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A Weak Limit Theorem for Generalized Jiřina Processes

Part of: Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  14 July 2016

Yuqiang Li
Yuqiang Li*
Affiliation:
East China Normal University
*
∗Postal address: School of Finance and Statistics, East China Normal University, Shanghai 200241, China. Email address: [email protected]
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Abstract

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In this paper we prove that a sequence of scaled generalized Jiřina processes can converge weakly to a nonlinear diffusion process with Lévy jumps under certain conditions.

Keywords

Generalized branching processesweak convergencenonlinear diffusion processes

MSC classification

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J60: Diffusion processes 60E07: Infinitely divisible distributions; stable distributions
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 2 , June 2009 , pp. 453 - 462
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.Google Scholar
[2] Feller, W. (1951). Diffusion processes in genetics. In Proc. 2nd Berkeley Symp. Math. Statist. Prob., University of California Press, Berkeley, CA, pp. 227–246.Google Scholar
[3] Höpfner, R. (1985). On some classes of population-size-dependent Galton–Watson processes. J. Appl. Prob. 22, 25–36.Google Scholar
[4] Ispány, M., Pap, G. and Zuijlen, M. C. A. (2005). Fluctuation limit of branching processes with immigration and estimation of the means. Adv. Appl. Prob. 37, 523–538.CrossRefGoogle Scholar
[5] Jiřina, M. (1960). Stochastic branching processes with continuous-state space. Czech. Math. J. 8, 292–313.CrossRefGoogle Scholar
[6] Jiřina, M. (1969). On Feller's branching diffusion processes. Časopis. Pěst. Mat. 94, 84–90.Google Scholar
[7] Kawazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Theory Prob. Appl. 16, 34–51.CrossRefGoogle Scholar
[8] Klebaner, F. (1984). Geometric rate of growth in population-size-dependent branching processes. J. Appl. Prob. 21, 40–49.Google Scholar
[9] Li, Y. (2006). On a continuous-state population-size-dependent branching process and its extinction. J. Appl. Prob. 43, 195–207.CrossRefGoogle Scholar
[10] Li, Y. (2009). Uniqueness of solutions of martingale problems with degenerate coefficients. Preprint. Available at http://www.paper.edu.cn/en/paper.php?serial_number=200901-298.Google Scholar
[11] Li, Z.-H. (2000). Orenstein–Uhlenbeck type processes and branching processes with immigration. J. Appl. Prob. 37, 627–634.Google Scholar
[12] Li, Z. (2006). A limit theorem for discrete Galton–Watson branching processes with immigration. J. Appl. Prob. 43, 289–295.Google Scholar
[13] Lipow, C. (1977). Limiting diffusions for population-size-dependent branching processes. J. Appl. Prob. 14, 14–24.Google Scholar
[14] Rosenkranz, G. (1985). Diffusion approximation of controlled branching processes with random environments. Stoch. Anal. Appl. 3, 363–377.CrossRefGoogle Scholar
[15] Stroock, D. (1975). Diffusion processes associated Lévy generators. Z. Wahrscheinlichkeitsth. 32, 209–244.Google Scholar