# Patterned random matrices

##### By: Bose, Arup.

Material type: BookPublisher: Florida CRC Press 2018Description: xxi, 267p. With index.ISBN: 9781138591462.Subject(s): Probabilities | Random matrices | Random variables | Linear - Multilinear algebra | Probability theory - ApplicationsDDC classification: 512.9434 Summary: 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/9781138591462Item type | Current location | Item location | Collection | Call number | Status | Date due | Barcode |
---|---|---|---|---|---|---|---|

Books | Vikram Sarabhai Library General Stacks | Slot 1368 (0 Floor, East Wing) | Non-fiction | 512.9434 B6P2 (Browse shelf) | Available | 199347 |

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512.896 G3M2 Matrix algebra for engineers | 512.897 S8I61 Introduction to linear algebra | 512.9434 B6L2 Large covariance and autocovariance matrices | 512.9434 B6P2 Patterned random matrices | 512.9434 B6R2 Random circulant matrices | 512.9434 M2M2 Matrix differential calculus with applications in statistics and econometrics | 512.9434 T2T6 Topics in random matrix theory |

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

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

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