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

By Matthias Reitzner, Matthias Schulte. 2011

140 citations recorded by the source corpus.

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-It\^{o} 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.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/5f888fbee8d6c15e74ab08513c701c760b5e5141

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Citations in the corpus

Normal Approximation of Poisson Functionals in Kolmogorov Distance (2012, methodology): “ase of the described setting is that the Poisson point process depends on a real valued intensity parameter. The following corollary deals with this situation and is the counterpart of Theorem 5.2 in [11] for the Kolmogorov distance. Corollary 4.3 Let ηt be a Poisson point process with intensity mea”
Second-order properties and central limit theorems for geometric functionals of Boolean models (2013, methodology): “ric functionals of a Boolean model by proving a general result on the normal approximation of Poisson functionals. Our approach is based on recent ﬁndings in [23, 25] and uses similar arguments as in [28]. Throughout this section, let ηbe a Poisson process on a measurable space (X,X) with a σ-ﬁnite ”
Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization (2014, unknown): “We refer the reader to [36, 39] for the first bounds of this type in the context of CLTs in the Wasserstein and smoother distances, and to [46] for the first panoply of results indicating that Malliavin-Stein techniques can be of great value for deriving quantitative CLTs in a geometric setting.”
Lectures on the Poisson Process (2017, methodology): “In preparing this manuscript we have benefited from comments on earlier versions from Daryl Daley, Fabian Gieringer, Christian Hirsch, Daniel Hug, Olav Kallenberg, Paul Keeler, Martin Möhle, Franz Nestmann, Jim xii Preface Pitman, Matthias Schulte, Tomasz Rolski, Dietrich Stoyan, Christoph Thäle, ”
Normal approximation for stabilizing functionals (2017, methodology): “Bounds for the normal approximation of such random variables were deduced, for example, in [35] and [21] for the Wasserstein distance and in [43] and [14] for the Kolmogorov distance.”
Moments and Central Limit Theorems for Some Multivariate Poisson Functionals (2012, methodology): “ (ℓ) t are ﬁnite sums of random variables of type (4.1). In fact, under some additional conditions, any Poisson functional with ﬁnite Wiener-Itoˆ chaos expansion can be represented in such a way, cf. [16]. We prepare the proof of Theorem 5.2 by the following lemma: Lemma 5.5. Let Xand Ybe ℓ-dimensio”
Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs (2011, methodology): “The following statement, based on the results proved by Reitzner and Schulte in [30], collects two crucial properties of U -statistics.”