Gamma limits and U-statistics on the Poisson space

Abstract

Using Stein's method and the Malliavin calculus of variations, we derive explicit estimates for the Gamma approximation of functionals of a Poisson measure. In particular, conditions are presented under which the distribution of a sequence of multiple Wiener-Ito stochastic integrals with respect to a compensated Poisson measure converges to a Gamma distribution. As an illustration, we present a quantitative version and a non-central extension of a classical theorem by de Jong in the case of degenerate U-statistics of order two. Several multidimensional extensions, in particular allowing for mixed or hybrid limit theorems, are also provided.

References in corpus (2)

• Noncentral convergence of multiple integrals
• Nourdin-Peccati analysis on Wiener and Wiener-Poisson space for general distributions

Cited by in corpus (2)

• Poisson convergence on the free Poisson algebra
• Stable limit theorems on the Poisson space