On large deviations in testing simple hypotheses for locally stationary Gaussian processes

We derive a large deviation result for the log-likelihood ratio for testing simple hypotheses in locally stationary Gaussian processes. This result allows us to find explicitly the rates of exponential decay of the error probabilities of type I and type II for Neyman?CPearson tests. Furthermore, we obtain the analogue of classical results on asymptotic efficiency of tests such as Stein??s lemma and the Chernoff bound, as well as the more general Hoeffding bound concerning best possible joint exponential rates for the two error probabilities.