Moment Estimator for Random Vectors with Heavy Tails

If a set of independent, identically distributed random vectors has heavy tails, so that the covariance matrix does not exist, there is no reason to expect that the sample covariance matrix conveys useful information. On the contrary, this paper shows that the eigenvalues and eigenvectors of the sample covariance matrix contain detailed information about the probability tails of the data. The eigenvectors indicate a set of marginals which completely determine the moment behavior of the data, and the eigenvalues can be used to estimate the tail thickness of each marginal. The paper includes an example application to a data set from finance.