30 research outputs found
A Peccati-Tudor type theorem for Rademacher chaoses
No full textIn this article, we prove that in the Rademacher setting, a random vector with chaotic components is close in distribution to a centred Gaussian vector, if both the maximal influence of the associated kernel and the fourth cumulant of each component is small. In particular, we recover the univariate case recently established in D\"obler and Krokowski (2017). Our main strategy consists in a novel adaption of the exchangeable pairs couplings initiated in Nourdin and Zheng (2017), as well as its combination with estimates via chaos decomposition
A Peccati-Tudor type theorem for Rademacher chaoses
No full textIn this article, we prove that in the Rademacher setting, a random vector with chaotic components is close in distribution to a centered Gaussian vector, if both the maximal influence of the associated kernel and the fourth cumulant of each component is small. In particular, we recover the univariate case recently established in Döbler and Krokowski (2019). Our main strategy consists in a novel adaption of the exchangeable pairs couplings initiated in Nourdin and Zheng (2017), as well as its combination with estimates via chaos decomposition
Quantitative central limit theorems for the parabolic Anderson model driven by colored noises
No full textIn this paper, we study the spatial averages of the solution to the parabolic Anderson model driven by a space-time Gaussian homogeneous noise that is colored in both time and space. We establish quantitative central limit theorems (CLT) of this spatial statistics under some mild assumptions, by using the Malliavin-Stein approach. The highlight of this paper is the obtention of rate of convergence in the colored-in-time setting, where one can not use Ito calculus due to the lack of martingale structure. In particular, modulo highly technical computations, we apply a modified version of second-order Gaussian Poincaré inequality to overcome this lack of martingale structure and our work improves the results by Nualart-Zheng (Electron. J. Probab. 2020) and Nualart-Song-Zheng (ALEA, Lat. Am. J. Probab. Math. Stat. 2021).</p
Almost sure central limit theorem for the hyperbolic Anderson model with Lévy white noise
No full textIn this paper, we present an almost sure central limit theorem (ASCLT) for the hyperbolic Anderson model (HAM) with a Lévy white noise in a finite-variance setting, complementing a recent work by Balan and Zheng [Trans. Amer. Math. Soc. 377 (2024), pp. 4171–4221] on the (quantitative) central limit theorems for the solution to the HAM. We provide two different proofs: one uses the Clark-Ocone formula and takes advantage of the martingale structure of the white-in-time noise, while the other is obtained by combining the second-order Gaussian Poincaré inequality with Ibragimov and Lifshits’ method of characteristic functions. Both approaches are different from the one developed in the PhD thesis of C. Zheng [Multi-dimensional Malliavin-Stein method on the Poisson space and its applications to limit theorems (PhD dissertation), Université Pierre et Marie Curie, Paris VI, 2011], allowing us to establish the ASCLT without lengthy computations of star contractions. Moreover, the second approach is expected to be useful for similar studies on SPDEs with colored-in-time noises, when the former approach, based on Itô calculus, is not applicable
Almost sure central limit theorem for the hyperbolic Anderson model with L\'evy white noise
No full textIn this paper, we present an almost sure central limit theorem (ASCLT) for
the hyperbolic Anderson model (HAM) with a L\'evy white noise in a
finite-variance setting, complementing a recent work by Balan and Zheng
(\emph{Trans.~Amer.~Math.~Soc.}, 2024) on the (quantitative) central limit
theorems for the solution to the HAM. We provide two different proofs: one uses
the Clark-Ocone formula and takes advantage of the martingale structure of the
white-in-time noise, while the other is obtained by combining the second-order
Gaussian Poincar\'e inequality with Ibragimov and Lifshits' method of
characteristic functions. Both approaches are different from the one developed
in the PhD thesis of C. Zheng (2011), allowing us to establish the ASCLT
without lengthy computations of star contractions. Moreover, the second
approach is expected to be useful for similar studies on SPDEs with
colored-in-time noises, whereas the former, based on It\^o calculus, is not
applicable.Comment: V2:15pages, minor revision; V1: 15page
Normal approximation and almost sure central limit theorem for non-symmetric Rademacher functionals
No full textpeer reviewedIn this work, we study the normal approximation and almost sure central limit theorems for some functionals of an independent sequence of Rademacher random variables. In particular, we provide a new chain rule that improves the one derived by Nourdin et al. (2010) and then we deduce the bound on Wasserstein distance for normal approximation using the (discrete) Malliavin–Stein approach. Besides, we are able to give the almost sure central limit theorem for a sequence of random variables inside a fixed Rademacher chaos using the Ibragimov–Lifshits criterio
Recent developments around the Malliavin-Stein approach (Fourth moment phenomena via exchangeable pairs)
No full textPart I is a survey, part II is a collection of papers
Almost sure central limit theorem for the hyperbolic Anderson model with Lévy white noise
No full textFirst author draf
Almost sure central limit theorems via chaos expansions and related results
No full textFirst author draf
