Large Covariance and Autocovariance Matrices

The material will be based on very recent advances in the theory and application of large covariance and autocovariance matrices. Technologies and methods in medical sciences, image processing, and other fields generate data where the dimension is large compared to the sample size and may also increase as the next set of measurements become available. Theoretical and practical study of such type of data has attracted recent attention of researchers since most of the methods in finite dimensional set up do not work in these cases, even asymptotically. This book will be mainly focused on the topics in high-dimensional situations.

Large Covariance and Autocovariance Matrices brings together a collection of recent results on sample covariance and autocovariance matrices in high-dimensional models and novel ideas on how to use them for statistical inference in one or more high-dimensional time series models. The prerequisites include knowledge of elementary multivariate analysis, basic time series analysis and basic results in stochastic convergence.

Part I is on different methods of estimation of large covariance matrices and auto-covariance matrices and properties of these estimators. Part II covers the relevant material on random matrix theory and non-commutative probability. Part III provides results on limit spectra and asymptotic normality of traces of symmetric matrix polynomial functions of sample auto-covariance matrices in high-dimensional linear time series models. These are used to develop graphical and significance tests for different hypotheses involving one or more independent high-dimensional linear time series.

The book should be of interest to people in econometrics and statistics (large covariance matrices and high-dimensional time series), mathematics (random matrices and free probability) and computer science (wireless communication). Parts of it can be used in post-graduate courses on high-dimensional statistical inference, high-dimensional random matrices and high-dimensional time series models. It should be particularly attractive to researchers developing statistical methods in high-dimensional time series models.

(with Snigdhansu Chatterjee) to be published by Hindustan Book Agency.

is a post-doctoral fellow at the Informatics Institute, University of Florida. After graduating from St. Xavier''s College, Kolkata, she obtained her master’s in 2012 and PhD in 2016 from the Indian Statistical Institute. Her thesis in high-dimensional covariance and auto-covariance matrices, written under the supervision of Dr. Bose, has received high acclaim.