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Central limit theorems for $$U$$-statistics of Poisson point processes. (English) Zbl 1293.60061
A $$U$$-statistic for a Poisson process is defined as the sum $$\sum f(x_1, \dots, x_k)$$ over all $$k$$-tuples of distinct points of the point process. Wiener-Itō chaos expansions are used to obtain an expression for the variance of the $$U$$-statistics. Using Malliavin calculus and Theorem 3.1 from G. Peccati et al. [Ann. Probab. 38, No. 2, 443–478 (2010; Zbl 1195.60037)], a central limit theorem is obtained for $$U$$-statistics of Poisson point processes with explicit bounds for the Wasserstein distance between the normalized $$U$$-statistic and a standard Gaussian random variable. This result is applied to establish a central limit theorem for geometric $$U$$-statistics complementing the work by R. Lachièze-Rey and G. Peccati [Stochastic Processes Appl. 123, No. 12, 4186–4218 (2013; Zbl 1294.60082)]. A central limit theorem with explicit bounds is also developed for the intersection process of Poisson hyperplanes, local $$U$$-statistics and the total edge length of a random geometric graph.

##### MSC:
 60H07 Stochastic calculus of variations and the Malliavin calculus 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60F05 Central limit and other weak theorems 60D05 Geometric probability and stochastic geometry 05C80 Random graphs (graph-theoretic aspects)
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