The goal of this project is to establish a systematic asymptotic theory for estimates of large dimensional covariance matrices in time series; a fundamental problem in high-dimensional inference. In particular, the investigator plans to study properties of sample covariances and sample covariance matrices for stationary processes; deal with consistent estimation of covariance matrices of stationary processes and its applications in prediction and other problems; and explore non-Gaussian features of random processes by estimating higher order cumulant tensors.
Covariance matrices play a fundamental role in various fields including environmental science, engineering, economics and finance. Estimation of covariance matrices is needed in analyzing, testing, monitoring and predicting of seismic, economic and financial and other time series. Results developed from this project can provide a theoretical foundation for estimation of covariance matrices and can potentially improve time series processing algorithms that are used in various applications.