arXiv subjects

Leonardo Maini

Publications and source records attributed to Leonardo Maini.

8 recordsLinked to original sources

arXiv2026

Phase Transitions in the Fluctuations of Functionals of Random Neural Networks

We establish central and non-central limit theorems for sequences of functionals of the Gaussian output of an infinitely-wide random neural network on the d-dimensional sphere . We show that the asymptotic behaviour of these functionals as the depth of the network increases depends crucially on the fixed points of the covariance function, resulting in three distinct limiting regimes: convergence to the same functional of a limiting Gaussian field, convergence to a Gaussian distribution, convergence to a distribution in the Qth Wiener chaos. Our proofs exploit tools that are now classical (Hermite expansions, Diagram Formula, Stein-Malliavin techniques), but also ideas which have never been used in similar contexts: in particular, the asymptotic behaviour is determined by the fixed-point structure of the iterative operator associated with the covariance, whose nature and stability governs the different limiting regimes.

math.PR↗

arXiv2026

Limit theorems for anisotropic functionals of stationary Gaussian fields with Gneiting covariance function

We study non-linear additive functionals of stationary Gaussian fields over anisotropically growing domains in $\mathbb{R}^d$, including spatiotemporal settings, and establish Gaussian and non-Gaussian limit theorems under non-separable covariance structures. We characterize the regimes in which the normalized functionals converge either to a Gaussian distribution or to a $2$-domain Rosenblatt distribution, depending on precise long-range dependence conditions. Our analysis covers covariance functions from the Gneiting class, which provides a canonical family of non-separable spatiotemporal models. A key structural result shows that such covariances are asymptotically separable in a precise cumulant sense, allowing us to identify explicitly the limiting distributions without imposing additional spectral assumptions. These results extend existing spatiotemporal limit theorems beyond separable and short-memory frameworks and provide a unified description of anisotropic long-range dependence phenomena.

math.PR↗

arXiv2025

Malliavin differentiability for the excursion measure of a Gaussian field

In this work, we investigate the Malliavin differentiability of the excursion volume of a Gaussian field. In particular, we prove that if the moments of the covariance kernel go to 0 polinomially, then we can determine exactly for which p the excursion measure has pth square integrable Malliavin derivative. While our approach can not be implemented for Gaussian fields indexed by discrete sets (where the moments do not vanish), when the Gaussian field is indexed by Rd and stationary, or indexed by Sd and isotropic, the condition on the polynomial decay of the mentioned moments can be inferred starting from simple and general conditions on the covariance function of the field.

math.PR↗

arXiv2025

Almost sure central limit theorems via chaos expansions and related results

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.

math.PR↗

arXiv2024

Limit theorems for $p$-domain functionals of stationary Gaussian fields

Fix an integer $p\geq 1$ and refer to it as the number of growing domains. For each $i\in\{1,\ldots,p\}$, fix a compact subset $D_i\subseteq\mathbb R^{d_i}$ where $d_1,\ldots,d_p\ge 1$. Let $d= d_1+\dots+d_{p}$ be the total underlying dimension. Consider a continuous, stationary, centered Gaussian field $B=(B_x)_{x\in \mathbb R^d}$ with unit variance. Finally, let $\varphi:\mathbb R \rightarrow \mathbb R$ be a measurable function such that $\mathrm E[\varphi(N)^2]<\infty$ for $N\sim N(0,1)$. In this paper, we investigate central and non-central limit theorems as $t_1,\ldots,t_p\to\infty$ for functionals of the form \[ Y(t_1,\dots,t_p):=\int_{t_1D_1\times\dots \times t_pD_p}\varphi(B_{x})dx. \] Firstly, we assume that the covariance function $C$ of $B$ is {\it separable} (that is, $C=C_1\otimes\ldots\otimes C_{p}$ with $C_i:\mathbb R^{d_i}\to\mathbb R$), and thoroughly investigate under what condition $Y(t_1,\dots,t_p)$ satisfies a central or non-central limit theorem when the same holds for $\int_{t_iD_i}\varphi(B^{(i)}_{x_i})dx_i$ for at least one (resp. for all) $i\in \{1,\ldots,p\}$, where $B^{(i)}$ stands for a stationary, centered, Gaussian field on $\mathbb R^{d_i}$ admitting $C_i$ for covariance function. When $\varphi$ is an Hermite polynomial, we also provide a quantitative version of the previous result, which improves some bounds from A. Reveillac, M. Stauch, and C. A. Tudor, Hermite variations of the fractional brownian sheet, Stochastics and Dynamics 12 (2012). Secondly, we extend our study beyond the separable case, examining what can be inferred when the covariance function is either in the Gneiting class or is additively separable.

