Locally Stationary Functional Time Series

TL;DR

This paper develops a framework for analyzing functional time series with changing dynamics over time, introducing local stationarity, spectral representations, and estimators for time-varying spectral density operators.

Contribution

It introduces the concept of local stationarity for functional data, extending spectral analysis and estimation techniques to nonstationary functional time series.

Findings

Functional local stationarity is established for time-varying ARMA processes.

A Cramér representation for weakly stationary functional processes is derived.

A consistent, asymptotically Gaussian estimator for the spectral density operator is proposed.

Abstract

The literature on time series of functional data has focused on processes of which the probabilistic law is either constant over time or constant up to its second-order structure. Especially for long stretches of data it is desirable to be able to weaken this assumption. This paper introduces a framework that will enable meaningful statistical inference of functional data of which the dynamics change over time. We put forward the concept of local stationarity in the functional setting and establish a class of processes that have a functional time-varying spectral representation. Subsequently, we derive conditions that allow for fundamental results from nonstationary multivariate time series to carry over to the function space. In particular, time-varying functional ARMA processes are investigated and shown to be functional locally stationary according to the proposed definition. As a side-result, we establish a Cram\'er representation for an important class of weakly stationary functional processes. Important in our context is the notion of a time-varying spectral density operator of which the properties are studied and uniqueness is derived. Finally, we provide a consistent nonparametric estimator of this operator and show it is asymptotically Gaussian using a weaker tightness criterion than what is usually deemed necessary.