by Arup Bose

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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"--

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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.

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". . . 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*

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.

**A unified framework****Empirical and limiting spectral distribution****Moment method****A metric for probability measures****Patterned matrices: A unified approach****Exercises****Common symmetric patterned matrices**-
**Wigner matrix****Toeplitz and Hankel matrices****Reverse Circulant matrix****Symmetric Circulant and related matrices****Additional properties of the LSDs****Exercises** **Patterned***XX***matrices**-
**A unified setup****Aspect ratio***y*= 0**Aspect ratio***y*= 0**Exercises** **Circulant matrices**-
**Normal approximation****Circulant matrix***k*-Circulant matrices**Exercises** **Wigner-type matrices**-
**Wigner-type matrix****Exercises** **Balanced Toeplitz and Hankel matrices**-
**Main results****Exercises** **Patterned band matrices**-
**LSD for band matrices****Proof****Exercises** **Triangular matrices**-
**General pattern****Triangular Wigner matrix** **Joint convergence of i.i.d. 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 **

GOR010228183

Patterned Random Matrices by Arup Bose

Arup Bose

Used - Like New

Hardback

Taylor & Francis Ltd

2018-05-17

269

1138591467

9781138591462

N/A

Book picture is for illustrative purposes only, actual binding, cover or edition may vary.

The book has been read, but looks new. The book cover has no visible wear, and the dust jacket is included if applicable. No missing or damaged pages, no tears, possible very minimal creasing, no underlining or highlighting of text, and no writing in the margins.