arXiv subjects

Simon Campese

Publications and source records attributed to Simon Campese.

14 recordsLinked to original sources

arXiv2025

Gaussian approximation on the Skorokhod space via Malliavin calculus and regularization

We introduce a carr\'e du champ operator for Banach-valued random elements, taking values in the projective tensor product, and use it to control the bounded Lipschitz distance between a Malliavin-smooth random element satisfying mild regularity assumptions and a Radon Gaussian taking values in the Skorokhod space equipped with the uniform topology. In the case where the random element is a Banach-valued multiple integral, the carr\'e du champ expression is further bounded by norms of the contracted integral kernel. The main technical tool is an integration by parts formula, which might be of independent interest. As a by-product, we recover a bound obtained recently by D\"uker and Zoubouloglou in the Hilbert space setting and complement it by providing contraction bounds.

math.PR↗

arXiv2025

A Caveat on Metrizing Convergence in Distribution on Hilbert Spaces

We consider Sobolev-type distances on probability measures over separable Hilbert spaces involving the Schatten-$p$ norms, which include as special cases a distance first introduced by Bourguin and Campese (2020) when $p=2$, and a distance introduced by Gin\'e and Leon (1980) when $p=\infty$. Our analysis shows that, unless $p=\infty$, these distances fail to metrize convergence in distribution in infinite dimensions. This clarifies several inconsistencies and misconceptions in the recent literature that arose from confusion between different types of distances.

math.PR↗

arXiv2023

Limit theorems for general functionals of Brownian local times

In this paper, we present the asymptotic theory for integrated functions of increments of Brownian local times in space. Specifically, we determine their first-order limit, along with the asymptotic distribution of the fluctuations. Our key result establishes that a standardized version of our statistic converges stably in law towards a mixed normal distribution. Our contribution builds upon a series of prior works by S. Campese, X. Chen, Y. Hu, W.V. Li, M.B. Markus, D. Nualart and J. Rosen \cite{C17,CLMR10,HN09,HN10,MR08,R11,R11b}, which delved into special cases of the considered problem, such as quadratic, cubic and polynomial cases. We establish the limit theorem for general functions that satisfy mild smoothness and growth conditions. This extends the scope beyond the polynomial cases studied in previous works, providing a more comprehensive understanding of the asymptotic properties of the considered functionals.

math.PR↗

arXiv2023

Functional Gaussian approximations on Hilbert-Poisson spaces

We develop a functional Stein-Malliavin method in a non-diffusive Poissonian setting, thus obtaining a) quantitative central limit theorems for approximation of arbitrary non-degenerate Gaussian random elements taking values in a separable Hilbert space and b) fourth moment bounds for approximating sequences with finite chaos expansion. Our results rely on an infinite-dimensional version of Stein's method of exchangeable pairs combined with the so-called Gamma calculus. Two applications are included: Brownian approximation of Poisson processes in Besov-Liouville spaces and a functional limit theorem for an edge-counting statistic of a random geometric graph.

math.PR↗

arXiv2020

Approximation of Hilbert-valued Gaussians on Dirichlet structures

We introduce a framework to derive quantitative central limit theorems in the context of non-linear approximation of Gaussian random variables taking values in a separable Hilbert space. In particular, our method provides an alternative to the usual (non-quantitative) finite dimensional distribution convergence and tightness argument for proving functional convergence of stochastic processes. We also derive four moments bounds for Hilbert-valued random variables with possibly infinite chaos expansion, which include, as special cases, all finite-dimensional four moments results for Gaussian approximation in a diffusive context proved earlier by various authors. Our main ingredient is a combination of an infinite-dimensional version of Stein's method as developed by Shih and the so-called Gamma calculus. As an application, rates of convergence for the functional Breuer-Major theorem are established.

math.PR↗

arXiv2018

Continuous Breuer-Major theorem: tightness and non-stationarity

Let $Y=(Y(t))_{t\geq0}$ be a zero-mean Gaussian stationary process with covariance function $ρ:\mathbb{R}\to\mathbb{R}$ satisfying $ρ(0)=1$. Let $f:\mathbb{R}\to\mathbb{R}$ be a square-integrable function with respect to the standard Gaussian measure, and suppose the Hermite rank of $f$ is $d\geq 1$. If $\int_\mathbb{R} |ρ(s)|^dds<\infty$, then the celebrated Breuer-Major theorem (in its continuous version) asserts that the finite-dimensional distributions of $Z_\varepsilon:=\sqrt{\varepsilon}\int_0^{\cdot/\varepsilon}f(Y(s))ds$ converge to those of $σW$ as $\varepsilon\to 0$, where $W$ is a standard Brownian motion and $σ$ is some explicit constant. Since its first appearance in 1983, this theorem has become a crucial probabilistic tool in different areas, for instance in signal processing or in statistical inference for fractional Gaussian processes. The goal of this paper is twofold. Firstly, we investigate the tightness in the Breuer-Major theorem. Surprisingly, this problem did not receive a lot of attention until now, and the best available condition due to Ben Hariz [1] is neither arguably very natural, nor easy-to-check in practice. In contrast, our condition very simple, as it only requires that $|f|^p$ must be integrable with respect to the standard Gaussian measure for some $p$ strictly bigger than 2. It is obtained by means of the Malliavin calculus, in particular Meyer inequalities. Secondly, and motivated by a problem of geometrical nature, we extend the continuous Breuer-Major theorem to the notoriously difficult case of self-similar Gaussian processes which are not necessarily stationary. An application to the fluctuations associated with the length process of a regularized version of the bifractional Browninan motion concludes the paper.

