Local inference for locally stationary time series based on the empirical spectral measure

The time varying empirical spectral measure plays a major role in the treatment of inference problems for locally stationary processes. The properties of the empirical spectral measure and related statistics are studied — both when its index function is fixed or when dependent on the sample size. In particular we prove a general central limit theorem. Several applications and examples are given including semiparametric Whittle estimation, local least squares estimation and spectral density estimation.