Asymptotic distributions of Wishart type products of random matrices

We study asymptotic distributions of large dimensional random matrices of the form $BB^{*}$, where $B$ is a product of $p$ rectangular random matrices, using free probability and combinatorics of colored labeled noncrossing partitions. These matrices are taken from the set of off-diagonal blocks of the family $\mathcal{Y}$ of independent Hermitian random matrices which are asymptotically free, asymptotically free against the family of deterministic diagonal matrices, and whose norms are uniformly bounded almost surely. This class includes unitarily invariant Hermitian random matrices with limit distributions given by compactly supported probability measures $\nu$ on the real line. We express the limit moments in terms of colored labeled noncrossing pair partitions, to which we assign weights depending on even free cumulants of $\nu$ and on asymptotic dimensions of blocks (Gaussianization). For products of $p$ independent blocks, we show that the limit moments are linear combinations of a new family of polynomials called generalized multivariate Fuss-Narayana polynomials. In turn, the product of two blocks of the same matrix leads to an example with rescaled Raney numbers.