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Testing LRD in the spectral domain for functional time series in manifolds

M. D. Ruiz-Medina, R. M. Crujeiras

TL;DR

The paper develops a spectral-domain test for long-range dependence in functional time series on manifolds, exploiting isometry-invariant spectral density operators and a weighted periodogram framework. A Central Limit Theorem yields an asymptotic Gaussian law for the test statistic under SRD, while a bias analysis under LRD supports consistency, with almost-sure divergence of the statistic when LRD is present. The authors implement the test via a random-projection approach to manage dimensionality and validate the method through simulations on spherical data, demonstrating correct size and strong power in finite samples. The contributions enable reliable inference for spatio-temporal processes on non-Euclidean spaces, with applications to climate, geophysics, and related fields, and provide a foundation for extensions to more general manifolds and data regimes.

Abstract

A statistical hypothesis test for long range dependence (LRD) is formulated in the spectral domain for functional time series in manifolds. The elements of the spectral density operator family are assumed to be invariant with respect to the group of isometries of the manifold. The proposed test statistic is based on the weighted periodogram operator. A Central Limit Theorem is derived to obtain the asymptotic Gaussian distribution of the proposed test statistic operator under the null hypothesis. The rate of convergence to zero, in the Hilbert--Schmidt operator norm, of the bias of the integrated empirical second and fourth order cumulant spectral density operators is obtained under the alternative hypothesis. The consistency of the test follows from the consistency of the integrated weighted periodogram operator under LRD. Practical implementation of our testing approach is based on the random projection methodology. A simulation study illustrates, in the context of spherical functional time series, the asymptotic normality of the test statistic under the null hypothesis, and its consistency under the alternative. The empirical size and power properties are also computed for different functional sample sizes, and under different scenarios.

Testing LRD in the spectral domain for functional time series in manifolds

TL;DR

The paper develops a spectral-domain test for long-range dependence in functional time series on manifolds, exploiting isometry-invariant spectral density operators and a weighted periodogram framework. A Central Limit Theorem yields an asymptotic Gaussian law for the test statistic under SRD, while a bias analysis under LRD supports consistency, with almost-sure divergence of the statistic when LRD is present. The authors implement the test via a random-projection approach to manage dimensionality and validate the method through simulations on spherical data, demonstrating correct size and strong power in finite samples. The contributions enable reliable inference for spatio-temporal processes on non-Euclidean spaces, with applications to climate, geophysics, and related fields, and provide a foundation for extensions to more general manifolds and data regimes.

Abstract

A statistical hypothesis test for long range dependence (LRD) is formulated in the spectral domain for functional time series in manifolds. The elements of the spectral density operator family are assumed to be invariant with respect to the group of isometries of the manifold. The proposed test statistic is based on the weighted periodogram operator. A Central Limit Theorem is derived to obtain the asymptotic Gaussian distribution of the proposed test statistic operator under the null hypothesis. The rate of convergence to zero, in the Hilbert--Schmidt operator norm, of the bias of the integrated empirical second and fourth order cumulant spectral density operators is obtained under the alternative hypothesis. The consistency of the test follows from the consistency of the integrated weighted periodogram operator under LRD. Practical implementation of our testing approach is based on the random projection methodology. A simulation study illustrates, in the context of spherical functional time series, the asymptotic normality of the test statistic under the null hypothesis, and its consistency under the alternative. The empirical size and power properties are also computed for different functional sample sizes, and under different scenarios.

Paper Structure

This paper contains 29 sections, 10 theorems, 108 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Hypothesis testing
  4. Preliminary results under SRD
  5. Second and fourth order bias asymptotics under LRD
  6. A test for LRD in functional time series on $\mathbb{M}_{d}$
  7. Practical implementation
  8. Simulation study
  9. Asymptotic Gaussian distribution of $\mathcal{S}_{B_{T}}$ under $H_{0}$
  10. Consistency of the test
  11. Example 1
  12. Example 2
  13. Example 3
  14. Almost surely divergence of $\mathcal{S}_{B_{T}}$ in $\mathcal{S}(L^{2}(\mathbb{M}_{d},d\nu, \mathbb{C}))$ norm under $H_{1}$
  15. Example 4
  16. ...and 14 more sections

Key Result

Lemma 1

Assume that $E\|X_{0}\|^{k}<\infty,$ for all $k\geq 2,$ and Then, for every frequencies $\omega_{j},$$j=1,\dots,J,$ with $J<\infty,$ where $\to_{D}$ denotes the convergence in distribution. Here, $\widehat{f}_{\omega_{j} },$$j=1,\dots,J,$ are jointly zero--mean complex Gaussian elements in $\mathcal{S}(L^{2}(\mathbb{M}_{d},d\nu, \mathbb{C}))$$=L^{2}(\mathbb{M}_{d}^{2},d\nu\otimes d\nu , \mathbb{C

Figures (12)

  • Figure 1: One realization at times $t=30, 130, 230, 330, 430, 530, 630, 730, 830,930$ of SPHARMA(1,1) process $\left(\lambda_{n}(\varphi_{1})=0.7\left(\frac{n+1}{n}\right)^{-3/2}\right.,$ and $\left.\lambda_{n}(\psi_{1})= (0.4)\left(\frac{n+1}{n}\right)^{-5/1.95},\ n=1,2,3,4,5,6,7,8\right),$ projected into the direct sum $\bigoplus_{n=1}^{8}\mathcal{H}_{n}$ of eigenspaces $\mathcal{H}_{n},$$n=1,\dots,8,$ of $\Delta_{2}$
  • Figure 2: Empirical projections of the probability measure of standardized $\mathcal{S}_{B_{T}},$$B_{T}=T^{-1/4},$ under $H_{0},$ into the eigenspaces $\mathcal{H}_{n}\otimes \mathcal{H}_{n},$ for $n=1,2,3,4,5,6,7,8,$ respectively displayed from the left to the right, and from the top to the bottom, for functional samples size $T=1000$ and $R=3000$ repetitions
  • Figure 3: Empirical projections of the probability measure of standardized $\mathcal{S}_{B_{T}},$$B_{T}=T^{-1/4},$ under $H_{0},$ into the eigenspaces $\mathcal{H}_{n}\otimes \mathcal{H}_{n},$ for $n=1,2,3,4,5,6,7,8,$ respectively displayed from the left to the right, and from the top to the bottom, for functional samples size $T=2000$ and $R=3000$ repetitions
  • Figure 4: Empirical projections of the probability measure of standardized $\mathcal{S}_{B_{T}},$$B_{T}=T^{-1/4},$ under $H_{0},$ into the eigenspaces $\mathcal{H}_{n}\otimes \mathcal{H}_{n},$ for $n=1,2,3,4,5,6,7,8,$ respectively displayed from the left to the right, and from the top to the bottom, for functional samples size $T=3000$ and $R=3000$ repetitions
  • Figure 5: Example 1. One sample realization at times $t=30, 130, 230, 330, 430, 530, 630, 730, 830,930$ of multifractionally integrated SPHARMA(1,1) process projected into $\bigoplus_{n=1}^{8}\mathcal{H}_{n}$
  • ...and 7 more figures

Theorems & Definitions (20)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Corollary 1
  • Lemma 3
  • Proposition 1
  • Theorem 2
  • Remark 1
  • Corollary 2
  • Theorem 3
  • ...and 10 more