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

• Published in: Advances in applied probability Vol. 53; no. 4; pp. 1023 - 1060
• Main Authors: Barczy, Mátyás, Palau, Sandra, Pap, Gyula
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.12.2021; Applied Probability Trust
• Subjects: Asymptotic methods; Asymptotic properties; Branching (mathematics); Conditional probability; Convergence; Decomposition; Eigenvalues; Eigenvectors; Immigration; Investigations; Markov processes; Normal distribution; Normality; Original Article; Original Articles; Probability; Random variables; Multi-type continuous-state and continuous-time branching processes with immigration; mixed normal distribution; 60J80; 60F15
• ISSN: 0001-8678; 1475-6064
• DOI: 10.1017/apr.2021.7

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 the 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.