Almost sure, L 1 - and L 2 -growth behavior of supercritical multi-type continuous state and continuous time branching p
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https://doi.org/10.1007/s11425-019-1552-1
Almost sure, L1- and L2-growth behavior of supercritical multi-type continuous state and continuous time branching processes with immigration M´aty´as Barczy1,∗ , Sandra Palau2 & Gyula Pap3 1MTA-SZTE
Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Szeged H-6720, Hungary; 2Department of Statistics and Probability, Instituto de Investigaciones en Matem´ aticas Aplicadas y en Sistemas, Universidad Nacional Aut´ onoma de M´ exico, Ciudad de M´ exico 04510, M´ exico; 3Bolyai Institute, University of Szeged, Szeged H-6720, Hungary Email: [email protected], [email protected], [email protected] Received March 8, 2019; accepted August 8, 2019
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. Keywords
multi-type continuous state and continuous time branching processes with immigration, almost
sure, L1 - and L2 -growth behaviour MSC(2010)
60J80, 60F15
Citation: 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, 2020, 63, https://doi.org/10.1007/s11425-019-1552-1
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Introduction
The description of the asymptotic behavior of branching processes without or with immigration has a long history. For multi-type Galton-Watson processes without immigration, see, e.g., Athreya and Ney [3, Sections 4–8 in Chapter V]. For supercritical multi-type Galton-Watson processes with immigration, see, e.g., Kaplan [13]. * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Barczy M et al.
2
Sci China Math
Let us consider a multi-type continuous state and continuous time branching process with immigration (CBI process) which can be represented as a pathwise unique strong solution of the stochastic differential equation (SDE) ∫
t
Xt = X0 +
e u )du + (β + BX
0
+
d ∫
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