 Joint convergence of independent patterned matrices
11 Autocovariance matrix
Probabilities
Random matrices
Random variables
Linear - Multilinear algebra
Probability theory - Applications
512.9434 B6P2
9781138591462
190507