Abstract
We study the distributional properties of jumps of multi-type continuous state and continuous time branching processes with immigration (multi-type CBI processes). We derive an expression for the distribution function of the first jump time of a multi-type CBI process with jump size in a given Borel set having finite total Lévy measure, which is defined as the sum of the measures appearing in the branching and immigration mechanisms of the multi-type CBI process in question. Using this we derive an expression for the distribution function of the local supremum of the norm of the jumps of a multi-type CBI process. Further, we show that if A is a nondegenerate rectangle anchored at zero and with total Lévy measure zero, then the probability that the local coordinate-wise supremum of jumps of the multi-type CBI process belongs to A is zero. We also prove that a converse statement holds.
Funding Statement
Mátyás Barczy was supported by the project TKP2021-NVA-09. Project no. TKP2021-NVA-09 has been implemented with the support provided by the Ministry of Culture and Innovation of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-NVA funding scheme.
Acknowledgments
We would like to thank Professor Zenghu Li for his constant help in discussing our questions about distributional properties of single-type CBI processes, and for sending us a preliminary draft version of the new second edition of his book [26]. We would like to thank the referee for his/her comments that helped us to improve the paper.
Citation
Mátyás Barczy. Sandra Palau. "Distributional properties of jumps of multi-type CBI processes." Electron. J. Probab. 29 1 - 39, 2024. https://doi.org/10.1214/24-EJP1125
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