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Asymptotic Equivalence of Locally Stationary Processes and Bivariate White Noise

Cristina Butucea, Alexander Meister, Angelika Rohde

TL;DR

The paper develops a general Gaussianization framework to prove Le Cam-type asymptotic equivalence between Gaussian experiments with high- or infinite-dimensional covariance structures and low- to moderate-dimensional Gaussian models. Central to the method are a pre-smoothing projection onto a low-rank subspace, a novel localization strategy, dimension reduction, and a high-dimensional Central Limit Theorem in total variation distance, augmented by a rigorous Edgeworth-type expansion. The authors apply this machinery to locally stationary Gaussian time series, showing asymptotic equivalence with a bivariate Gaussian white noise model whose drift is the time-varying log-spectral density, via a construction of circulant-type matrices and GOE perturbations to enable Gaussianization. The results bridge intricate dependency structures in high dimensions with tractable Gaussian limits, enabling nonparametric spectral-density estimation and broader Gaussian-process inference in time-series contexts with practical implications for high-dimensional covariance estimation and asymptotic decision theory.

Abstract

We consider a general class of statistical experiments, in which an $n$-dimensional centered Gaussian random variable is observed and its covariance matrix is the parameter of interest. The covariance matrix is assumed to be well-approximable in a linear space of lower dimension $K_n$ with eigenvalues uniformly bounded away from zero and infinity. We prove asymptotic equivalence of this experiment and a class of $K_n$-dimensional Gaussian models with informative expectation in Le Cam's sense when $n$ tends to infinity and $K_n$ is allowed to increase moderately in $n$ at a polynomial rate. For this purpose we derive a new localization technique for non-i.i.d. data and a novel high-dimensional Central Limit Law in total variation distance. These results are key ingredients to show asymptotic equivalence between the experiments of locally stationary Gaussian time series and a bivariate Wiener process with the log spectral density as its drift. Therein a novel class of matrices is introduced which generalizes circulant Toeplitz matrices traditionally used for strictly stationary time series.

Asymptotic Equivalence of Locally Stationary Processes and Bivariate White Noise

TL;DR

The paper develops a general Gaussianization framework to prove Le Cam-type asymptotic equivalence between Gaussian experiments with high- or infinite-dimensional covariance structures and low- to moderate-dimensional Gaussian models. Central to the method are a pre-smoothing projection onto a low-rank subspace, a novel localization strategy, dimension reduction, and a high-dimensional Central Limit Theorem in total variation distance, augmented by a rigorous Edgeworth-type expansion. The authors apply this machinery to locally stationary Gaussian time series, showing asymptotic equivalence with a bivariate Gaussian white noise model whose drift is the time-varying log-spectral density, via a construction of circulant-type matrices and GOE perturbations to enable Gaussianization. The results bridge intricate dependency structures in high dimensions with tractable Gaussian limits, enabling nonparametric spectral-density estimation and broader Gaussian-process inference in time-series contexts with practical implications for high-dimensional covariance estimation and asymptotic decision theory.

Abstract

We consider a general class of statistical experiments, in which an -dimensional centered Gaussian random variable is observed and its covariance matrix is the parameter of interest. The covariance matrix is assumed to be well-approximable in a linear space of lower dimension with eigenvalues uniformly bounded away from zero and infinity. We prove asymptotic equivalence of this experiment and a class of -dimensional Gaussian models with informative expectation in Le Cam's sense when tends to infinity and is allowed to increase moderately in at a polynomial rate. For this purpose we derive a new localization technique for non-i.i.d. data and a novel high-dimensional Central Limit Law in total variation distance. These results are key ingredients to show asymptotic equivalence between the experiments of locally stationary Gaussian time series and a bivariate Wiener process with the log spectral density as its drift. Therein a novel class of matrices is introduced which generalizes circulant Toeplitz matrices traditionally used for strictly stationary time series.
Paper Structure (23 sections, 15 theorems, 244 equations)

This paper contains 23 sections, 15 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. General Gaussianization Scheme
  3. Pre-Smoothing
  4. A Novel Approach to Localization
  5. Dimension Reduction in the Localized and Smoothed Experiment
  6. High-dimensional Central Limit Theorem in Total Variation Distance
  7. Upper Bound on the Fourier Tails
  8. High-dimensional Edgeworth Expansion
  9. Piecing Together the Ingredients for Gaussianization
  10. Transformation of the Covariance Matrices
  11. Matrix Experiment with GOE Noise
  12. Re-Globalization
  13. Asymptotic Equivalence of Locally Stationary Times Series and Bivariate White Noise
  14. Locally Stationary Time Series
  15. Useful Matrices for Locally Stationary Processes
  16. ...and 8 more sections

Key Result

Lemma 1

Grant (eq:condMk) and assume that $\lim_{n\to\infty} \, K_n^2 \, \gamma_n^2 / n\, = \, 0$. Then Moreover all eigenvalues of $\Gamma$, $\Gamma_\theta$ and $\tilde{\Gamma}_\theta$ admit the lower bound $c \rho^2$ for any fixed $c\in (0,2)$ when $n$ is sufficiently large (uniformly with respect to $\theta$).

Theorems & Definitions (16)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Lemma 5
  • Theorem 2
  • Remark 1
  • Lemma 6
  • Lemma 7
  • ...and 6 more