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Giovanni Peccati

Publications and source records attributed to Giovanni Peccati.

At least 55 records · Page 3Linked to original sources
arXiv2019

Approximation of Fractional Local Times: Zero Energy and Derivatives

We consider empirical processes associated with high-frequency observations of a fractional Brownian motion (fBm) $X$ with Hurst parameter $H\in (0,1)$, and derive conditions under which these processes verify a (possibly uniform) law of large numbers, as well as a second order (possibly uniform) limit theorem. We devote specific emphasis to the `zero energy' case, corresponding to a kernel whose integral on the real line equals zero. Our asymptotic results are associated with explicit rates of convergence, and are expressed either in terms of the local time of $X$ or of its \blue{derivatives}: in particular, the full force of our finding applies to the `rough range' $0< H < 1/3$, on which the previous literature has been mostly silent. The {\color{black}use of the derivatives} of local times for studying the fluctuations of high-frequency observations of a fBm is new, and is the main technological breakthrough of the present paper. Our results are based on the use of Malliavin calculus and Fourier analysis, and extend and complete several findings in the literature, e.g. by Jeganathan (2004, 2006, 2008) and Podolskij and Rosenbaum (2018).

math.PR↗
arXiv2019

Concentration of the Intrinsic Volumes of a Convex Body

The intrinsic volumes are measures of the content of a convex body. This paper uses probabilistic and information-theoretic methods to study the sequence of intrinsic volumes of a convex body. The main result states that the intrinsic volume sequence concentrates sharply around a specific index, called the central intrinsic volume. Furthermore, among all convex bodies whose central intrinsic volume is fixed, an appropriately scaled cube has the intrinsic volume sequence with maximum entropy.

math.MG↗
arXiv2018

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 carré-du-champ operators. Our results substantially complete the fourth moment theorems recently obtained by Döbler and Peccati (2018) and Döbler, Vidotto and Zheng (2018). An important tool for achieving our goals is a novel product formula for multiple integrals under minimal conditions.

math.PR↗
arXiv2017

The Gamma Stein equation and non-central de Jong theorems

We study the Stein equation associated with the one-dimensional Gamma distribution, and provide novel bounds, allowing one to effectively deal with test functions supported by the whole real line. We apply our estimates to derive new quantitative results involving random variables that are non-linear functionals of random fields, namely: (i) a non-central quantitative de Jong theorem for sequences of degenerate U-statistics satisfying minimal uniform integrability conditions, significantly extending previous findings by de Jong (1990), Nourdin, Peccati and Reinert (2010) and Döbler and Peccati (2016), (ii) a new Gamma approximation bound on the Poisson space, refining previous estimates by Peccati and Thäle (2013), and (iii) new Gamma bounds on a Gaussian space, strengthening estimates by Nourdin and Peccati (2009). As a by-product of our analysis, we also deduce a new inequality for Gamma approximations via exchangeable pairs, that is of independent interest.

math.PR↗
arXiv2017

Quantitative limit theorems for local functionals of arithmetic random waves

We consider Gaussian Laplace eigenfunctions on the two-dimensional flat torus (arithmetic random waves), and provide explicit Berry-Esseen bounds in the 1-Wasserstein distance for the normal and non-normal high-energy approximation of the associated Leray measures and total nodal lengths, respectively. Our results provide substantial extensions (as well as alternative proofs) of findings by Oravecz, Rudnick and Wigman (2007), Krishnapur, Kurlberg and Wigman (2013), and Marinucci, Peccati, Rossi and Wigman (2016). Our techniques involve Wiener-Ito chaos expansions, integration by parts, as well as some novel estimates on residual terms arising in the chaotic decomposition of geometric quantities that can implicitly be expressed in terms of the coarea formula.

