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

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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 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 L1 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 L1 limit. Moreover, under a second order moment condition on the branching and immigration mechanisms, we prove L2 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.

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Acknowledgements

The first author was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. The second author was supported by the Royal Society Newton International Fellowship and the EU-funded Hungarian (Grant No. EFOP-3.6.1-16-2016-00008). The authors thank the referees for their comments that helped them to improve the paper.

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Authors and Affiliations

  1. MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Szeged, H-6720, Hungary

    Mátyás Barczy

  2. Department of Statistics and Probability, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Ciudad de México, 04510, México

    Sandra Palau

  3. Bolyai Institute, University of Szeged, Szeged, H-6720, Hungary

    Gyula Pap

Authors
  1. Mátyás Barczy
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  2. Sandra Palau
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  3. Gyula Pap
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Correspondence to Mátyás Barczy.

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Barczy, M., Palau, S. & Pap, G. Almost sure, L1- and L2-growth behavior of supercritical multi-type continuous state and continuous time branching processes with immigration. Sci. China Math. 63, 2089–2116 (2020). https://doi.org/10.1007/s11425-019-1552-1

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  • DOI: https://doi.org/10.1007/s11425-019-1552-1

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