The Bernstein-von Mises Theorem for Stationary Processes

In the literature of time series analysis since Whittle (1953), many authors (for example, Dunsmuir and Hannan (1976), Dunsmuir (1979), and Hosoya and Taniguchi (1982)) have considered an approach using Whittle’s log-likelihood, which is an approximation of Gaussian log-likelihood of the data, and have developed the asymptotic properties of an estimator that maximizes Whittle’s loglikelihood. The Whittle likelihood is useful because it is easy to compute, and the use of the periodogram transforms dependent data into asymptotically independent data. Hence, there has been considerable interest in the further development of the theory in other directions. Monti (1997) applied the empirical likelihood approach to Whittle’s likelihood for constructing confidence regions. Choudhuri et al. (2004) showed that the actual joint distribution of the periodograms, at certain frequencies for a Gaussian time series, is mutually contiguous with the corresponding Whittle measure. Contiguity plays vital roles in estimation and testing theory. The Bernstein-von Mises theorem is one of the fundamental results in the asymptotic theory of Bayesian inference, and gives the convergence of the posterior density to normal. For Markov processes this result was obtained by Borwanker et al. (1971). Applications of this theorem lead to various results on the asymptotic behavior of Bayes estimates. This paper discusses a Bayes approach to stationary time series. We give the asymptotic properties of the posterior density under Whittle measure. Then the Bernstein-von Mises theorems for shortand long-memory stationary processes are shown. In Section 2 we present our main results. These results enable us to elucidate the asymptotic behavior of Bayes estimates. Also some examples will be given. Proofs are relegated to Section 3.