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 {\it almost sure central limit theorem} (ASCLT) only under mild conditions on the covariance function of the underlying stationary Gaussian fields. In this setting, we additionally derive a {\it quantitative central limit theorem} with rate of convergence in quadratic Wasserstein distance, and show certain regularity property for the said integral functionals. In particular, we solve an open question on the {\it 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 $q$) for the $q$-th moment 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. 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.