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 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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