## Electronic Journal of Statistics

### 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.

#### Article information

Source
Electron. J. Statist., Volume 12, Number 1 (2018), 107-170.

Dates
Received: May 2017
First available in Project Euclid: 15 January 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1516006818

Digital Object Identifier
doi:10.1214/17-EJS1384

Mathematical Reviews number (MathSciNet)
MR3746979

Zentralblatt MATH identifier
06841001

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62M15: Spectral analysis

#### Citation

van Delft, Anne; Eichler, Michael. Locally stationary functional time series. Electron. J. Statist. 12 (2018), no. 1, 107--170. doi:10.1214/17-EJS1384. https://projecteuclid.org/euclid.ejs/1516006818

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