Patterned Random 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 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.

". . . 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

• 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 XX matrices 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