Abstract
Strongly critical branching processes with immigration (BPI) are investigated in varying environment (VE) when the time-varying offspring mean converges to 1 fast enough. Under natural assumptions a diffusion approximation is derived when either the offspring or the immigration variances are strictly positive. In the case of asymptotically vanishing offspring variances a fluctuation limit theorem is proved.
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Acknowledgements
The author would like to thank the anonymous reviewer for drawing his attention to the close connection between branching processes with immigration and controlled branching processes and to the papers González and del Puerto [3] and González et al. [4].
This paper was finished when the author was a researcher at the Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences.
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Appendix
Appendix
In the proofs we need the following result about convergence of random step processes towards a diffusion process, see Ispány and Pap [6, Corollary 2.2].
Theorem 5.3
Let \(\beta: \mathbb{R}_{+} \times \mathbb{R} \rightarrow \mathbb{R}\) and \(\gamma: \mathbb{R}_{+} \times \mathbb{R} \rightarrow \mathbb{R}\) be continuous functions. Assume that uniqueness in the sense of probability law holds for the SDE
with initial value \(\mathcal{U}_{0} = u_{0}\) for all \(u_{0} \in \mathbb{R}\), where \((\mathcal{W}_{t})_{t\in \mathbb{R}_{+}}\) is a standard Wiener process. Let \((\mathcal{U}_{t})_{t\in \mathbb{R}_{+}}\) be a solution of (5.39) with initial value \(\mathcal{U}_{0} = 0\).
For each \(n \in \mathbb{N}\) , let \((U_{k}^{n})_{k\in \mathbb{N}}\) be a sequence of random variables adapted to a filtration \((\mathcal{F}_{k}^{n})_{k\in \mathbb{Z}_{+}}\) . Let \(\mathcal{U}_{t}^{n}:=\sum _{ k=1}^{\lfloor nt\rfloor }U_{k}^{n}\) , \(t \in \mathbb{R}_{+},n \in \mathbb{N}\) . Suppose \(\mathrm{E}\big(\vert U_{k}^{n}\vert ^{2}\big) <\infty\) for all \(n,k \in \mathbb{N}\) . Suppose that, for each T > 0,
Then \(\boldsymbol{\mathcal{U}}^{n}\stackrel{\mathcal{L}}{\longrightarrow }\boldsymbol{\mathcal{U}}\) as n →∞.
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Ispány, M. (2016). Some Asymptotic Results for Strongly Critical Branching Processes with Immigration in Varying Environment. In: del Puerto, I., et al. Branching Processes and Their Applications. Lecture Notes in Statistics(), vol 219. Springer, Cham. https://doi.org/10.1007/978-3-319-31641-3_5
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DOI: https://doi.org/10.1007/978-3-319-31641-3_5
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