Moderate deviations on Poisson chaos

Abstract: This paper deals with U-statistics of Poisson processes and multiple Wiener-It^o integrals on the Poisson space. Via sharp bounds on the cumulants for both classes of random variables, moderate deviation principles, concentration inequalities and normal approximation bounds with Cram\'er correction are derived. It is argued that the results obtained in this way are in a sense best possible and cannot be improved systematically. Applications in stochastic geometry and to functionals of Ornstein-Uhlenbeck-L\'evy processes are investigated.