On the information-theoretic limits of graphical model selection for Gaussian time series

We consider the problem of inferring the conditional independence graph (CIG) of a multivariate stationary dicrete-time Gaussian random process based on a finite length observation. Using information-theoretic methods, we derive a lower bound on the error probability of any learning scheme for the underlying process CIG. This bound, in turn, yields a minimum required sample-size which is necessary for any algorithm regardless of its computational complexity, to reliably select the true underlying CIG. Furthermore, by analysis of a simple selection scheme, we show that the information-theoretic limits can be achieved for a subclass of processes having sparse CIG. We do not assume a parametric model for the observed process, but require it to have a sufficiently smooth spectral density matrix (SDM).