math.PR↗
arXiv2016

Quantitative de Jong theorems in any dimension

We develop a new quantitative approach to a multidimensional version of the well-known {\it de Jong's central limit theorem} under optimal conditions, stating that a sequence of Hoeffding degenerate $U$-statistics whose fourth cumulants converge to zero satisfies a CLT, as soon as a Lindeberg-Feller type condition is verified. Our approach allows one to deduce explicit (and presumably optimal) Berry-Esseen bounds in the case of general $U$-statistics of arbitrary order $d\geq1$. One of our main findings is that, for vectors of $U$-statistics satisfying de Jong' s conditions and whose covariances admit a limit, componentwise convergence systematically implies joint convergence to Gaussian: this is the first instance in which such a phenomenon is described outside the frameworks of homogeneous chaoses and of diffusive Markov semigroups.

math.PR↗
arXiv2016

Non-Universality of Nodal Length Distribution for Arithmetic Random Waves

"Arithmetic random waves" are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013)). In this paper we find that their nodal length converges to a non-universal (non-Gaussian) limiting distribution, depending on the angular distribution of lattice points lying on circles. Our argument has two main ingredients. An explicit derivation of the Wiener-Itô chaos expansion for the nodal length shows that it is dominated by its $4$th order chaos component (in particular, somewhat surprisingly, the second order chaos component vanishes). The rest of the argument relies on the precise analysis of the fourth order chaotic component.

math-ph↗
arXiv2016

A Stein deficit for the logarithmic Sobolev inequality

We provide explicit lower bounds for the deficit in the Gaussian logarithmic Sobolev inequality in terms of differential operators that are naturally associated with the so-called Stein characterization of the Gaussian distribution. The techniques are based on a crucial use of the representation of the relative Fisher information, along the Ornstein-Uhlenbeck semigroup, in terms of the Minimal Mean-Square Error from information theory.

math.PR↗
arXiv2016

Quantitative stable limit theorems on the Wiener space

We use Malliavin operators in order to prove quantitative stable limit theorems on the Wiener space, where the target distribution is given by a possibly multidimensional mixture of Gaussian distributions. Our findings refine and generalize previous works by Nourdin and Nualart [J. Theoret. Probab. 23 (2010) 39-64] and Harnett and Nualart [Stochastic Process. Appl. 122 (2012) 3460-3505], and provide a substantial contribution to a recent line of research, focussing on limit theorems on the Wiener space, obtained by means of the Malliavin calculus of variations. Applications are given to quadratic functionals and weighted quadratic variations of a fractional Brownian motion.

math.PR↗
arXiv2016

On the rate of convergence in de Finetti's representation theorem

A consequence of de Finetti's representation theorem is that for every infinite sequence of exchangeable 0-1 random variables $(X_k)_{k\geq1}$, there exists a probability measure $μ$ on the Borel sets of $[0,1]$ such that $\bar X_n = n^{-1} \sum_{i=1}^n X_i$ converges weakly to $μ$. For a wide class of probability measures $μ$ having smooth density on $(0,1)$, we give bounds of order $1/n$ with explicit constants for the Wasserstein distance between the law of $\bar X_n$ and $μ$. This extends a recent result {by} Goldstein and Reinert \cite{goldstein2013stein} regarding the distance between the scaled number of white balls drawn in a Pólya-Eggenberger urn and its limiting distribution. We prove also that, in the most general cases, the distance between the law of $\bar X_n$ and $μ$ is bounded below by $1/n$ and above by $1/\sqrt{n}$ (up to some multiplicative constants). For every $δ\in [1/2,1]$, we give an example of an exchangeable sequence such that this distance is of order $1/n^δ$.

