We study Statistical Consistency of an approximate subspace identification procedure for the infinite dimensional a posteriori model of a frequency estimation problem in an Empirical Bayesian framework. By first imposing a natural uniform prior probability density on the unknown frequencies, the estimation of the hyperparameters of the a priori distribution can be accomplished by a sequence of subspace identification techniques. These techniques exploit the special structure of the covariance matrix of the a posteriori process which is discovered by making a connection with classical results on energy concentration in deterministic signal processing. The convergence of the spectrum of the subspace estimates to the (nonrational) spectral density of the a posteriori process has an analytic counterpart in the approximation of symmetric positive definite Toeplitz matrices by submatrices of finite rank. This is proven by a weak-sense convergence theorem for Toeplitz spectra.