Multivariate second order Poincaré inequalities for Poisson functionals

Abstract

Given a vector F = (F1, . . . , Fm) of Poisson functionals F1, . . . , Fm, we investigate the proximity between F and an m-dimensional centered Gaussian random vector NΣ with covariance matrix Σ ∈ R m×m . Apart from finding proximity bounds for the d2-and d3-distances, based on classes of smooth test functions, we obtain proximity bounds for the dconvex-distance, based on the less tractable test functions comprised of indicators of convex sets. The bounds for all three distances are shown to be of the same order, which is presumably optimal. The bounds are multivariate counterparts of the univariate second order Poincaré inequalities and, as such, are expressed in terms of integrated moments of first and second order difference operators. The derived second order Poincaré inequalities for indicators of convex sets are made possible by a new bound on the second derivatives of the solution to the Stein equation for the multivariate normal distribution. We present applications to the multivariate normal approximation of first order Poisson integrals and of statistics of Boolean models.

Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance

“…Let f t = f h t be the solution of the Stein's equation ( 10) associated with h = h t . In [5] (see also [29]), it is shown that max…”

“…As anticipated, to prove Theorem 1.2, we shall combine the somewhat classical smoothing estimate (14) with a remarkable bound by Schulte and Yukich [29].…”

“…Inequalities in the spirit of (9) were derived, for example, in [18] (in the context of limit theorems for homogeneous sums) and [27] (in the framework of multivariate normal approximations on the Poisson space)-see also [29] and the references therein.…”

“…1 The aim of the present note is to establish explicit bounds on the quantity d c (F, G), in the special case where F is a vector of smooth functionals of an infinite-dimensional Gaussian field, and G = N is a m-dimensional centered Gaussian vector with covariance > 0. Our main tool is the so-called Malliavin-Stein method for probabilistic approximations [17], that we will combine with some powerful recursive estimates on d c , recently derived in [29] in the context of multidimensional second-order Poincaré inequalities on the Poisson space-see Lemma 2.1.…”

Multivariate approximation in total variation using local dependence

“…15) which follows because the area of D α α α ∩ E is of order O(λ −1 ) and the intensity of Ξ is of order O(λ). Note that the bound (4.15) and all the upper bounds below are uniform in α α α and β β β.First, with (4.15) in mind, we obtainβ β β∈Dα α α∩E E{ς β β β | ς α α α = 1}ν(dβ β β) ≤ ν(D α α α ∩ E ) = O(1), β β β∈Dα α α∩E θ β β β ν(dβ β β) ≤ ν(D α α α ∩ E ) = O(1),which, together with (4.14), imply thatα α α∈E β β β∈Dα α α∩E {E(ς α α α ς β β β ) + θ α α α θ β β β }ν(dβ β β)ν(dα α α) = α α α∈E β β β∈Dα α α∩E {E [ς β β β | ς α α α = 1] + θ β β β } θ α α α ν(dβ β β)ν(dα α α) = O(1) α α α∈E θ α α α ν(α α α) = O(λ 1/2 ).…”