This paper discusses the asymptotic properties of the posterior density under Whittle measure. The Bernstein-von Mises theorem is shown for short-and longmemory stationary processes. Applications to Bayesian inference for time series are provided.

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In this article, we investigate an optimal property of the maximum likelihood estimator of Gaussian locally stationary processes by the second-order approximation. In the case where the model is correctly specified, it is shown that appropriate modifications of the maximum likelihood estimator for Gaussian locally stationary processes is second-order asymptotically efficient. We also discuss second-order robustness properties. Copyright 2009 The Author. Journal compilation 2009 Blackwell Publishing Ltd