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Almost sure central limit theorems via chaos expansions and related results

Leonardo Maini, Maurizia Rossi, Guangqu Zheng

Abstract

In this work, we investigate the asymptotic behavior of integral functionals of stationary Gaussian random fields as the integration domain tends to be the whole space. More precisely, using the Wiener chaos expansion and Malliavin-Stein method, we establish an almost sure central limit theorem (ASCLT) only under mild conditions on the covariance function of the underlying stationary Gaussian field. In this setting, we additionally derive a quantitative central limit theorem with rate of convergence in Wasserstein distance, and show certain regularity property for the said integral functionals (the latter under weaker conditions). In particular, we solved an open question on the Malliavin differentiability of the excursion volume of Berry's random wave model. As a key consequence of our analysis, we obtain the exact asymptotic rate (as a function of the exponent) for moments of Bessel functions, thus confirming a conjecture based on existing numerical simulations. In the end, we provide two applications of our result: (i) ASCLT in the context of Breuer-Major central limit theorems, (ii) ASCLT for Berry's random wave model. It is worth stressing that our approach does not require any knowledge on the regularity properties of random variables (e.g., Malliavin differentiability) and hence not only complements the existing literature, but also leads to novel results that are of independent interest.

Almost sure central limit theorems via chaos expansions and related results

Abstract

In this work, we investigate the asymptotic behavior of integral functionals of stationary Gaussian random fields as the integration domain tends to be the whole space. More precisely, using the Wiener chaos expansion and Malliavin-Stein method, we establish an almost sure central limit theorem (ASCLT) only under mild conditions on the covariance function of the underlying stationary Gaussian field. In this setting, we additionally derive a quantitative central limit theorem with rate of convergence in Wasserstein distance, and show certain regularity property for the said integral functionals (the latter under weaker conditions). In particular, we solved an open question on the Malliavin differentiability of the excursion volume of Berry's random wave model. As a key consequence of our analysis, we obtain the exact asymptotic rate (as a function of the exponent) for moments of Bessel functions, thus confirming a conjecture based on existing numerical simulations. In the end, we provide two applications of our result: (i) ASCLT in the context of Breuer-Major central limit theorems, (ii) ASCLT for Berry's random wave model. It is worth stressing that our approach does not require any knowledge on the regularity properties of random variables (e.g., Malliavin differentiability) and hence not only complements the existing literature, but also leads to novel results that are of independent interest.

Paper Structure

This paper contains 12 sections, 19 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Breuer-Major type CLTs
  3. Berry's random wave model
  4. Main results
  5. Preliminaries
  6. Functions of regular variation
  7. Basics on Malliavin-Stein method
  8. Auxiliary results on ASCLT
  9. Main proofs
  10. Proofs of QCLT and ASCLT
  11. Proofs of Corollaries
  12. Technical results

Key Result

Theorem 1.1

Let $\varphi$ be as in HerExp with Hermite rank $R\geq 1$. Let $\mathcal{C}$ be the covariance function as in C00. Recall the definition def_Y of $Y_t$. Then, the following statements hold. (i) [Breuer-Major's theorem] Assume $\mathcal{C}\in L^R( \mathbb{R}^d)$. Then, and $\dfrac{Y_t - \mathbb{E}[Y_t]}{t^{d/2} }$ converges in law to $\mathcal{N}(0, \sigma^2)$ as $t\to+\infty$. See, e.g., BM83CNN2

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.9
  • Remark 1.10
  • Remark 1.11
  • Lemma 1.12
  • Remark 1.13
  • Corollary 1.14
  • ...and 37 more