  Under a first order moment condition on the immigration mechanism, we show
that an appropriately scaled supercritical and irreducible multi-type
continuous state and continuous time branching process with immigration (CBI
process) converges almost surely. If an $x \log(x)$ moment condition on the
branching mechanism does not hold, then the limit is zero. If this $x \log(x)$
moment condition holds, then we prove $L_1$ convergence as well. The projection
of the limit on any left non-Perron eigenvector of the branching mean matrix is
vanishing. If, in addition, a suitable extra power moment condition on the
branching mechanism holds, then we provide the correct scaling for the
projection of a CBI process on certain left non-Perron eigenvectors of the
branching mean matrix in order to have almost sure and $L_1$ limit. Moreover,
under a second order moment condition on the branching and immigration
mechanisms, we prove $L_2$ convergence of an appropriately scaled process and
the above mentioned projections as well. A representation of the limits is also
provided under the same moment conditions.
