Locally Stationary Processes

TL;DR

This paper provides a comprehensive overview of locally stationary processes, covering time-varying autoregressive models, nonlinear and linear processes, spectral densities, and empirical spectral methods, highlighting recent theoretical developments.

Contribution

It offers a unified framework for understanding locally stationary processes, emphasizing spectral analysis and empirical methods, and extends existing theory to nonlinear and linear cases.

Findings

Detailed analysis of time-varying autoregressive processes

Development of spectral density and likelihood theory for locally stationary processes

Discussion of empirical spectral processes and their applications

Abstract

The article contains an overview over locally stationary processes. At the beginning time varying autoregressive processes are discussed in detail - both as as a deep example and an important class of locally stationary processes. In the next section a general framework for time series with time varying finite dimensional parameters is discussed with special emphasis on nonlinear locally stationary processes. Then the paper focusses on linear processes where a more general theory is possible. First a general definition for linear processes is given and time varying spectral densities are discussed in detail. Then the Gaussian likelihood theory is presented for locally stationary processes. In the next section the relevance of empirical spectral processes for locally stationary time series is discussed. Empirical spectral processes play a major role in proving theoretical results and provide a deeper understanding of many techniques. The article concludes with an overview of other results for locally stationary processes.