Lachieze-Rey, Raphael; Peccati, Giovanni Fine Gaussian fluctuations on the Poisson space. I: Contractions, cumulants and geometric random graphs. (English) Zbl 1295.60015 Electron. J. Probab. 18, Paper No. 32, 32 p. (2013). 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. Reviewer: Nicolas Privault (Singapore) Cited in 1 ReviewCited in 29 Documents 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 Keywords:central limit theorems; contractions; Malliavin calculus; Poisson limit theorems; Poisson space; random graphs; Stein’s method; \(U\)-statistics; Wasserstein distance; Wiener chaos × Cite Format Result Cite Review PDF Full Text: DOI arXiv