Abstract
Using Malliavin operators together with an interpolation technique inspired by Arratia, Goldstein and Gordon (1989), we prove a new inequality on the Poisson space, allowing one to measure the distance between the laws of a general random vector, and of a target random element composed of Gaussian and Poisson random variables. Several consequences are deduced from this result, in particular: (1) new abstract criteria for multidimensional stable convergence on the Poisson space, (2) a class of mixed limit theorems, involving both Poisson and Gaussian limits, (3) criteria for the asymptotic independence of U-statistics following Gaussian and Poisson asymptotic regimes. Our results generalize and unify several previous findings in the field. We provide an application to joint sub-graph counting in random geometric graphs.
Highlights
Introduction and frameworkThe aim of this paper is to prove and apply a new probabilistic inequality, involving vectors of random variables that are functionals of a Poisson measure defined on a general abstract space
This estimate – which is formally stated in formula (2.9) below – is expressed in terms of Malliavin operators, and basically allows one to measure the distance between the laws of a general random element and of a random vector whose components are in part Gaussian and in part Poisson random variables
As we shall abundantly illustrate in the sequel, the inequality (2.9) is a genuine ‘portmanteau statement’ – in the sense that it can be used to directly deduce a number of disparate results about the convergence of random variables defined on a Poisson space, as well as to recover known ones
Summary
The aim of this paper is to prove and apply a new probabilistic inequality, involving vectors of random variables that are functionals of a Poisson measure defined on a general abstract space. – Mixed limits: Our results allow one to prove quantitative limit theorems (that is, limit theorems with explicit information on the rate of convergence), where the target distribution is a multidimensional combination of independent Gaussian and Poisson components – Multi-dimensional Poisson convergence: A particular choice of parameters in our main estimates allows one to deduce multidimensional Poisson approximation results, having a stable nature – in the classic sense of [2, 52]. This generalizes the one-dimensional findings of [35]. An Appendix contains basic notions about Malliavin operators and contractions
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