math.PR↗

arXiv2018

Four moments theorems on Markov chaos

We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumption we make on the Pearson distribution is that it admits four moments. While in general one cannot use moments to establish convergence to a heavy-tailed distributions, we provide a context in which only the first four moments suffices. These results are obtained by proving a general carré du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions. For elements of a Markov chaos, this bound can be reduced to just the first four moments.

math.PR↗

arXiv2017

Free quantitative fourth moment theorems on Wigner space

We prove a quantitative Fourth Moment Theorem for Wigner integrals of any order with symmetric kernels, generalizing an earlier result from Kemp et al. (2012). The proof relies on free stochastic analysis and uses a new biproduct formula for bi-integrals. A consequence of our main result is a Nualart-Ortiz-Latorre type characterization of convergence in law to the semicircular distribution for Wigner integrals. As an application, we provide Berry-Esseen type bounds in the context of the free Breuer-Major theorem for the free fractional Brownian motion.

math.OA↗

arXiv2015

A limit theorem for moments in space of the increments of Brownian local time

We proof a limit theorem for moments in space of the increments of Brownian local time. As special cases for the second and third moments, previous results by Chen et al. (Ann. Prob. 38, 2010, no. 1) and Rosen (Stoch. Dyn. 11, 2011, no. 1), which were later reproven by Hu and Nualart (Electron. Commun. Probab. 14, 2009; Electron. Commun. Probab. 15, 2010) and Rosen (Séminaire de Probabilités XLIII, Springer, 2011) are included. Furthermore, a conjecture of Rosen for the fourth moment is settled. In comparison to the previous methods of proof, we follow a fundamentally different approach by exclusively working in the space variable of the Brownian local time, which allows to give a unified argument for arbitrary orders. The main ingredients are Perkins' semimartingale decomposition, the Kailath-Segall identity and an asymptotic Ray-Knight Theorem by Pitman and Yor.

math.PR↗

arXiv2015

Fourth Moment Theorems for complex Gaussian approximation

We prove a bound for the Wasserstein distance between vectors of smooth complex random variables and complex Gaussians in the framework of complex Markov diffusion generators. For the special case of chaotic eigenfunctions, this bound can be expressed in terms of certain fourth moments of the vector, yielding a quantitative Fourth Moment Theorem for complex Gaussian approximation on complex Markov diffusion chaos. This extends results of Azmoodeh, Campese, Poly (2014) and Campese, Nourdin, Peccati (2015) for the real case. Our main ingredients are a complex version of the so called $Γ$-calculus and Stein's method for the multivariate complex Gaussian distribution.

math.PR↗

arXiv2015

Multivariate Gaussian approximations on Markov chaoses

We prove a version of the multidimensional Fourth Moment Theorem for chaotic random vectors, in the general context of diffusion Markov generators. In addition to the usual componentwise convergence and unlike the infinite-dimensional Ornstein-Uhlenbeck generator case, another moment-type condition is required to imply joint convergence of of a given sequence of vectors.

math.PR↗

arXiv2015

Approximate Normality of High-Energy Hyperspherical Eigenfunctions

The Berry heuristic has been a long standing \emph{ansatz} about the high energy (i.e. large eigenvalues) behaviour of eigenfunctions (see Berry 1977). Roughly speaking, it states that under some generic boundary conditions, these eigenfunctions exhibit Gaussian behaviour when the eigenvalues grow to infinity. Our aim in this paper is to make this statement quantitative and to establish some rigorous bounds on the distance to Gaussianity, focussing on the hyperspherical case (i.e., for eigenfunctions of the Laplace-Beltrami operator on the normalized $d$-dimensional sphere - also known as spherical harmonics). Some applications to non-Gaussian models are also discussed.

math.PR↗

arXiv2013

Optimal Convergence Rates and One-Term Edgeworth Expansions for Multidimensional Functionals of Gaussian Fields

We develop techniques for determining the exact asymptotic speed of convergence in the multidimensional normal approximation of smooth functions of Gaussian fields. As a by-product, our findings yield exact limits and often give rise to one-term generalized Edgeworth expansions increasing the speed of convergence. Our main mathematical tools are Malliavin calculus, Stein's method and the Fourth Moment Theorem. This work can be seen as an extension of the results of arXiv:0803.0458 to the multi-dimensional case, with the notable difference that in our framework covariances are allowed to fluctuate. We apply our findings to exploding functionals of Brownian sheets, vectors of Toeplitz quadratic functionals and the Breuer-Major Theorem.

math.PR↗

arXiv2013

Fourth Moment Theorems for Markov Diffusion Generators

Inspired by the insightful article arXiv:1210.7587, we revisit the Nualart-Peccati-criterion arXiv:math/0503598 (now known as the Fourth Moment Theorem) from the point of view of spectral theory of general Markov diffusion generators. We are not only able to drastically simplify all of its previous proofs, but also to provide new settings of diffusive generators (Laguerre, Jacobi) where such a criterion holds. Convergence towards gamma and beta distributions under moment conditions is also discussed.

math.PR↗