On the CLT for discrete Fourier transforms of functional time series

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• 作者：Clé ; ment Cerovecki ; ^{clement.cerovecki@gmail.com} ; Siegfried Hö ; rmann
• 关键词：Central limit theorem ; Functional time series ; Fourier transform ; Periodogram ; Stationarity
• 刊名：Journal of Multivariate Analysis
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：154
• 期：Complete
• 页码：282-295
• 全文大小：494 K
• 卷排序：154

文摘

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)(Xt:t≥1) defined, for all θ∈(−π,π]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 θθ if the process (Xt)(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)Sn(θn), when θn→θ0θn→θ0 is a sequence of fundamental frequencies. In particular we also deduce the regular CLT (θ=0θ=0) under new and very mild assumptions. We show that our results apply to the most commonly studied functional time series.