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A functional central limit theorem for spatial birth and death processes

Part of: Limit theorems Special processes Stochastic analysis Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Xin Qi
Xin Qi*
Affiliation:
University of Wisconsin-Madison
*
∗ Current address: 1199 Whitney Avenue, Apt. 507, Hamden, CT 06517, USA. Email address: xinqi@stat.wisc.edu
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Abstract

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We give a functional central limit theorem for spatial birth and death processes based on the representation of such processes as solutions of stochastic equations. For any bounded and integrable function in Euclidean space, we define a family of processes which is obtained by integrals of this function with respect to the centered and scaled spatial birth and death process with constant death rate. We prove that this family converges weakly to a Gaussian process as the scale parameter goes to infinity. We do not need the birth rates to have a finite range of interaction. Instead, we require that the birth rates have a range of interaction that decays polynomially. In order to show the convergence of the finite-dimensional distributions of the above processes, we extend Penrose's multivariate spatial central limit theorem. An example of the asymptotic normalities of the time-invariance estimators for the birth rates of spatial point processes is given.

Keywords

Spatial birth and death processPoisson random measurestochastic equationfunctional central limit theorem

MSC classification

Primary: 60G55: Point processes
Secondary: 60F17: Functional limit theorems; invariance principles 60H20: Stochastic integral equations 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 3 , September 2008 , pp. 759 - 797
Copyright
Copyright © Applied Probability Trust 2008 

Footnotes

Supported in part by the National Science Foundation under grant DMS 05-03983.

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