Functional Poisson approximation in Kantorovich–Rubinstein distance with applications to U-statistics and stochastic geometry

Abstract

A Poisson or a binomial process on an abstract state space and a symmetric function f acting on k-tuples of its points are considered. They induce a point process on the target space of f . The main result is a functional limit theorem which provides an upper bound for an optimal transportation distance between the image process and a Poisson process on the target space. The technical background are a version of Stein's method for Poisson process approximation, a Glauber dynamics representation for the Poisson process and the Malliavin formalism. As applications of the main result, error bounds for approximations of U-statistics by Poisson, compound Poisson and stable random variables are derived, and examples from stochastic geometry are investigated.

Inhomogeneous random graphs, isolated vertices, and Poisson approximation

“…Our bound has an extra factor 1 ∧ α −1/2 in front, which may make it better when α is large. Also, unlike [5] we do not make any topological assumptions on the measurable space X. As remarked just after the statement of Theorem 3.1, it is possible to extend that result to the case where the measure λ is σ-finite, and hence to extend the above argument likewise, but we do not go into details here.…”

Malliavin and Dirichlet structures for independent random variables

“…This works well only if there exists a Malliavin gradient on the space on which X is defined (see for instance [15]). That is to say, that up to now, this approach is restricted to functionals of Rademacher [27], Poisson [15,34] or Gaussian random variables [32] or processes [11,12]. Then, strangely enough, the first example of applications of the Stein's method which was the CLT, cannot be handled through this approach.…”

Gaussian polytopes: A cumulant-based approach

“…which in view of Proposition 2.1 and Remark 3.2 (iv) in [15] (see also Lemma 3.2 in [10]) implies that P (λ) converges in distribution to a Poisson point process P on R d−1 × R whose intensity measure has density as in (5)…”