Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T09:20:09.087Z Has data issue: false hasContentIssue false
  • English
  • Français

A functional central limit theorem for spatial birth and death processes

Part of: Special processes Limit theorems Stochastic processes Stochastic analysis

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: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

Baddeley, A. J. (2000). Time-invariance estimating equations. Bernoulli 6, 783–808.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1985). Markov Processes: Characterization and Convergence. John Wiley, New York.Google Scholar
Garcia, N. L. and Kurtz, T. G. (2006). Spatial birth and death processes as solutions of stochastic equations. Alea 1, 281–303.Google Scholar
Holley, R. and Stroock, D. W. (1979). Central limit phenomena of various interacting systems. Ann. Math. 110, 333–393.Google Scholar
Kesten, H. and Lee, S. (1996). The central limit theorem for weighted minimal spanning trees on random points. Ann. Appl. Prob. 6, 495–527.Google Scholar
Kurtz, T. and Li, S. (2003). Time-invariance modeling and estimation for spatial point processes: general theory. Res. Rep., University of Wisconsin-Madison. Available at http://www.math.wisc.edu/∼kurtz/papers/shun.pdf.Google Scholar
Lee, S. (1997). The central limit theorem for Euclidean minimal spanning trees. I. Ann. Appl. Probab. 7, 996–1020.Google Scholar
McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. Ann. Prob. 2, 620–628.Google Scholar
Mitoma, I. (1983). Tightness of probabilities on c(0,1; ψ′) and d(0,1; ψ′). Ann. Prob. 4, 989–999.Google Scholar
Penrose, M. D. (2001). A central limit theorem with applications to percolation, epidemics and Boolean models. Ann. Probab. 11, 1005–1041.Google Scholar
Penrose, M. D. (2005). Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Prob. 33, 1945–1945.Google Scholar
Penrose, M. D. (2008). Existence and spatial limit theorems for lattice and continuum particle systems. Prob. Surveys 5, 1–36.CrossRefGoogle Scholar
Qi, X. (2007). The central limit theorems for space-time point processes. , University of Wisconsin-Madison.Google Scholar
Revuz, D. and Yor, M. (2004). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.Google Scholar
Ripley, B. D. (1979). Algorithm AS 137: Simulating spatial patterns: dependent samples from a multivariate density. Appl. Statist. 28, 109–112.Google Scholar