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

TL;DR

This paper studies the long-term behavior of projections of supercritical multi-type continuous state and time branching processes with immigration, showing asymptotic mixed normality and convergence of type frequencies under certain moment conditions.

Contribution

It establishes asymptotic mixed normality and convergence results for projections of multi-type branching processes with immigration under specific moment assumptions.

Findings

Projections on certain eigenvectors are asymptotically mixed normal.

Under random scaling, asymptotic normality is achieved.

Relative frequencies of types converge almost surely under immigration.

Abstract

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.