Stein normal approximation for multidimensional Poisson random measures by third cumulant expansions

We derive normal approximation bounds by the Stein method for stochastic integrals with respect to a Poisson random measure over R, d ≥ 2. This approach relies on third cumulant Edgeworth-type expansions based on derivation operators defined by the Malliavin calculus for Poisson random measures. The use of third cumulants can exhibit faster convergence rates than the standard BerryEsseen rate for some sequences of Poisson stochastic integrals.