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A Multilevel birth-death particle system and its continuous diffusion

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Yadong Wu
Yadong Wu*
Affiliation:
Carleton University
*
∗ Postal address: Department of Mathematics and Statistics, Carleton University, Ottawa, Canada K15 5B6.
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Abstract

In this paper we introduce a multilevel birth-death particle system and consider its diffusion approximation which can be characterized as a M([R+)-valued process. The tightness of rescaled processes is proved and we show that the limiting M(R+)-valued process is the unique solution of the M([R+)-valued martingale problem for the limiting generator. We also study the moment structures of the limiting diffusion process.

Keywords

MULTILEVEL BIRTH-DEATH PROCESSTIGHTNESSMARTINGALE PROBLEMDIFFUSION APPROXIMATIONMEASURE-VALUED PROCESSMOMENT STRUCTURE

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J60: Diffusion processes 60J57: Multiplicative functionals 60F05: Central limit and other weak theorems

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 3 , September 1993 , pp. 549 - 569
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

Supported partially by a scholarship from the Faculty of Graduate Studies and Research of Carleton University, Ottawa, and by a NSERC Canada grant to D. Dawson.

References

[1] Dawson, D. A. (1972) Stochastic evolution equations. Math. Biosci. 15, 287–316.Google Scholar
[2] Dawson, D. A. (1975) Stochastic evolution equations and related measure-valued processes. J. Multivariate Anal. 5, 1–52.Google Scholar
[3] Dawson, D. A. (1977) The critical measure diffusion. Z. Wahrscheinlichkeitsth. 40, 125–145.Google Scholar
[4] Dawson, D. and Hochberg, K (1991). A multilevel branching model. Adv. Appl. Prob. 23, 701–715.Google Scholar
[5] Dawson, D. and Ivanoff, B. (1978) Branching diffusions and random measures. In Branching Processes, ed. Joffe, A. and Ney, P., pp. 61–104. Marcel Dekker, New York.Google Scholar
[6] El Karoui, N. and Roelly, S. (1991) Propriétés de martingales, explosion et représentation de Lévy-Khintchine d'une classe de processus de branchement à valeurs mesures. Stoch. Proc. Appl. 38, 239–266.Google Scholar
[7] Ethier, S. N. and Kurtz, T. G. (1985) Markov processes: Characterization and Convergence. Wiley, New York.Google Scholar
[8] Fitzsimmons, P. J. (1988) Construction and regularity of measure-valued branching processes. Israel J. Math. 464, 337–361.CrossRefGoogle Scholar
[9] Gorostiza, L. (1988). The demographic variation process of branching random fields. J. Multivariate Anal. 25, 174–200.Google Scholar
[10] Gorostiza, L. G. and Lopez-Mimbela, J. A. (1990) The multitype measure branching process. Adv. Appl. Prob. 22, 49–67.Google Scholar
[11] Karlin, S. (1975) A First Cource in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
[12] Konno, N. and Shiga, T. (1988) Stochastic differential equations for some measure-valued diffusions. Prob. Theory Rel. Fields 78, 201–225.Google Scholar
[13] ØKsendal, B. (1985) Stochastic Differential Equations, an Introduction with Applications. Springer-Verlag, New York.Google Scholar
[14] Roelly, S. and Rouault, A. (1990) Construction et propriétés de martingales des branchements spatiaux interactifs. Internat. Statist. Rev. 58, 43–65.Google Scholar
[15] Stroock, O. W. and Varadhan, S. R. S. (1979) Multidimensional Diffusion Processes. Springer-Verlag, New York.Google Scholar
[16] Watanabe, S. (1968) A limit theorem of branching processes and continuous state branching. J. Math. Kyoto Univ. 8, 141–167.Google Scholar
[17] Wu, Y. (1991) Dynamic Particle Systems and Multilevel Measure Branching Processes. Ph.D. thesis, Carleton University.Google Scholar