STA 290 Seminar: Johannes Heiny

STA 290 Seminar Series

DATE:             Tuesday April 10th, 4:10pm

LOCATION:    MSB 1143, Statistics Seminar Room

SPEAKER:      Johannes Heiny, Dept Mathematics, University of Aarhus (Denmark)

TITLE:             The spectrum of high-dimensional sample correlation matrices”

ABSTRACT:   In this talk, we consider the sample correlation matrix R associated to n observations of a p-dimensional time series. In our framework, we allow that p/n may tend to 0 or a positive constant. If the time series has a finite fourth moment, we show that the sample correlation matrix can be approximated by its sample covariance counterpart for a wide variety of models.

This result is very important for data analysts who use principal component analysis to detect some structure in high-dimensional time series. From a theoretical point of view, it allows to derive a plethora of ancillary results for functionals of the eigenvalues of R. For instance, we determine the almost sure behavior of the largest and smallest eigenvalues, and the limiting spectral distribution of R.

Finally, we discuss the case of time series with infinite fourth moment and determine the optimal moment conditions for the convergence of the empirical spectral distributions to their usual limits.