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New Berry-Esseen bounds for functionals of binomial point processes. (English) Zbl 1374.60023
Summary: We obtain explicit Berry-Esseen bounds in the Kolmogorov distance for the normal approximation of nonlinear 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 obtained by S. Chatterjee [Ann. Probab. 36, No. 4, 1584–1610 (2008; Zbl 1159.62009)], concerning normal approximations in the Wasserstein 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 coverage processes.

60F05 Central limit and other weak theorems
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60D05 Geometric probability and stochastic geometry
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