A Note on the Asymptotic Variance of Sample Roots

We derive the asymptotic distribution of the eigenvalues of a sample covari- ance matrix with distinct roots. Our theorem can accommodate the situation in which the population covariance matrix is estimated via its sample analogue as well as the more general case in which it is estimated via a pN-consistent extremum estimator. The sample roots will have a Normal distribution in a large sample with a covariance matrix that is easy to compute. We con- duct Monte Carlo experiments that show that standard errors based on our derived asymptotic distribution accurately approximate standard errors in the empirical distribution.