Locally Stationary Functional Time Series

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 Cramér 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.

References in corpus (6)

• Statistical inference for time-varying ARCH processes
• Common functional principal components
• Nonparametric regression for locally stationary time series
• Functional dynamic factor models with application to yield curve forecasting
• Locally Stationary Functional Time Series
• Data-adaptive estimation of time-varying spectral densities

Cited by in corpus (13)

• Locally Stationary Functional Time Series
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• Time Series Analysis and Modeling to Forecast: a Survey
• A nonparametric test for stationarity in functional time series
• A note on Herglotz's theorem for time series on function spaces
• A note on quadratic forms of stationary functional time series under mild conditions
• Testing equality of spectral density operators for functional linear processes
• Fourier-type monitoring procedures for strict stationarity
• On the estimation of locally stationary functional time series
• Adaptive Frequency Band Analysis for Functional Time Series
• Factor Models for High-Dimensional Functional Time Series
• Yield curve and macroeconomy interaction: evidence from the non-parametric functional lagged regression approach
• Confidence surfaces for the mean of locally stationary functional time series