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2019 Stable central limit theorems for super Ornstein-Uhlenbeck processes
Yan-Xia Ren, Renming Song, Zhenyao Sun, Jianjie Zhao
Electron. J. Probab. 24: 1-42 (2019). DOI: 10.1214/19-EJP396


In this paper, we study the asymptotic behavior of a supercritical $(\xi ,\psi )$-superprocess $(X_{t})_{t\geq 0}$ whose underlying spatial motion $\xi $ is an Ornstein-Uhlenbeck process on $\mathbb{R} ^{d}$ with generator $L = \frac{1} {2}\sigma ^{2}\Delta - b x \cdot \nabla $ where $\sigma , b >0$; and whose branching mechanism $\psi $ satisfies Grey’s condition and a perturbation condition which guarantees that, when $z\to 0$, $\psi (z)=-\alpha z + \eta z^{1+\beta } (1+o(1))$ with $\alpha > 0$, $\eta >0$ and $\beta \in (0, 1)$. Some law of large numbers and $(1+\beta )$-stable central limit theorems are established for $(X_{t}(f) )_{t\geq 0}$, where the function $f$ is assumed to be of polynomial growth. A phase transition arises for the central limit theorems in the sense that the forms of the central limit theorem are different in three different regimes corresponding to the branching rate being relatively small, large or critical at a balanced value.


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Yan-Xia Ren. Renming Song. Zhenyao Sun. Jianjie Zhao. "Stable central limit theorems for super Ornstein-Uhlenbeck processes." Electron. J. Probab. 24 1 - 42, 2019.


Received: 9 March 2019; Accepted: 15 November 2019; Published: 2019
First available in Project Euclid: 18 December 2019

zbMATH: 07149381
MathSciNet: MR4049077
Digital Object Identifier: 10.1214/19-EJP396

Primary: 60F05, 60J68


Vol.24 • 2019
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