- ホーム
- > 洋書
- > 英文書
- > Science / Mathematics
Full Description
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.
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.
Contents
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
