Normal Approximation of Poisson Functionals in Kolmogorov Distance

Abstract

Peccati, Solè, Taqqu, and Utzet recently combined Stein's method and Malliavin calculus to obtain a bound for the Wasserstein distance of a Poisson functional and a Gaussian random variable. Convergence in the Wasserstein distance always implies convergence in the Kolmogorov distance at a possibly weaker rate. But there are many examples of central limit theorems having the same rate for both distances. The aim of this paper is to show this behaviour for a large class of Poisson functionals, namely so-called U-statistics of Poisson point processes. The technique used by Peccati et al. is modified to establish a similar bound for the Kolmogorov distance of a Poisson functional and a Gaussian random variable. This bound is evaluated for a U-statistic, and it is shown that the resulting expression is up to a constant the same as it is for the Wasserstein distance.

Multivariate second order Poincaré inequalities for Poisson functionals

“…In [24], the univariate main result of [23] for the d W -distance is extended to vectors of Poisson functionals and the d 2 -and the d 3 -distances are considered. Evaluating these multivariate Malliavin-Stein bounds in the same way one evaluates in [16] the univariate bounds from [23] and [7,30] to derive (1.4) and (1.5), one obtains the following multivariate second order Poincaré inequalities.…”

“…When F ∈ dom D, E F = 0, and VarF = 1, the main results of [16] [16] for exact formulas). The proximity bounds (1.4) and (1.5), whose proofs rely on previous Malliavin-Stein bounds in [23] and [7,30], respectively, are second order Poincaré inequalities, as described in [16]. The reason for this name is that the 'first order' Poincaré inequality…”

“…(1.5) may be evaluated since the first two difference operators have a clear interpretation via the operation of adding additional points. This is the advantage of these findings over Malliavin-Stein bounds for normal approximation of Poisson functionals which either require the knowledge of the chaos expansion of F (see, for example, [7,12,23,30]) or which involve bounds expressed in terms of gradient operators and conditional expectations as in [25]. Inequality (1.5) yields rates of normal approximation for some classic problems in stochastic geometry and some non-linear functionals of Poisson-shot-noise processes [16], as well as for functionals of convex hulls of random samples in a smooth convex body, statistics of nearest neighbors graphs, the number of maximal points in a random sample, and estimators of surface area and volume arising in set approximation [15].…”

“…A second order Poincaré inequality [5] measures the closeness of F to a Gaussian random variable, where distance is given by some specified metric on the space of distribution functions. The paper [16] establishes second order Poincaré inequalities for Poisson functionals F , with bounds given in terms of integrated moments of first and second order difference operators, which are an outcome of the research on the Malliavin-Stein method for Poisson functionals in the recent years; see, for example, [7,23,30] and the book [22]. The bounds from [16] can be usefully applied to yield rates of normal convergence for various functionals of Poisson processes, including those represented as a sum of stabilizing score functions [15].…”

Fourth moment theorems on the Poisson space in any dimension

“…This yields a Peccati-Tudor type theorem, as well as an optimal improvement in the univariate case.Finally, a transfer principle "from-Poisson-to-Gaussian" is derived, which is closely related to the universality phenomenon for homogeneous multilinear sums.where d TV denotes the total variation distance between the laws of two real random variables. The techniques developed in [26] have also been adapted to non-Gaussian spaces which admit a Malliavin calculus structure: for instance, the papers [16,34,36,41] deal with the Poisson space case, whereas [17,18,30,44] develop the corresponding techniques for sequences of independent Rademacher random variables. The question FOURTH MOMENT THEOREMS 3 about general fourth moment theorems on these spaces, however, has remained open in general, until the two recent articles [13] and [11].…”