Poisson process Fock space representation, chaos expansion and covariance inequalities

By Günter Last, Mathew D. Penrose. 2009

126 citations recorded by the source corpus.

We consider a Poisson process η on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of η. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener–Itô chaos expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincaré inequality is a special case. Further applications are covariance identities for Poisson processes on (strictly) ordered spaces and Harris–FKG-inequalities for monotone functions of η.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c4a969c9fadfc52b99718820a8012450753dd5b1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Normal Approximation of Poisson Functionals in Kolmogorov Distance (2012, background): “3 in [8]).”
Central limit theorems for $U$-statistics of Poisson point processes (2011, methodology): “To prove our central limit theorems, we first use a result of Last and Penrose [19], to expand a U -statistic in a Wiener–Itô chaos expansion as a finite sum of multiple Wiener–Itô integrals.”
Second-order properties and central limit theorems for geometric functionals of Boolean models (2013, methodology): “,...,xnF:=D n x1,...,xnf(η), n∈N,x1,...,xn∈X, where Dn is the nth iterated diﬀerence operator used in Section 3. If f˜ is another representative of F, then the multivariate Mecke equation (see, e.g., [15], (2.10)) implies that Dn x1,...,xnf(η)=D n x1,...,xn f˜(η) P-a.s. and for λn-a.e. (x1,...,xn)∈X”
Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization (2014, background): “The Poincaré inequality (see [26, 55] and the references therein) states that the variance of a square-integrable Poisson functional F can be bounded as VarF ≤ E ∫”
Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs (2011, methodology): “7) follows from an application of the results proved by Last and Penrose in [11].”
Lectures on the Poisson Process (2017, background): “Ito [57] for homogeneous Poisson processes on the line, and in [90] for general Poisson processes.”
Quantitative results on the corrector equation in stochastic homogenization (2014, background): “In the case of the Poisson point process, a corresponding covariance estimate holds as well and is known as the Harris–FKG inequality, see [25, 21].”
Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry (2014, background): “e integrable and symmetric function fn with respect to the signed random measure ζ−M. Moreover, the series in (5.9) converges in L2(Pζ) and is called the Wiener–Itˆo chaos expansion of F; we refer to [29] for further details. We can now deﬁne the Ornstein–Uhlenbeck generator Lon the Poisson space by”
The scaling limit of Poisson-driven order statistics with applications in geometric probability (2012, background): “nd only if τ = 1, in which case the mentioned Weibull distribution is nothing than an exponential distribution with parameter β. The Wiener-Itˆo chaos decomposition as outlined by Last and Penrose in [16] has stimulated a number of applications in geometric probability and stochastic geometry, that ”