On the CLT for discrete Fourier transforms of functional time series

Clément Cerovecki, Siegfried Hörmann

Research output: Contribution to journalArticlepeer-review


The purpose of this paper is to derive sharp conditions for the asymptotic normality of a discrete Fourier transform of a functional time series (Xt:t≥1) defined, for all θ∈(−π,π], by Sn(θ)=Xte−iθ+⋯+Xte−inθ. Assuming that the function space is a Hilbert space we prove that a Central Limit Theorem (CLT) holds for almost all frequencies θ if the process (Xt) is stationary, ergodic and purely non-deterministic. Under slightly stronger assumptions we formulate versions which provide a CLT for fixed frequencies as well as for Sn(θn), when θn→θ0 is a sequence of fundamental frequencies. In particular we also deduce the regular CLT (θ=0) under new and very mild assumptions. We show that our results apply to the most commonly studied functional time series.
Original languageEnglish
Pages (from-to)282-295
Number of pages14
JournalJournal of Multivariate Analysis
Publication statusPublished - 2017
Externally publishedYes


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