Moments and central limit theorems for some multivariate Poisson functionals

Guenter Last, M D Penrose, Matthias Schulte, Christoph Thaele

Research output: Contribution to journalArticle

25 Citations (Scopus)

Abstract

This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-Ito integrals with respect to the compensated Poisson process. Second, a multivariate central limit theorem is shown for a vector whose components admit a finite chaos expansion of the type of a Poisson U-statistic. The approach is based on recent results of Peccati et al. (2010),combining Malliavin calculus and Stein's method, and also yields Berry-Esseen type bounds. As applications, moment formulae and central limit theorems for general geometric functionals of intersection processes associated with a stationary Poisson process of k-dimensional flats in Rd are discussed.

Original languageEnglish
Pages (from-to)348-364
Number of pages19
JournalAdvances in Applied Probability
Volume46
Issue number2
DOIs
Publication statusPublished - Jun 2014

Fingerprint

Poisson process
Central limit theorem
Siméon Denis Poisson
Moment
Chaos theory
Itô Integral
Stein's Method
Wiener Integral
Chaos Expansion
Measurable space
Malliavin Calculus
Statistics
Cumulants
Stationary Process
Statistic
Intersection
Arbitrary

Keywords

  • Berry-Esseen type bounds; central limit theorem; intersection process; multiple Wiener-Ito integral; Poisson process; Poisson flat process; product formula; stochastic geometry; Wiener-Ito chaos expansion

Cite this

Moments and central limit theorems for some multivariate Poisson functionals. / Last, Guenter ; Penrose, M D; Schulte, Matthias; Thaele, Christoph.

In: Advances in Applied Probability, Vol. 46, No. 2, 06.2014, p. 348-364.

Research output: Contribution to journalArticle

Last, Guenter ; Penrose, M D ; Schulte, Matthias ; Thaele, Christoph. / Moments and central limit theorems for some multivariate Poisson functionals. In: Advances in Applied Probability. 2014 ; Vol. 46, No. 2. pp. 348-364.
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