On the limiting behavior of estimates based on sample covariances

The paper addresses the issue of estimating the parameters of stationary time series from finite or infinite sets of sample covariances. Several known results concerning finite sets of sample covariances are briefly reviewed. It is then shown that for certain Gaussian processes an efficient estimation of the process parameters is possible if the number of sample covariances used by the estimation algorithm tends to infinity. Next we examine stationary processes whose innovations are non-Gaussian. It is shown that the achievable accuracy of the parameter estimates based on the sample covariances is the same as that for the Gaussian case. Thus, parameter estimates based on sample covariances are insensitive to the process distribution and as a result, these estimates are generally statistically inefficient.