Concentration of measure and generalized product of random vectors with an application to Hanson-Wright-like inequalities

Starting from concentration of measure hypotheses on m random vectors Z1,ldots, Zm, this article provides an expression of the concentration of functionals φ(Z1,ldots, Zm) where the variations of φ on each variable depend on the product of the norms (or semi-norms) of the other variables (as if φ were a product). We illustrate the importance of this result through various generalizations of the Hanson-Wright concentration inequality as well as through a study of the random matrix XDXT and its resolvent Q = (Ip - frac1nXDXT)-1, where X and D are random, which have fundamental interest in statistical machine learning applications.