Asymptotic behavior of projections of supercritical multi-type continuous-state and continuous-time branching processes with immigration

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