Multiple integral representation for functionals of Dirichlet processes

We point out that a proper use of the Hoeffding--ANOVA decomposition for symmetric statistics of finite urn sequences, previously introduced by the author, yields a decomposition of the space of square-integrable functionals of a Dirichlet--Ferguson process, written $L^2(D)$, into orthogonal subspaces of multiple integrals of increasing order. This gives an isomorphism between $L^2(D)$ and an appropriate Fock space over a class of deterministic functions. By means of a well-known result due to Blackwell and MacQueen, we show that each element of the $n$th orthogonal space of multiple integrals can be represented as the $L^2$ limit of $U$-statistics with degenerate kernel of degree $n$. General formulae for the decomposition of a given functional are provided in terms of linear combinations of conditioned expectations whose coefficients are explicitly computed. We show that, in simple cases, multiple integrals have a natural representation in terms of Jacobi polynomials. Several connections are established, in particular with Bayesian decision problems, and with some classic formulae concerning the transition densities of multiallele diffusion models, due to Littler and Fackerell, and Griffiths. Our results may also be used to calculate the best approximation of elements of $L^2(D)$ by means of $U$-statistics of finite vectors of exchangeable observations.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ5169 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Gaussian Approximations of Multiple Integrals

Fix an integer k, and let I(l), l=1,2,..., be a sequence of k-dimensional vectors of multiple Wiener-It\^o integrals with respect to a general Gaussian process. We establish necessary and sufficient conditions to have that, as l diverges, the law of I(l) is asymptotically close (for example, in the sense of Prokhorov's distance) to the law of a k-dimensional Gaussian vector having the same covariance matrix as I(l). The main feature of our results is that they require minimal assumptions (basically, boundedness of variances) on the asymptotic behaviour of the variances and covariances of the elements of I(l). In particular, we will not assume that the covariance matrix of I(l) is convergent. This generalizes the results proved in Nualart and Peccati (2005), Peccati and Tudor (2005) and Nualart and Ortiz-Latorre (2007). As shown in Marinucci and Peccati (2007b), the criteria established in this paper are crucial in the study of the high-frequency behaviour of stationary fields defined on homogeneous spaces.Comment: 15 page

Stein's method meets Malliavin calculus: a short survey with new estimates

We provide an overview of some recent techniques involving the Malliavin calculus of variations and the so-called ``Stein's method'' for the Gaussian approximations of probability distributions. Special attention is devoted to establishing explicit connections with the classic method of moments: in particular, we use interpolation techniques in order to deduce some new estimates for the moments of random variables belonging to a fixed Wiener chaos. As an illustration, a class of central limit theorems associated with the quadratic variation of a fractional Brownian motion is studied in detail.Comment: 31 pages. To appear in the book "Recent advances in stochastic dynamics and stochastic analysis", published by World Scientifi

Universal Gaussian fluctuations of non-Hermitian matrix ensembles: from weak convergence to almost sure CLTs

In the paper [25], written in collaboration with Gesine Reinert, we proved a universality principle for the Gaussian Wiener chaos. In the present work, we aim at providing an original example of application of this principle in the framework of random matrix theory. More specifically, by combining the result in [25] with some combinatorial estimates, we are able to prove multi-dimensional central limit theorems for the spectral moments (of arbitrary degrees) associated with random matrices with real-valued i.i.d. entries, satisfying some appropriate moment conditions. Our approach has the advantage of yielding, without extra effort, bounds over classes of smooth (i.e., thrice differentiable) functions, and it allows to deal directly with discrete distributions. As a further application of our estimates, we provide a new "almost sure central limit theorem", involving logarithmic means of functions of vectors of traces.Comment: 40 pages. This is an expanded version of a paper formerly called "Universal Gaussian fluctuations of non-Hermitian matrix ensembles", by the same authors. Sections 1.4 and 5 (about almost sure central limit theorems) are ne

Optimal Berry-Esseen bounds on the Poisson space

We establish new lower bounds for the normal approximation in the Wasserstein distance of random variables that are functionals of a Poisson measure. Our results generalize previous findings by Nourdin and Peccati (2012, 2015) and Bierm\'e, Bonami, Nourdin and Peccati (2013), involving random variables living on a Gaussian space. Applications are given to optimal Berry-Esseen bounds for edge counting in random geometric graphs

Fourth moment theorems on the Poisson space: analytic statements via product formulae

We prove necessary and sufficient conditions for the asymptotic normality of multiple integrals with respect to a Poisson measure on a general measure space, expressed both in terms of norms of contraction kernels and of variances of carr\'e-du-champ operators. Our results substantially complete the fourth moment theorems recently obtained by D\"obler and Peccati (2018) and D\"obler, Vidotto and Zheng (2018). An important tool for achieving our goals is a novel product formula for multiple integrals under minimal conditions.Comment: 14 page

Noncentral convergence of multiple integrals

Fix $\nu>0$, denote by $G(\nu/2)$ a Gamma random variable with parameter $\nu/2$ and let $n\geq2$ be a fixed even integer. Consider a sequence $\{F_k\}_{k\geq1}$ of square integrable random variables belonging to the $n$th Wiener chaos of a given Gaussian process and with variance converging to $2\nu$. As $k\to\infty$, we prove that $F_k$ converges in distribution to $2G(\nu/2)-\nu$ if and only if $E(F_k^4)-12E(F_k^3)\to12\nu^2-48\nu$.Comment: Published in at http://dx.doi.org/10.1214/08-AOP435 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org