101 research outputs found
Perturbation analysis of Poisson processes
We consider a Poisson process on a general phase space. The
expectation of a function of can be considered as a functional of the
intensity measure of . 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 . 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 -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 be a Boolean model based on a stationary Poisson process of
compact, convex particles in Euclidean space . Let denote a
compact, convex observation window. For a large class of functionals ,
formulas for mean values of 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 for increasing observation window , including
convergence rates. Our approach is based on the Fock space representation
associated with . 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
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