  Under a fourth order moment condition on the branching and a second order
moment condition on the immigration mechanisms, we show that an appropriately
scaled projection of a supercritical and irreducible continuous state and
continuous time branching process with immigration on certain left non-Perron
eigenvectors of the branching mean matrix is asymptotically mixed normal. With
an appropriate random scaling, under some conditional probability measure, we
prove asymptotic normality as well. In case of a non-trivial process, under a
first order moment condition on the immigration mechanism, we also prove the
convergence of the relative frequencies of distinct types of individuals on a
suitable event; for instance, if the immigration mechanism does not vanish,
then this convergence holds almost surely.
