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Limit theorems for anisotropic functionals of stationary Gaussian fields with Gneiting covariance function

Nikolai Leonenko, Leonardo Maini, Ivan Nourdin, Francesca Pistolato

Abstract

We study non-linear additive functionals of stationary Gaussian fields over anisotropically growing domains in $\mathbb{R}^d$, including spatiotemporal settings, and establish Gaussian and non-Gaussian limit theorems under non-separable covariance structures. We characterize the regimes in which the normalized functionals converge either to a Gaussian distribution or to a $2$-domain Rosenblatt distribution, depending on precise long-range dependence conditions. Our analysis covers covariance functions from the Gneiting class, which provides a canonical family of non-separable spatiotemporal models. A key structural result shows that such covariances are asymptotically separable in a precise cumulant sense, allowing us to identify explicitly the limiting distributions without imposing additional spectral assumptions. These results extend existing spatiotemporal limit theorems beyond separable and short-memory frameworks and provide a unified description of anisotropic long-range dependence phenomena.

Limit theorems for anisotropic functionals of stationary Gaussian fields with Gneiting covariance function

Abstract

We study non-linear additive functionals of stationary Gaussian fields over anisotropically growing domains in , including spatiotemporal settings, and establish Gaussian and non-Gaussian limit theorems under non-separable covariance structures. We characterize the regimes in which the normalized functionals converge either to a Gaussian distribution or to a -domain Rosenblatt distribution, depending on precise long-range dependence conditions. Our analysis covers covariance functions from the Gneiting class, which provides a canonical family of non-separable spatiotemporal models. A key structural result shows that such covariances are asymptotically separable in a precise cumulant sense, allowing us to identify explicitly the limiting distributions without imposing additional spectral assumptions. These results extend existing spatiotemporal limit theorems beyond separable and short-memory frameworks and provide a unified description of anisotropic long-range dependence phenomena.
Paper Structure (26 sections, 19 theorems, 167 equations, 1 figure)

This paper contains 26 sections, 19 theorems, 167 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Overview of the proof
  4. Further remarks and example
  5. Asymptotic separability
  6. Plan of the paper
  7. Preliminaries
  8. Chaotic decompositions and fourth moment theorems
  9. A Hilbert space associated with the covariance kernel.
  10. Wiener chaoses and multiple integrals.
  11. Contractions and the Fourth Moment Theorem.
  12. Cyclic products, and characteristic functions in the second chaos.
  13. Rosenblatt distributions
  14. $2$-domain Rosenblatt distribution
  15. Miscellanea
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let us consider $Y(t)$ as in eq:Yt. Suppose Assumptions ass:gneiting and ass:KEY hold. Let $R\ge2$ be the Hermite rank of $\varphi$. If $C\in L^R(\mathbb R^{d})$, then, as $t\to\infty$, Moreover, $\mathop{\mathrm{\mathbb{V}ar}}\nolimits(Y(t))\sim \ell \ t_1(t)^{d_1}t_2(t)^{d_2}$, where If $C_1\in L^{R}(\mathbb R^{d_1})$, and $C_2\notin L^{R-1}(\mathbb R^{d_2})$, but $C_2$ is slowly or regularly

Figures (1)

  • Figure 1.1: Four non-critical regimes of the variance of $Y_{t_1,t_2}$ (up to constants) and limiting distribution of $\widetilde{Y}_{t_1,t_2}$, when the Hermite rank of $\varphi$ is $R=2$, and both $C_1$ and $C_2$ are regularly varying with parameters $-\rho_1$ and $-\rho_2$, respectively, with constant slowly varying components. In magenta, the line $\rho_2 = \frac{d_1d_2}{2(d_1-\rho_1)}$. The limiting distribution is everywhere Gaussian, except for $(\rho_1,\rho_2)$ in the gray region delimited by the magenta and cyan curves, where the limiting distribution is a random variable in the second chaos.

Theorems & Definitions (60)

  • Definition 1.1: Regularly varying function
  • Remark 1.2
  • Remark 1.3: The case $R=1$
  • Theorem 1.1
  • Remark 1.4
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.5: Criticality
  • Remark 1.6
  • Remark 1.7
  • ...and 50 more