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A central limit theorem for super-Brownian motion with super-Brownian immigration

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Wen-Ming Hong  and
Zeng-Hu Li
Wen-Ming Hong*
Affiliation:
Fudan University
Zeng-Hu Li*
Affiliation:
Beijing Normal University
*
∗Postal address: Department of Mathematics, Fudan University, Shanghai 200433, Peoples Republic of China. Email address: [email protected]
∗∗Postal address: Department of Mathematics, Beijing Normal University, Beijing 100875, People's Republic of China. Email address: [email protected].
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Abstract

We prove a central limit theorem for the super-Brownian motion with immigration governed by another super-Brownian. The limit theorem leads to Gaussian random fields in dimensions d ≥ 3. For d = 3 the field is spatially uniform; for d ≥ 5 its covariance is given by the potential operator of the underlying Brownian motion; and for d = 4 it involves a mixture of the two kinds of fluctuations.

Keywords

Super-Brownian motionimmigrationrandom mediumcentral limit theorem

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F05: Central limit and other weak theorems
Type
Short Communications
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1218 - 1224
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

This research supported by the National Natural Science Foundation of China (Grant No. 19361060 and Grant No. 19671011) and the Mathematical Center of the State Education Commission.

References

Dawson, D. A. (1993). Measure-valued Markov processes. In Lecture Notes in Math. 1541. Springer, Berlin, pp. 1–260.Google Scholar
Dawson, D. A., and Fleischmann, K. (1997). A continuous super-Brownian motion in a super-Brownian medium. J. Theor. Prob. 10, 213–276.Google Scholar
Dynkin, E. B. (1991). Branching particle systems and superprocesses. Ann. Prob. 19, 1157–1194.CrossRefGoogle Scholar
Iscoe, I. (1986). A weighted occupation time for a class of measure-valued critical branching Brownian motion. Prob. Theor. Rel. Fields 71, 85–116.Google Scholar
Konno, N., and Shiga, T. (1988). Stochastic partial differential equations for some measure-valued diffusions. Prob. Theor. Rel. Fields 79, 34–51.Google Scholar
Li, Z. H. (1992). Measure-valued branching processes with immigration. Stoch. Proc. Appl. 43, 249–264.Google Scholar
Li, Z. H. (1996). Immigration structures associated with Dawson–Watanabe superprocesses. Stoch. Proc. Appl. 62, 73–86.CrossRefGoogle Scholar
Li, Z. H. (1999). Some central limit theorems for super Brownian motions. Act. Math. Sci. (Ser. A) 19, 121–126.CrossRefGoogle Scholar
Li, Z. H., and Shiga, T. (1995). Measure-valued branching diffusions: immigrations, excursions and limit theorems. J. Math. Kyoto Univ. 35, 233–274.Google Scholar