Almost sure, L_1- and L_2-growth behavior of supercritical multi-type continuous state and continuous time branching processes with immigration

TL;DR

This paper establishes almost sure, L_1, and L_2 convergence behaviors of supercritical multi-type continuous state and time branching processes with immigration, under various moment conditions, and provides limit representations.

Contribution

It extends the understanding of convergence behaviors of multi-type CBI processes under different moment conditions, including new limit representations and scaling results.

Findings

Almost sure convergence under first order moment condition.

L_1 convergence when the $x \, \log(x)$ moment condition holds.

L_2 convergence under second order moment conditions.

Abstract

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.