### Abstract

Original language | English |
---|---|

Pages (from-to) | 282-295 |

Number of pages | 14 |

Journal | Journal of multivariate analysis |

Volume | 154 |

Publication status | Published - 2017 |

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### Cite this

*Journal of multivariate analysis*,

*154*, 282-295.

**On the CLT for discrete Fourier transforms of functional time series.** / Cerovecki, Clément; Hörmann, Siegfried.

Research output: Contribution to journal › Article › Research › peer-review

*Journal of multivariate analysis*, vol. 154, pp. 282-295.

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TY - JOUR

T1 - On the CLT for discrete Fourier transforms of functional time series

AU - Cerovecki, Clément

AU - Hörmann, Siegfried

N1 - DOI: 10.1016/j.jmva.2016.11.006

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

M3 - Article

VL - 154

SP - 282

EP - 295

JO - Journal of multivariate analysis

JF - Journal of multivariate analysis

SN - 0047-259x

ER -