  We provide a new extension of Breiman's Theorem on computing tail
probabilities of a product of random variables to a multivariate setting. In
particular, we give a complete characterization of regular variation on cones
in $[0,\infty)^d$ under random linear transformations. This allows us to
compute probabilities of a variety of tail events, which classical multivariate
regularly varying models would report to be asymptotically negligible. We
illustrate our findings with applications to risk assessment in financial
systems and reinsurance markets under a bipartite network structure.
