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

Barczy, Mátyás and Palau, Sandra and Pap, Gyula (2020) Almost sure, L_1- and L_2-growth behavior of supercritical multi-type continuous state and continuous time branching processes with immigration. Science China Mathematics, 63 (10). pp. 2089-2116. ISSN 1674-7283

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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 xlog(x) moment condition on the branching mechanism does not hold, then the limit is zero. If this xlog(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.

Item Type:	Article
Additional Information:	Mátyás Barczy is supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Sandra Palau was supported by the Royal Society Newton International Fellowship and by the EU-funded Hungarian grant EFOP-3.6.1-16-2016-00008.
Uncontrolled Keywords:	multi-type continuous state and continuous time branching processes with immigration, almost sure, L_1- and L_2-growth behaviour
Subjects:	Q Science / természettudomány > QA Mathematics / matematika
Depositing User:	Dr Mátyás Barczy
Date Deposited:	08 Mar 2021 15:48
Last Modified:	03 Apr 2023 07:09
URI:	http://real.mtak.hu/id/eprint/122012

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• Almost sure, L_1- and L_2-growth behavior of supercritical multi-type continuous state and continuous time branching processes with immigration. (deposited 01 Sep 2020 14:24)
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