math.PR↗

arXiv2023

Asymptotic covariances for functionals of weakly stationary random fields

Let $(A_x)_{x\in\mathbb{R}^d}$ be a locally integrable, centered, weakly stationary random field, i.e. $\mathbb{E}[A_x]=0$, ${\rm Cov}(A_x,A_y)=K(x-y)$, $\forall x,y\in\mathbb{R}^d$, with measurable covariance function $K:\mathbb{R}^d\rightarrow\mathbb{R}$. Assuming only that $w_t:=\int_{\{|z|\le t\}}K(z)dz$ is regularly varying (which encompasses the classical assumptions found in the literature), we compute $$\lim_{t\rightarrow\infty}{\rm Cov}\left(\frac{\int_{tD}A_x dx}{t^{d/2}w_t^{1/2}}, \frac{\int_{tL}A_y dy}{t^{d/2}w_t^{1/2}}\right)$$ for $D,L\subseteq \mathbb{R}^d$ belonging to a certain class of compact sets. As an application, we combine this result with existing limit theorems to obtain multi-dimensional limit theorems for non-linear functionals of stationary Gaussian fields, in particular proving new results for the Berry's random wave model. At the end of the paper, we also show how the problem for $A$ with a general continuous covariance function $K$ can be reduced to the same problem for a radial, continuous covariance function $K_{\text{iso}}$. The novel ideas of this work are mainly based on regularity conditions for (cross) covariograms of Euclidean sets and standard properties of regularly varying functions.

math.PR↗

arXiv2023

Fluctuations of Polyspectra in Spherical and Euclidean Random Wave Models

We consider polynomial transforms (polyspectra) of Berry's model -- the Euclidean Random Wave model -- and of Random Hyperspherical Harmonics. We determine the asymptotic behavior of variance for polyspectra of any order in the high-frequency limit. In particular, we are able to treat polyspectra of any odd order $q\geq 5$, whose asymptotic behavior was left as a conjecture in the case of Random Hyperspherical Harmonics by Marinucci and Wigman (\emph{Comm. Math. Phys.} 2014). To this end, we exploit a relation between the variance of polyspectra and the distribution of uniform random walks on Euclidean space with finitely many steps, which allows us to rely on technical results in the latter context.

math.PR↗

arXiv2022

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

Let $B=(B_x)_{x\in\mathbb{R}^d}$ be a collection of $N(0,1)$ random variables forming a real-valued continuous stationary Gaussian field on $\mathbb{R}^d$, and set $C(x-y)=\mathbb{E}[B_xB_y]$. Let $\varphi:\mathbb{R}\to\mathbb{R}$ be such that $\mathbb{E}[\varphi(N)^2]<\infty$ with $N\sim N(0,1)$, let $R$ be the Hermite rank of $\varphi$, and consider $Y_t = \int_{tD} \varphi(B_x)dx$, $t>0$, with $D\subset \mathbb{R}^d$ compact. Since the pioneering works from the 80s by Breuer, Dobrushin, Major, Rosenblatt, Taqqu and others, central and noncentral limit theorems for $Y_t$ 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 $t\to\infty$ the fluctuations of $Y_t$ 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 $R\geq 2$. We show in this paper that this intuition forged over the last forty years can be wrong, and not only marginally or in critical cases. We will indeed bring to light a variety of situations where $ Y_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 $C\not\in L^R(\mathbb{R}^d)$. Our main mathematical tools are the Malliavin-Stein method and Fourier analysis techniques.

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