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Patterned Random Matrices
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Patterned Random Matrices

by Arup Bose

Book Details

Format Paperback / Softback
ISBN-10 036773446X
ISBN-13 9780367734466
Publisher Taylor & Francis Ltd
Imprint Chapman & Hall/CRC
Country of Manufacture GB
Country of Publication GB
Publication Date Dec 18th, 2020
Print length 269 Pages
Weight 453 grams
Product Classification: Algebra Probability & statistics
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This book focuses on the limit spectral distribution (LSD) of patterned random matrices and provides a comprehensive variety of LSD results. It is accessible to first or second years Master's students and uses very elementary techniques. It is suitable for a beginner in random matrices with some probability background.

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.

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