Stationarity is a crucial assumption in classical time series analysis. The theory of oscillatory spectra (Priestley, 1965) represents an attempt to overcome the resulting limitations for modeling nonstationary time series. However, it is the more flexible concept of locally stationary processes (Dahlhaus, 1993) which, by extending the theory of oscillatory spectra, provides a suitable framework for a general asymptotic theory of nonstationary processes. A fundamental characteristic of these processes is the time varying spectral density. While the literature on these models is already well developed, several questions of asymptotic inference remain open. Among them, two seem to stand out as most interesting: the possible asymptotic equivalence to a Gaussian white noise model, and the question of optimal exponential rates of large deviation type in testing and estimation problems. Since Le Cam developed the comparison of statistical experiments via their risk functions, many statistical models have been proved to be locally asymptotically normal, with the aim of establishing benchmarks for optimal procedures. For a parametric model of a time varying spectral density, local asymptotic normality has been established in the literature. However, for a better conceptual understanding of asymptotic inference, it is of interest to study the stronger property of asymptotic equivalence to a Gaussian white noise model, valid globally and over nonparametric function classes. As regards large deviation theory for locally stationary processes, some fragments are already available in the literature. A more fully developed theory can be envisaged, yielding not only testing results such as Stein's lemma and the Chernoff bound as special cases, but possibly also insights into the information geometry of these models based on the asymptotic Kullback-Leibler information.
Practitioners of statistics and data analysis very often assume that data show more or less similar behavior over different time periods, even when there is clear evidence to the contrary. For instance, this phenomenon can be observed in records of atmospheric turbulence, seismic signals from earthquakes, or speech signals analyzed in biological research. There is a need to develop more refined statistical methods for these cases. An ingenious theoretical solution to this problem has been proposed in the literature, based on the assumption that if data change over time, they often do not do so abruptly, but in a smooth way. This phenomenon is called local stationarity. If the series exhibits these "smooth changes", existing statistical methods can be adapted to smoothly change over time as well, considerably extending the scope of data analysis. The current proposal aims at a more thorough mathematical-statistical investigation of these locally stationary models. It also has a major educational component, as it is intended to accompany the collaboration between the principal investigator and a promising young scientist who has previously been supported with full tuition and stipend from the government of Mexico.
