The Annals of Probability

Central limit theorems for $U$-statistics of Poisson point processes

Matthias Reitzner and Matthias Schulte

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Abstract

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.

Article information

Source
Ann. Probab., Volume 41, Number 6 (2013), 3879-3909.

Dates
First available in Project Euclid: 20 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1384957778

Digital Object Identifier
doi:10.1214/12-AOP817

Mathematical Reviews number (MathSciNet)
MR3161465

Zentralblatt MATH identifier
1293.60061

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60F05: Central limit and other weak theorems
Secondary: 60G55: Point processes 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

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

Citation

Reitzner, Matthias; Schulte, Matthias. Central limit theorems for $U$-statistics of Poisson point processes. Ann. Probab. 41 (2013), no. 6, 3879--3909. doi:10.1214/12-AOP817. https://projecteuclid.org/euclid.aop/1384957778


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