Estimation of large covariance matrices using results on large deviations

Abstract

It appears that the general theory of large deviations has become an important part of probability theory, especially in the field of finance and insurance mathematics.<br />We have successfully used the theory of large deviations to show how regularized estimator of large covariance matrices converge to the population covariance matrix of multivariate normal i.i.d. stochastic processes, if the matrices are well-conditioned as long as long as log(p)/n tends to zero (p is the dimension of the random variable, n the size of the sample).<br />Based on an article of Bickel and Levina, we have not managed to establish a convergence result using theorems for large deviations from a book of Saulis and Statulevicius for stationary processes and we leave this question unanswered. My contribution to the topic is on the one hand to identify some flaws of the mentioned article and on the other to give an indication how the results could eventually be generalized to stationary processes instead of only assuming an identically and independently distributed Gaussian process. Since, we have showed that the banded estimator converges to the population covariance matrix and the Cholesky factor converges to the inverse of the population covariance matrix under certain conditions, no one should use the sample covariance matrix anymore in the case of p >n. The just mentioned results can bring a significant improvement in the finance industry, where it is necessary to have a reliable estimator of the population covariance matrix, especially in the field of portfolio optimization, where most often the number of assets is much larger than the number of observations.