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Central limit theorems for non-linear functionals of Gaussian fields via Wiener chaos decomposition

Fabio Coppini, Wioletta M. Ruszel

TL;DR

The paper studies non linear functionals of stationary Gaussian fields on $\mathbb{Z}^d$ and proves a Breuer–Major type central limit theorem for the field of transformed values $\Phi_N(j)=H(X_j)$ tested against smooth domains. Using a Wiener chaos decomposition and the fourth moment theorem, it derives distributional convergence of the field to a Gaussian white noise in $\mathcal{H}^{-\alpha}(D)$ under a Hermite rank based summability condition on the covariance. It then applies the main result to powers of the discrete Gaussian Free Field, showing that even powers converge to white noise (under suitable dimension) while odd powers converge to a continuous Gaussian Free Field with explicit covariance, including a precise normalization constant. The paper also provides a tightness argument and situates the lattice results within the broader Breuer–Major framework, offering a robust approach to fluctuations of nonlinear Gaussian functionals on lattices with potential applications to GFF-related models.

Abstract

We review and present some known results for non-linear functionals of Gaussian variables in the context of discrete Gaussian fields defined on the $d$ dimensional lattice. Our main result is a Central Limit Theorem in the spirit of the classical Breuer-Major theorem, together with applications to the powers of the Gaussian Free Field. Notably, we show that even powers of the discrete Gaussian Free Field converge to the Gaussian white noise, while odd powers converge to a continuous Gaussian Free Field with explicit covariance. The proofs are based on the Wiener chaos decomposition and the fourth moment theorem (Nualart-Peccati, 2005), and include a tightness result. Even if these tools are well-known in the literature, their application to Gaussian fields on the lattice appears to be new.

Central limit theorems for non-linear functionals of Gaussian fields via Wiener chaos decomposition

TL;DR

The paper studies non linear functionals of stationary Gaussian fields on and proves a Breuer–Major type central limit theorem for the field of transformed values tested against smooth domains. Using a Wiener chaos decomposition and the fourth moment theorem, it derives distributional convergence of the field to a Gaussian white noise in under a Hermite rank based summability condition on the covariance. It then applies the main result to powers of the discrete Gaussian Free Field, showing that even powers converge to white noise (under suitable dimension) while odd powers converge to a continuous Gaussian Free Field with explicit covariance, including a precise normalization constant. The paper also provides a tightness argument and situates the lattice results within the broader Breuer–Major framework, offering a robust approach to fluctuations of nonlinear Gaussian functionals on lattices with potential applications to GFF-related models.

Abstract

We review and present some known results for non-linear functionals of Gaussian variables in the context of discrete Gaussian fields defined on the dimensional lattice. Our main result is a Central Limit Theorem in the spirit of the classical Breuer-Major theorem, together with applications to the powers of the Gaussian Free Field. Notably, we show that even powers of the discrete Gaussian Free Field converge to the Gaussian white noise, while odd powers converge to a continuous Gaussian Free Field with explicit covariance. The proofs are based on the Wiener chaos decomposition and the fourth moment theorem (Nualart-Peccati, 2005), and include a tightness result. Even if these tools are well-known in the literature, their application to Gaussian fields on the lattice appears to be new.

Paper Structure

This paper contains 12 sections, 9 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Motivation and main result
  3. Related works
  4. Applications to the powers of the discrete Gaussian Free Field
  5. Even powers
  6. Odd powers
  7. Preliminaries
  8. Isonormal Wiener processes representation
  9. Central limit theorem for the $q$-th components
  10. Proof of Theorem \ref{['thm:main']}
  11. Convergence of observables
  12. Tightness

Key Result

Theorem 1

Let $(X_i)_{i \in \mathbb{Z}^d}$ be a standard stationary Gaussian field with covariance $\rho(u) = \mathbb{E}[X_o X_u]$ for $u\in \mathbb{Z}^d$. Let $H: \mathbb{R}\to \mathbb{R}$ be a function with Hermite expansion $H = \sum_{q=m}^\infty c_q H_q$ and Hermite rank $m \geq 1$. Suppose that Let $\Phi_N$ be the field defined in Equationdef:langle Phi_N, then as $N\rightarrow \infty$ in $\mathcal{H

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2: breuer_central_1983
  • Remark 1
  • Remark 2
  • Theorem 3
  • proof
  • Theorem 4: nualart_central_2005
  • Proposition 1
  • proof
  • Lemma 1
  • ...and 7 more