On the CLT for discrete Fourier transforms of functional time series

Cerovecki, Clément; Hörmann, Siegfried

doi:10.1016/j.jmva.2016.11.006

Cited by 22 publications

(19 citation statements)

References 36 publications

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“…We can further bound this quantity by sup ℓ≥1 j>Nn θ ℓ,j . To show that this converges to zero we apply the tighness Lemma 14 in Cerovecki and Hörmann (2017) with p (n) j = θ n,j and p (0) j = θ 0,j . Finally, from the compactness of Θ we know that there exists a subsequence ( θ Nn ℓ n ℓ ) ℓ≥1 that converges in ℓ 2 to x, say, and observe that by definition…”

Section: Appendixmentioning

confidence: 99%

Functional GARCH models: The quasi-likelihood approach and its applications

Cerovecki

Francq

Hörmann

et al. 2019

Journal of Econometrics

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The increasing availability of high frequency data has initiated many new research areas in statistics. Functional data analysis (FDA) is one such innovative approach towards modelling time series data. In FDA, densely observed data are transformed into curves and then each (random) curve is considered as one data object. A natural, but still relatively unexplored, context for FDA methods is related to financial data, where high-frequency trading currently takes a significant proportion of trading volumes. Recently, articles on functional versions of the famous ARCH and GARCH models have appeared. Due to their technical complexity, existing estimators of the underlying functional parameters are moment based-an approach which is known to be relatively inefficient in this context. In this paper, we promote an alternative quasi-likelihood approach, for which we derive consistency and asymptotic normality results. We support the relevance of our approach by simulations and illustrate its use by forecasting realised volatility of the S&P100 Index.

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Section: Appendixmentioning

confidence: 99%

Functional GARCH models: The quasi-likelihood approach and its applications

Cerovecki

Francq

Hörmann

et al. 2019

Journal of Econometrics

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“…By Theorem 4 of [9], for all ω ∈ (−π, π], if (X n ) n∈Z is a linear process with ∑ ∞ i=1 ψ j L < ∞, then 1 √ n S n (ω) converges in distribution as n → ∞ to a complex Gaussian random element with covariance operator…”

Section: )mentioning

confidence: 99%

“…Theorem 1 and 4 of [9] infer the following duality between C X;h and F X [ω], with A[z] as in (6.5) and adjoint A[z] * :…”

Section: )mentioning

confidence: 99%

An innovations algorithm for the prediction of functional linear processes

Klepsch

Klüppelberg

2017

Journal of Multivariate Analysis

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When observations are curves over some natural time interval, the field of functional data analysis comes into play. Functional linear processes account for temporal dependence in the data. The prediction problem for functional linear processes has been solved theoretically, but the focus for applications has been on functional autoregressive processes. We propose a new computationally tractable linear predictor for functional linear processes. It is based on an application of the Multivariate Innovations Algorithm to finite-dimensional subprocesses of increasing dimension of the infinite-dimensional functional linear process. We investigate the behavior of the predictor for increasing sample size. We show that, depending on the decay rate of the eigenvalues of the covariance and the spectral density operator, the resulting predictor converges with a certain rate to the theoretically best linear predictor.

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“…. , n i , be two independent samples of curves, satisfying model (4). For i ∈ {1, 2} and for (u, v) ∈ [0, 1] 2 , denote by c i (u, v) the kernels of the long run covariance operators 2πF…”

Section: Bootstrap Validitymentioning

confidence: 99%

Moving block and tapered block bootstrap for functional time series with an application to the $K$-sample mean problem

Pilavakis¹,

Paparoditis²,

Sapatinas³

2019

Bernoulli

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We consider infinite-dimensional Hilbert space-valued random variables that are assumed to be temporal dependent in a broad sense. We prove a central limit theorem for the moving block bootstrap and for the tapered block bootstrap, and show that these block bootstrap procedures also provide consistent estimators of the long run covariance operator. Furthermore, we consider block bootstrap-based procedures for fully functional testing of the equality of mean functions between several independent functional time series. We establish validity of the block bootstrap methods in approximating the distribution of the statistic of interest under the null and show consistency of the block bootstrap-based tests under the alternative. The finite sample behaviour of the procedures is investigated by means of simulations. An application to a real-life dataset is also discussed. n )) − nE(X n ⊗ X n ) HS = o P (1), in probability. The Tapered Block BootstrapThe TBB procedure is a modification of the block bootstrap procedure considered in Section 2.2 which is obtained by introducing a tapering of the random elements X t . The tapering function downweights the endpoints of each block B i , towards zero, i.e., towards the mean function of X t . The pseudo observations are then obtained by choosing, with replacement, k appropriately scaled and tapered blocks of length b of centered observations and joining them together.

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On the CLT for discrete Fourier transforms of functional time series

Abstract: We consider a strictly stationary functional time series. Our target is to study the weak convergence of the discrete Fourier transforms under sharp conditions. As a side-result we obtain the regular CLT for partial sums under mild assumptions.

Cited by 22 publications

References 36 publications

Functional GARCH models: The quasi-likelihood approach and its applications

Functional GARCH models: The quasi-likelihood approach and its applications

An innovations algorithm for the prediction of functional linear processes

Moving block and tapered block bootstrap for functional time series with an application to the $K$-sample mean problem

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