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Fine Gaussian fluctuations on the Poisson space. I: Contractions, cumulants and geometric random graphs. (English) Zbl 1295.60015

The authors derive normal approximation results based on the Stein method for finite sums of multiple stochastic integrals with respect to a Poisson random measure. These results are then applied to asymptotic edge-counting problems in random geometric graphs, by representing their edge counts as \(U\)-statistics written as linear combinations of first- and second-order Poisson stochastic integrals. As a result, asymptotic distribution estimates for renormalized edge counts are obtained in different convergence regimes as the intensity of the underlying Poisson random measure tends to infinity. This provides in particular a characterization of those geometric random graphs whose edge-counting statistics exhibit asymptotic Gaussianity.

MSC:

60D05 Geometric probability and stochastic geometry
60F05 Central limit and other weak theorems
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60H07 Stochastic calculus of variations and the Malliavin calculus