Likelihood asymptotics of stationary Gaussian arrays
Carsten H. Chong, Fabian Mies
TL;DR
This work develops a local asymptotic normality theory for triangular arrays of stationary Gaussian time series depending on a multidimensional parameter, allowing non-diagonal rate matrices and spectral densities with diverse near-zero behavior. The authors prove a general LAN theorem under broad regularity conditions and correct a gap in Toeplitz-method lemmas that had underpinned earlier results, thereby establishing asymptotic efficiency of the MLE in the array context. They illustrate the theory with three applications: high-frequency infill data for mixed fractional Brownian motion, joint high-frequency and long-span inference for fractional OU processes, and mildly integrated AR models with sample-size dependent parameters. The results extend classical Gaussian LAN theory to complex array settings, enabling efficient inference in high-frequency statistics, econometrics, and related fields where spectral densities exhibit non-uniform integrability or multi-scale behavior.
Abstract
This paper develops an asymptotic likelihood theory for triangular arrays of stationary Gaussian time series depending on a multidimensional unknown parameter. We give sufficient conditions for the associated sequence of statistical models to be locally asymptotically normal in Le Cam's sense, which in particular implies the asymptotic efficiency of the maximum likelihood estimator. Unique features of the array setting covered by our theory include potentially nondiagonal rate matrices as well as spectral densities that satisfy different power-law bounds at different frequencies and may fail to be uniformly integrable. To illustrate our theory, we study efficient estimation for Gaussian processes sampled at high frequency and for a class of autoregressive models with moderate deviations from a unit root.