A note on Herglotz's theorem for time series on function spaces

Anne van Delft*, Michael Eichler

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review


In this article, we prove Herglotz's theorem for Hilbert-valued time series. This requires the notion of an operator-valued measure, which we shall make precise for our setting. Herglotz's theorem for functional time series allows to generalize existing results that are central to frequency domain analysis on the function space. In particular, we use this result to prove the existence of a functional Cramer representation of a large class of processes, including those with jumps in the spectral distribution and long-memory processes. We furthermore obtain an optimal finite dimensional reduction of the time series under weaker assumptions than available in the literature. The results of this paper therefore enable Fourier analysis for processes of which the spectral density operator does not necessarily exist. (C) 2019 Elsevier B.V. All rights reserved.
Original languageEnglish
Pages (from-to)3687-3710
Number of pages24
JournalStochastic Processes and Their Applications
Issue number6
Publication statusPublished - Jun 2020


  • Functional data analysis
  • Spectral analysis
  • Time series

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