Multidimensional Stein's method for Gamma approximation

Let F(v) be the centered Gamma law with parameter v > 0 and let us denote by PY the probability distribution of a random vector Y. We develop a multidimensional variant of the Stein's method for Gamma approximation that allows to obtain bounds for the Wasserstein distance between the probability distribution of an arbitrary random vector (X, Y) in R x R-n and the probability distribution F(v) circle times PY. In the case of random vectors with components in Wiener chaos, these bounds lead to some interesting criteria for the joint convergence of a sequence ((X-n, Y-n), n >= 1) to F(v) circle times PY, by assuming that (X-n, n > 1) converges in law, as n ->infinity, to F(v) and (Y-n, n > 1) converges in law, as n ->infinity, to an arbitrary random vector Y. We illustrate our criteria by two concrete examples.