The talk gives a brief introduction to multiple stochastic integrals with respect to Gaussian and Poisson random measures. We will present and recall the seminal concept of a `four moments theorem', the first of its kind discovered by Nualart and Peccati (2005). Roughly speaking, a four moments theorem for a sequence of multiple stochastic integrals asserts that the convergence in distribution of such a sequence to a specific distribution is equivalent to the convergence of the first four moments of this sequence to the corresponding moments of that limiting distribution. For Gaussian integrals, both the situation of a normal and of a Gamma limiting law have been investigated so far. For Poisson integrals, yet, this has been only done for the normal approximation. We focus on the Gamma approximation for Poisson integrals, show a four moments theorem for integrals of order q=2 and q=4 and show also the limitations of the classical methods to prove a more general result concerning higher orders. As an application of our findings, we present a universality result for homogeneous sums.

The talk is based on a joint work with Christoph Thäle, Ruhr-University Bochum, available at arxiv.org/abs/1502.01568.