Moderate deviations of empirical periodogram and non-linear functionals of moving average processes

A moderate deviation principle for non-linear functionals, with at most quadratic growth, of moving average processes (or linear processes) is established. The main assumptions on the moving average process are a Logarithmic Sobolev Inequality for the driving random variables and the continuity, or some (weaker) integrability condition on the spectral density (covering some cases of long range dependence). We also obtain the moderate deviation estimate for the empirical periodogram, exhibiting an interesting new form of the rate function, i.e. with a correction term compared to the Gaussian rate functional. As statistical applications we provide the moderate deviation estimates of the least square and the Yule-Walker estimators of the parameter of a stationary autoregressive process and of the Neyman-Pearson likelihood ratio test in the Gaussian case. (c) 2005 Elsevier SAS. All rights reserved.