Perturbation analysis of Poisson processes

We consider a Poisson process $\Phi$ on a general phase space. The expectation of a function of $\Phi$ can be considered as a functional of the intensity measure $\lambda$ of $\Phi$. Extending earlier results of Molchanov and Zuyev [Math. Oper. Res. 25 (2010) 485-508] on finite Poisson processes, we study the behaviour of this functional under signed (possibly infinite) perturbations of $\lambda$. In particular, we obtain general Margulis-Russo type formulas for the derivative with respect to non-linear transformations of the intensity measure depending on some parameter. As an application, we study the behaviour of expectations of functions of multivariate L\'evy processes under perturbations of the L\'evy measure. A key ingredient of our approach is the explicit Fock space representation obtained in Last and Penrose [Probab. Theory Related Fields 150 (2011) 663-690].Comment: Published in at http://dx.doi.org/10.3150/12-BEJ494 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Stochastic analysis for Poisson processes

This survey is a preliminary version of a chapter of the forthcoming book "Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-It\^o Chaos Expansions and Stochastic Geometry" edited by Giovanni Peccati and Matthias Reitzner. The paper develops some basic theory for the stochastic analysis of Poisson process on a general $\sigma$-finite measure space. After giving some fundamental definitions and properties (as the multivariate Mecke equation) the paper presents the Fock space representation of square-integrable functions of a Poisson process in terms of iterated difference operators. This is followed by the introduction of multivariate stochastic Wiener-It\^o integrals and the discussion of their basic properties. The paper then proceeds with proving the chaos expansion of square-integrable Poisson functionals, and defining and discussing Malliavin operators. Further topics are products of Wiener-It\^o integrals and Mehler's formula for the inverse of the Ornstein-Uhlenbeck generator based on a dynamic thinning procedure. The survey concludes with covariance identities, the Poincar\'e inequality and the FKG-inequality

Second-order properties and central limit theorems for geometric functionals of Boolean models

Let $Z$ be a Boolean model based on a stationary Poisson process $\eta$ of compact, convex particles in Euclidean space ${\mathbb{R}}^d$. Let $W$ denote a compact, convex observation window. For a large class of functionals $\psi$, formulas for mean values of $\psi(Z\cap W)$ are available in the literature. The first aim of the present work is to study the asymptotic covariances of general geometric (additive, translation invariant and locally bounded) functionals of $Z\cap W$ for increasing observation window $W$, including convergence rates. Our approach is based on the Fock space representation associated with $\eta$. For the important special case of intrinsic volumes, the asymptotic covariance matrix is shown to be positive definite and can be explicitly expressed in terms of suitable moments of (local) curvature measures in the isotropic case. The second aim of the paper is to prove multivariate central limit theorems including Berry-Esseen bounds. These are based on a general normal approximation result obtained by the Malliavin--Stein method.Comment: Published at http://dx.doi.org/10.1214/14-AAP1086 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org