Spectral central limit theorem for additive functionals of isotropic and stationary Gaussian fields

Let B=(Bx)x∈Rd be a collection of N(0,1) random variables forming a real-valued continuous stationary Gaussian field on Rd, and set C(x−y)=EBxBy. Let φ:R→R be such that Eφ(N)20 with D⊂Rd compact. Since the pioneering works from the 1980s by Breuer, Dobrushin, Major, Rosenblatt, Taqqu and others, central and noncentral limit theorems for Yt have been constantly refined, extended and applied to an increasing number of diverse situations, to such an extent that it has become a field of research in its own right. The common belief, representing the intuition that specialists in the subject have developed over the last four decades, is that as t→∞ the fluctuations of Yt around its mean are, in general (i.e., except possibly in very special cases), Gaussian when B has short memory, and non-Gaussian when B has long memory and R≥2. We show in this paper that this intuition forged over the last 40 years can be wrong, and not only marginally or in critical cases. We will indeed bring to light a variety of situations where Yt admits Gaussian fluctuations in a long memory context. To achieve this goal, we state and prove a spectral central limit theorem, which extends the conclusion of the celebrated Breuer–Major theorem to situations where C∉LR(Rd). Our main mathematical tools are the Malliavin–Stein method and Fourier analysis techniques.