  Let F ($\nu$) be the centered Gamma law with parameter $\nu$ > 0 and let us
denote by P Y 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 second Wasserstein distance between the
probability distribution of an arbitrary random vector (X, Y) in R x R n and
the probability distribution F ($\nu$) $\otimes$ P Y. 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 $\ge$ 1) to F
($\nu$) $\otimes$ P Y , by assuming that (X n , n $\ge$ 1) converges in law, as
n $\rightarrow$ $\infty$, to F ($\nu$) and (Y n , n $\ge$ 1) converges in law,
as n $\rightarrow$ $\infty$, to an arbitrary random vector Y. We illustrate our
criteria by two concrete examples.
