Translator Disclaimer
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

Abstract

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.

Citation

Download Citation

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. https://doi.org/10.1214/19-EJP396

Information

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

Subjects:
Primary: 60F05, 60J68

JOURNAL ARTICLE
42 PAGES


SHARE
Vol.24 • 2019
Back to Top