Abstract
Given a vector $F=(F_1,\dots,F_m)$ of Poisson functionals $F_1,\dots,F_m$, we investigate the proximity between $F$ and an $m$-dimensional centered Gaussian random vector $N_\Sigma$ with covariance matrix $\Sigma\in\mathbb{R}^{m\times m}$. Apart from finding proximity bounds for the $d_2$- and $d_3$-distances, based on classes of smooth test functions, we obtain proximity bounds for the $d_{convex}$-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 Poincar\'e inequalities and, as such, are expressed in terms of integrated moments of first and second order difference operators. The derived second order Poincar\'e 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.
Highlights
Introduction and main resultsRoughly speaking, a first order Poincaré inequality for a random variable F measures the closeness of F to its mean
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
A main contribution of this paper is to provide such bounds via a new estimate on the second derivatives of the solution to the Stein equation for the multivariate normal distribution, which could be helpful for the multivariate normal approximation of other types of random vectors as well and, might be of independent interest
Summary
A first order Poincaré inequality for a random variable F measures the closeness of F to its mean. For many Poisson functionals F the second order Poincaré inequalities (1.4) and (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]. It seems that none of these findings could be applied to systematically achieve the normal approximation bounds for Poisson functionals given by our main results
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