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

Multidimensional limit theorems for homogeneous sums: a general transfer principle

The aim of the present paper is to establish the multidimensional counterpart of the \textit{fourth moment criterion} for homogeneous sums in independent leptokurtic and mesokurtic random variables (that is, having positive and zero fourth cumulant, respectively), recently established in \cite{NPPS} in both the classical and in the free setting. As a consequence, the transfer principle for the Central limit Theorem between Wiener and Wigner chaos can be extended to a multidimensional transfer principle between vectors of homogeneous sums in independent commutative random variables with zero third moment and with non-negative fourth cumulant, and homogeneous sums in freely independent non-commutative random variables with non-negative fourth cumulant.

math.PR↗
arXiv2015

New Kolmogorov bounds for functionals of binomial point processes

We obtain explicit Berry-Esseen bounds in the Kolmogorov distance for the normal approximation of non-linear functionals of vectors of independent random variables. Our results are based on the use of Stein's method and of random difference operators, and generalise the bounds recently obtained by Chatterjee (2008), concerning normal approximations in the Wasserstein distance. In order to obtain lower bounds for variances, we also revisit the classical Hoeffding decompositions, for which we provide a new proof and a new representation. Several applications are discussed in detail: in particular, new Berry-Esseen bounds are obtained for set approximations with random tessellations, as well as for functionals of covering processes.

math.PR↗
arXiv2015

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 Biermé, 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.

math.PR↗
arXiv2015

Concentration Bounds for Geometric Poisson Functionals: Logarithmic Sobolev Inequalities Revisited

We prove new concentration estimates for random variables that are functionals of a Poisson measure defined on a general measure space. Our results are specifically adapted to geometric applications, and are based on a pervasive use of a powerful logarithmic Sobolev inequality proved by L. Wu (2000), as well as on several variations of the so-called Herbst argument. We provide several applications, in particular to edge counting and more general length power functionals in random geometric graphs, as well as to the convex distance for random point measures recently introduced by M. Reitzner (2013).

math.PR↗
arXiv2015

Squared chaotic random variables: new moment inequalities with applications

We prove a new family of inequalities involving squares of random variables belonging to the Wiener chaos associated with a given Gaussian field. Our result provides a substantial generalisation, as well as a new analytical proof, of an estimate by Frenkel (2007), and also constitute a natural real counterpart to an inequality established by Arias-de-Reyna (1998) in the framework of complex Gaussian vectors. We further show that our estimates can be used to deduce new lower bounds on homogeneous polynomials, thus partially improving results by Pinasco (2012), as well as to obtain a novel probabilistic representation of the remainder in Hadamard inequality of matrix analysis.

math.PR↗
arXiv2014

Gaussian Phase Transitions and Conic Intrinsic Volumes: Steining the Steiner Formula

Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, such as linear inverse problems with convex constraints, and constrained statistical inference. It is a well-known fact that, given a closed convex cone $C\subset \mathbb{R}^d$, its conic intrinsic volumes determine a probability measure on the finite set $\{0,1,...d\}$, customarily denoted by $\mathcal{L}(V_C)$. The aim of the present paper is to provide a Berry-Esseen bound for the normal approximation of ${\cal L}(V_C)$, implying a general quantitative central limit theorem (CLT) for sequences of (correctly normalised) discrete probability measures of the type $\mathcal{L}(V_{C_n})$, $n\geq 1$. This bound shows that, in the high-dimensional limit, most conic intrinsic volumes encountered in applications can be approximated by a suitable Gaussian distribution. Our approach is based on a variety of techniques, namely: (1) Steiner formulae for closed convex cones, (2) Stein's method and second order Poincaré inequality, (3) concentration estimates, and (4) Fourier analysis. Our results explicitly connect the sharp phase transitions, observed in many regularised linear inverse problems with convex constraints, with the asymptotic Gaussian fluctuations of the intrinsic volumes of the associated descent cones. In particular, our findings complete and further illuminate the recent breakthrough discoveries by Amelunxen, Lotz, McCoy and Tropp (2014) and McCoy and Tropp (2014) about the concentration of conic intrinsic volumes and its connection with threshold phenomena. As an additional outgrowth of our work we develop total variation bounds for normal approximations of the lengths of projections of Gaussian vectors on closed convex sets.

math.PR↗