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Criterion for unlimited growth of critical multidimensional stochastic models

Part of: Markov processes

Published online by Cambridge University Press:  11 January 2017

Etienne Adam
Etienne Adam*
Affiliation:
Centre de Mathématiques Appliquées
*
* Postal address: Centre de Mathématiques Appliquées, Ecole Polytechnique, CNRS, Université Paris-Saclay, route de Saclay, 91128 Palaiseau, France. Email address: etienne.adam@polytechnique.edu
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Abstract

We give a criterion for unlimited growth with positive probability for a large class of multidimensional stochastic models. As a by-product, we recover the necessary and sufficient conditions for recurrence and transience for critical multitype Galton–Watson with immigration processes and also significantly improve some results on multitype size-dependent Galton–Watson processes.

Keywords

Lyapunov functionmartingalestochastic difference equationcritical multitype Galton–Watson process with immigrationmultitype size-dependent Galton–Watson process

MSC classification

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 4 , December 2016 , pp. 972 - 988
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] González, M., Martínez, R. and Mota, M. (2005). On the unlimited growth of a class of homogeneous multitype Markov chains. Bernoulli 11, 559–570.CrossRefGoogle Scholar
[2] Höpfner, R. (1985). On some classes of population-size-dependent Galton–Watson processes. J. Appl. Prob. 22, 25–36.CrossRefGoogle Scholar
[3] Horn, R. A. and Johnson, C. R. (YEAR). Matrix Analysis, 2nd edn. Cambridge University Press.Google Scholar
[4] Jagers, P. and Sagitov, S. (2000). The growth of general population-size-dependent branching processes year by year. J. Appl. Prob. 37, 1–14.Google Scholar
[5] Kawazu, K. (1976). On multitype branching processes with immigration. In Proceedings of the Third Japan–USSR Symposium on Probability Theory (Tashkent, 1975; Lecture Notes Math. 550), Springer, Berlin, pp.270–275.CrossRefGoogle Scholar
[6] Kersting, G. (1984). On population-size-dependent branching processes. Adv. Appl. Prob. 23, 614–625.CrossRefGoogle Scholar
[7] Klebaner, F. C. (1984). On population-size-dependent branching processes. Adv. Appl. Prob. 16, 30–55.CrossRefGoogle Scholar
[8] Klebaner, F. C. (1989). Linear growth in near-critical population-size-dependent multitype Galton–Watson processes. J. Appl. Prob. 26, 431–445.CrossRefGoogle Scholar
[9] Klebaner, F. C. (1991). Asymptotic behavior of near-critical multitype branching processes. J. Appl. Prob. 28, 512–519, 962.Google Scholar
[10] Lamperti, J. (1960). Criteria for the recurrence or transience of stochastic process. I. J. Math. Anal. Appl. 1, 314–330.CrossRefGoogle Scholar
[11] Lin, Z. and Bai, Z. (2010). Probability Inequalities. Science Press, Beijing.Google Scholar
[12] Seneta, E. (2006). Non-Negative Matrices and Markov Chains, 2nd edn. Springer, New York.Google Scholar