Open Access
November 2013 Central limit theorems for $U$-statistics of Poisson point processes
Matthias Reitzner, Matthias Schulte
Ann. Probab. 41(6): 3879-3909 (November 2013). DOI: 10.1214/12-AOP817


A $U$-statistic of a Poisson point process is defined as the sum $\sum f(x_{1},\ldots,x_{k})$ over all (possibly infinitely many) $k$-tuples of distinct points of the point process. Using the Malliavin calculus, the Wiener–Itô chaos expansion of such a functional is computed and used to derive a formula for the variance. Central limit theorems for $U$-statistics of Poisson point processes are shown, with explicit bounds for the Wasserstein distance to a Gaussian random variable. As applications, the intersection process of Poisson hyperplanes and the length of a random geometric graph are investigated.


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Matthias Reitzner. Matthias Schulte. "Central limit theorems for $U$-statistics of Poisson point processes." Ann. Probab. 41 (6) 3879 - 3909, November 2013.


Published: November 2013
First available in Project Euclid: 20 November 2013

zbMATH: 1293.60061
MathSciNet: MR3161465
Digital Object Identifier: 10.1214/12-AOP817

Primary: 60F05 , 60H07
Secondary: 60D05 , 60G55

Keywords: $U$-statistic , central limit theorem , Malliavin calculus , Poisson point process , Stein’s method , Wiener–Itô chaos expansion

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 6 • November 2013
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