I will present some generalisations and ramifications of a fundamental Theorem by P. de Jong (1987, 1990), characterising the asymptotic normality of degenerate U-statistics by means of a drastic simplification of the method of moments. In particular, I will focus on several ramifications of such a result in the framework of multiple integrals with respect to a random measure, both in a commutative and non-commutative setting.
Based on several joint works, e.g. with Ch. Döbler, T. Kemp, I. Nourdin, D. Nualart, S. Bourguin and R. Speicher.