NEW KOLMOGOROV BOUNDS FOR FUNCTIONALS OF BINOMIAL POINT PROCESSES

Raphaël Lachièze-Rey (1) , Giovanni Peccati (2)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
2 FSTC - Faculté des Sciences, de la Technologie et de la Communication

We obtain explicit Berry-Esseen bounds in the Kolmogorov distance for the normal approximation of non-linear functionals of vectors of independent random variables. Our results are based on the use of Stein's method and of random difference operators, and generalise the bounds recently obtained by Chatterjee (2008), concerning normal approximations in the Wasser-stein distance. In order to obtain lower bounds for variances, we also revisit the classical Hoeffding decompositions, for which we provide a new proof and a new representation. Several applications are discussed in detail: in particular, new Berry-Esseen bounds are obtained for set approximations with random tessellations, as well as for functionals of covering processes.

Mots clés: Berry-Esseen Bounds; Binomial Processes; Covering Processes; Random Tessella-tions; Stochastic Geometry; Stein's method

Domaines: Probabilités [math.PR]

Raphaël Lachièze-Rey, Giovanni Peccati. NEW KOLMOGOROV BOUNDS FOR FUNCTIONALS OF BINOMIAL POINT PROCESSES. 2015. ⟨hal-01155629v1⟩