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Solesne Bourguin

Publications and source records attributed to Solesne Bourguin.

33 records · Page 2Linked to original sources
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↗
arXiv2016

On high-frequency limits of $U$-statistics in Besov spaces over compact manifolds

In this paper, quantitative bounds in high-frequency central limit theorems are derived for Poisson based $U$-statistics of arbitrary degree built by means of wavelet coefficients over compact Riemannian manifolds. The wavelets considered here are the so-called needlets, characterized by strong concentration properties and by an exact reconstruction formula. Furthermore, we consider Poisson point processes over the manifold such that the density function associated to its control measure lives in a Besov space. The main findings of this paper include new rates of convergence that depend strongly on the degree of regularity of the control measure of the underlying Poisson point process, providing a refined understanding of the connection between regularity and speed of convergence in this framework.

math.ST↗
arXiv2016

On normal approximations for the two-sample problem on multidimensional tori

In this paper, quantitative central limit theorems for $U$-statistics on the $q$-dimensional torus defined in the framework of the two-sample problem for Poisson processes are derived. In particular, the $U$-statistics are built over tight frames defined by wavelets, named toroidal needlets, enjoying excellent localization properties in both harmonic and frequency domains. The Berry-Esséen type bounds associated with the normal approximations for these statistics are obtained by means of the so-called Stein-Malliavin techniques on the Poisson space, as introduced by Peccati, Solé, Taqqu, Utzet (2011) and further developed by Peccati, Zheng (2010) and Bourguin, Peccati (2014). Particular cases of the proposed framework allow to consider the two-sample problem on the circle as well as the local two-sample problem on $\mathbb{R}^q$ through a local homeomorphism argument.

math.PR↗
arXiv2015

Vector-valued semicircular limits on the free Poisson chaos

In this note, we prove a multidimensional counterpart of the central limit theorem on the free Poisson chaos recently proved by Bourguin and Peccati (2014). A noteworthy property of convergence toward the semicircular distribution on the free Poisson chaos is obtained as part of the limit theorem: component-wise convergence of sequences of vectors of multiple integrals with respect to a free Poisson random measure toward the semicircular distribution implies joint convergence. This result complements similar findings for the Wiener chaos by Peccati and Tudor (2005), the classical Poisson chaos by Peccati and Zheng (2010) and the Wigner chaos by Nourdin, Peccati and Speicher (2013).

math.OA↗
arXiv2015

Poisson convergence on the free Poisson algebra

Based on recent findings by Bourguin and Peccati, we give a fourth moment type condition for an element of a free Poisson chaos of arbitrary order to converge to a free (centered) Poisson distribution. We also show that free Poisson chaos of order strictly greater than one do not contain any non-zero free Poisson random variables. We are also able to give a sufficient and necessary condition for an element of the first free Poisson chaos to have a free Poisson distribution. Finally, depending on the parity of the considered free Poisson chaos, we provide a general counterexample to the naive universality of the semicircular Wigner chaos established by Deya and Nourdin as well as a transfer principle between the Wigner and the free Poisson chaos.

math.OA↗
arXiv2014

Gaussian approximations of nonlinear statistics on the sphere

We show how it is possible to assess the rate of convergence in the Gaussian approximation of triangular arrays of $U$-statistics, built from wavelets coefficients evaluated on a homogeneous spherical Poisson field of arbitrary dimension. For this purpose, we exploit the Stein-Malliavin approach introduced in the seminal paper by Peccati, Solé, Taqqu and Utzet (2011); we focus in particular on statistical applications covering evaluation of variance in non-parametric density estimation and Sobolev tests for uniformity.

math.PR↗
arXiv2013

Semicircular limits on the free Poisson chaos: counterexamples to a transfer principle

We establish a class of sufficient conditions, ensuring that a sequence of multiple integrals with respect to a free Poisson measure converges to a semicircular limit. We use this result to construct a set of explicit counterexamples, showing that the transfer principle between classical and free Brownian motions (recently proved by Kemp, Nourdin, Peccati and Speicher (2012)) does not extend to the framework of Poisson measures. Our counterexamples implicitly use kernels appearing in the classical theory of random geometric graphs. Several new results of independent interest are obtained as necessary steps in our analysis, in particular: (i) a multiplication formula for free Poisson multiple integrals, (ii) diagram formulae and spectral bounds for these objects, and (iii) a counterexample to the general universality of the Gaussian Wiener chaos in a classical setting.

math.OA↗
arXiv2013

Portmanteau inequalities on the Poisson space: mixed regimes and multidimensional clustering

Using Malliavin operators together with an interpolation technique inspired by Arratia, Goldstein and Gordon (1989), we prove a new inequality on the Poisson space, allowing one to measure the distance between the laws of a general random vector, and of a target random element composed of Gaussian and Poisson random variables. Several consequences are deduced from this result, in particular: (1) new abstract criteria for multidimensional stable convergence on the Poisson space, (2) a class of mixed limit theorems, involving both Poisson and Gaussian limits, (3) criteria for the asymptotic independence of $U$-statistics obeying to Gaussian and Poisson asymptotic regimes. Our results generalize and unify several previous findings in the field. We provide an application to joint sub-graph counting in random geometric graphs.

math.PR↗
arXiv2012

Asymptotic Cramér type decomposition for Wiener and Wigner integrals

We investigate generalizations of the Cramér theorem. This theorem asserts that a Gaussian random variable can be decomposed into the sum of independent random variables if and only if they are Gaussian. We prove asymptotic counterparts of such decomposition results for multiple Wiener integrals and prove that similar results are true for the (asymptotic) decomposition of the semicircular distribution into free multiple Wigner integrals.

math.OA↗
arXiv2011

Malliavin Calculus and Self Normalized Sums

We study the self-normalized sums of independent random variables from the perspective of the Malliavin calculus. We give the chaotic expansion for them and we prove a Berry-Esséen bound with respect to several distances.

math.PR↗
arXiv2011

Cramér theorem for Gamma random variables

In this paper we discuss the following problem: given a random variable $Z=X+Y$ with Gamma law such that $X$ and $Y$ are independent, we want to understand if then $X$ and $Y$ {\it each} follow a Gamma law. This is related to Cramér's theorem which states that if $X$ and $Y$ are independent then $Z=X+Y$ follows a Gaussian law if and only if $X$ {\it and} $Y$ follow a Gaussian law. We prove that Cramér's theorem is true in the Gamma context for random variables leaving in a Wiener chaos of fixed order but the result is not true in general. We also give an asymptotic variant of our result.

math.PR↗
arXiv2010

Asymptotic theory for fractional regression models via Malliavin calculus

We study the asymptotic behavior as $n\to \infty$ of the sequence $$S_{n}=\sum_{i=0}^{n-1} K(n^α B^{H_{1}}_{i}) (B^{H_{2}}_{i+1}-B^{H_{2}}_{i})$$ where $B^{H_{1}}$ and $B^{H_{2}}$ are two independent fractional Brownian motions, $K$ is a kernel function and the bandwidth parameter $α$ satisfies certain hypotheses in terms of $H_{1}$ and $H_{2}$. Its limiting distribution is a mixed normal law involving the local time of the fractional Brownian motion $B^{H_{1}}$. We use the techniques of the Malliavin calculus with respect to the fractional Brownian motion.

math.PR↗