Poisson process Fock space representation, chaos expansion and covariance inequalities

Details Poisson process Fock space representation, chaos expansion and covariance inequalities Poisson process Fock space representation, chaos expansion and covariance inequalities

2009-09-17

arxiv.org/abs/0909.3205v1

Mathematics

Probability

25 pages

Scientific paper

We consider a Poisson process $\eta$ on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of $\eta$. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener-Ito 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 Poincare 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 $\eta$.

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