Statistical inference for functions of the covariance matrix in stationary Gaussian vector time series

We consider inference for functions of the marginal covariance matrix under a general class of stationary multivariate temporal Gaussian models. The main application which motivated this work involves the estimation of configurational entropy from molecular dynamics simulations in computational chemistry, where current methods of entropy estimation involve calculations based on the sample covariance matrix. The class of Gaussian models we consider, referred to as Gaussian Independent Principal Components models, is characterised as follows: the temporal sequence corresponding to each principal component (PC) is permitted to have general (temporal) dependence structure, but sequences corresponding to distinct PCs are assumed independent. In many contexts, this model class has the potential to achieve a good balance between flexibility and tractability: distinct PCs are permitted to have different, and quite general, dependence structures, but, as we shall see, estimation and large-sample inference are quite feasible, even in high-dimensional settings. We derive the limiting large-sample Gaussian distribution for the sample covariance matrix, and also results for functions of the sample covariance matrix, which provide a basis for approximate inference procedures, including confidence calculations for scalar quantities of interest. The results are applied to the molecular dynamics application, and the asymptotic properties of a configurational entropy estimator are given. Rotation and translation are removed by initial Procrustes registration, so that entropy is calculated from the size-and-shape of the configuration. An improved estimator based on maximum likelihood is suggested, and some further applications are also discussed.