206 research outputs found
Invariance of Poisson measures under random transformations
We prove that Poisson measures are invariant under (random) intensity
preserving transformations whose finite difference gradient satisfies a cyclic
vanishing condition. The proof relies on moment identities of independent
interest for adapted and anticipating Poisson stochastic integrals, and is
inspired by the method applied in [22] on the Wiener space, although the
corresponding algebra is more complex than in the Wiener case. The examples of
application include transformations conditioned by random sets such as the
convex hull of a Poisson random measure
Girsanov identities for Poisson measures under quasi-nilpotent transformations
We prove a Girsanov identity on the Poisson space for anticipating
transformations that satisfy a strong quasi-nilpotence condition. Applications
are given to the Girsanov theorem and to the invariance of Poisson measures
under random transformations. The proofs use combinatorial identities for the
central moments of Poisson stochastic integrals.Comment: Published in at http://dx.doi.org/10.1214/10-AOP640 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Mixing of Poisson random measures under interacting transformations
We derive sufficient conditions for the mixing of all orders of interacting
transformations of a spatial Poisson point process, under a zero-type condition
in probability and a generalized adaptedness condition. This extends a
classical result in the case of deterministic transformations of Poisson
measures. The approach relies on moment and covariance identities for Poisson
stochastic integrals with random integrands
Stein approximation for functionals of independent random sequences
We derive Stein approximation bounds for functionals of uniform random
variables, using chaos expansions and the Clark-Ocone representation formula
combined with derivation and finite difference operators. This approach covers
sums and functionals of both continuous and discrete independent random
variables. For random variables admitting a continuous density, it recovers
classical distance bounds based on absolute third moments, with better and
explicit constants. We also apply this method to multiple stochastic integrals
that can be used to represent U-statistics, and include linear and quadratic
functionals as particular cases
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