Almost sure, L1- and L2-growth behavior of supercritical multi-type continuous state and continuous time branching processes with immigration

• Published in: Science China. Mathematics Vol. 63; no. 10; pp. 2089 - 2116
• Main Authors: Barczy, Mátyás, Palau, Sandra, Pap, Gyula
• Format: Journal Article
• Language: English
• Published: Beijing Science China Press 01.10.2020; Springer Nature B.V
• Subjects: Applications of Mathematics; Convergence; Eigenvectors; Immigration; Mathematics; Mathematics and Statistics; growth behaviour; and; multi-type continuous state and continuous time branching processes with immigration; almost sure; 60J80; 60F15
• ISSN: 1674-7283; 1869-1862
• DOI: 10.1007/s11425-019-1552-1

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.