Let $\mathit{B}={({\mathit{B}}_{\mathit{x}})}_{\mathit{x}\in {\mathbb{R}}^{\mathit{d}}}$ be a collection of $\mathit{N}(0,1)$ random variables forming a real-valued continuous stationary Gaussian field on ${\mathbb{R}}^{\mathit{d}}$, and set $\mathit{C}(\mathit{x}-\mathit{y})=\mathbb{E}[{\mathit{B}}_{\mathit{x}}{\mathit{B}}_{\mathit{y}}]$. Let $\mathit{\phi }:\mathbb{R}\to \mathbb{R}$ be such that $\mathbb{E}[\mathit{\phi }{(\mathit{N})}^2]<\infty$ with $\mathit{N}\sim \mathit{N}(0,1)$, let R be the Hermite rank of φ, and consider ${\mathit{Y}}_{\mathit{t}}={\int }_{\mathit{t}\mathit{D}}\mathit{\phi }({\mathit{B}}_{\mathit{x}})\mathit{d}\mathit{x}$, $\mathit{t}>0$ with $\mathit{D}\subset {\mathbb{R}}^{\mathit{d}}$ compact.

Since the pioneering works from the 1980s by Breuer, Dobrushin, Major, Rosenblatt, Taqqu and others, central and noncentral limit theorems for ${\mathit{Y}}_{\mathit{t}}$ 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 $\mathit{t}\to \infty$ the fluctuations of ${\mathit{Y}}_{\mathit{t}}$ 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 $\mathit{R}\ge 2$.

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 ${\mathit{Y}}_{\mathit{t}}$ 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 $\mathit{C}\notin {\mathit{L}}^{\mathit{R}}({\mathbb{R}}^{\mathit{d}})$. Our main mathematical tools are the Malliavin–Stein method and Fourier analysis techniques.