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Rank-dependent Galton‒Watson processes and their pathwise duals

Part of: Markov processes

Published online by Cambridge University Press:  01 February 2019

Serik Sagitov  and
Jonas Jagers
Serik Sagitov*
Affiliation:
Chalmers University of Technology and University of Gothenburg
Jonas Jagers*
Affiliation:
Chalmers University of Technology
*
Department of Mathematical Sciences, Chalmers University of Technology, 412 96 Gothenburg, Sweden. Email address: [email protected]
Department of Mathematical Sciences, Chalmers University of Technology, 412 96 Gothenburg, Sweden. Email address: [email protected]
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Abstract

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We introduce a modified Galton‒Watson process using the framework of an infinite system of particles labelled by (x,t), where x is the rank of the particle born at time t. The key assumption concerning the offspring numbers of different particles is that they are independent, but their distributions may depend on the particle label (x,t). For the associated system of coupled monotone Markov chains, we address the issue of pathwise duality elucidated by a remarkable graphical representation in which the trajectories of the primary Markov chains and their duals coalesce to form forest graphs on a two-dimensional grid.

Keywords

Graphical representationSiegmund's dualitydual forestlinear-fractional reproductionbirth‒death process

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue A: Branching and Applied Probability , December 2018 , pp. 229 - 239
Copyright
Copyright © Applied Probability Trust 2018 

References

[1]Asmussen, S. and Hering, H. (1983).Branching Processes.Birkhäuser,Boston, MA.Google Scholar
[2]Asmussen, S. and Sigman, K. (1996).Monotone stochastic recursions and their duals.Prob. Eng. Inf. Sci. 10,1–20.Google Scholar
[3]Athreya, K. B. and Ney, P. E. (1972).Branching Processes.Springer,New York.Google Scholar
[4]Bansaye, V. and Simatos, F. (2015).On the scaling limits of Galton‒Watson processes in varying environments.Electron. J. Prob. 20, 36pp.Google Scholar
[5]Bertoin, J. (2009).The structure of the allelic partition of the total population for Galton‒Watson processes with neutral mutations.Ann. Prob. 37,1502–1523.Google Scholar
[6]Braunsteins, P. and Hautphenne, S. (2017). Extinction in lower Hessenberg branching processes with countably many types. Preprint. Available at https://arxiv.org/abs/1706.02919v1.Google Scholar
[7]Diaconis, P. and Freedman, D. (1999).Iterated random functions.SIAM Rev. 41,45–76.Google Scholar
[8]Felipe, M. D. and Lambert, A. (2015).Time reversal dualities for some random forests.ALEA 12,399–426.Google Scholar
[9]Grey, D. R. (1988).Supercritical branching processes with density independent catastrophes.Math. Proc. Camb. Phil. Soc. 104,413–416.Google Scholar
[10]Grosjean, N. and Huillet, T. (2016).On a coalescence process and its branching genealogy.J. Appl. Prob. 53,1156–1165.Google Scholar
[11]Haccou, P.,Jagers, P. and Vatutin, V. A. (2005).Branching Processes: Variation, Growth and Extinction of Populations.Cambridge University Press.Google Scholar
[12]Jagers, P. (1974).Galton‒Watson processes in varying environments.J. Appl. Prob. 11,174–178.Google Scholar
[13]Jansen, S. and Kurt, N. (2014).On the notion(s) of duality for Markov processes.Prob. Surveys 11,59–120.Google Scholar
[14]Kendall, D. G. (1948).On the generalized `birth-and-death' process.Ann. Math. Statist. 19,1–15.Google Scholar
[15]Kersting, G. (2017). A unifying approach to branching processes in varying environments. Preprint. Available at https://arxiv.org/abs/1703.01960v6.Google Scholar
[16]Kimmel, M. and Axelrod, D. E. (2015).Branching Processes in Biology,2nd edn.Springer,New York.Google Scholar
[17]Klebaner, F. C. (1984).On population-size-dependent branching processes.Adv. Appl. Prob. 16,30–55.Google Scholar
[18]Klebaner, F. C.,Rösler, U. and Sagitov, S. (2007).Transformations of Galton‒Watson processes and linear fractional reproduction.Adv. Appl. Prob. 39,1036–1053.Google Scholar
[19]Klebaner, F. C. et al. (2011).Stochasticity in the adaptive dynamics of evolution: the bare bones.J. Biol. Dynam. 5,147–162.Google Scholar
[20]Mitov, K. V. and Omey, E. (2014).A branching process with immigration in varying environments.Commun. Statist. Theory Meth. 43,5211–5225.Google Scholar
[21]Möhle, M. (1999).The concept of duality and applications to Markov processes arising in neutral population genetics models.Bernoulli 5,761–777.Google Scholar
[22]Rahimov, I. (1995).Random Sums and Branching Stochastic Processes.Springer,New York.Google Scholar
[23]Sagitov, S. and France, T. (2017).Limit theorems for pure death processes coming down from infinity.J. Appl. Prob. 54,720–731.Google Scholar
[24]Sagitov, S. and Lindo, A. (2016).A special family of Galton-Watson processes with explosions. In Branching Processes and Their Applications (Lecture Notes Statist. 219),Springer, pp. 237–254.Google Scholar
[25]Sagitov, S. and Minuesa, C. (2017).Defective Galton-Watson processes.Stoch. Models 33,451–472.Google Scholar
[26]Sagitov, S. and Shaimerdenova, A. (2013).Extinction times for a birth-death process with weak competition.Lithuanian Math. J. 53,220–234.Google Scholar
[27]Siegmund, D. (1976).The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes.Ann. Prob. 4,914–924.Google Scholar
[28]Sturm, A. and Swart, J. M. (2018).Pathwise duals of monotone and additive Markov processes.J. Theoret. Prob. 31,932–983.Google Scholar
[29]Vatutin, V. A. (1977).A critical Galton-Watson branching process with emigration.Theory Prob. Appl. 22,465–481.Google Scholar
[30]Zubkov, A. M. (1970).A condition for the extinction of a bounded branching process.Math. Notes 8,472–477.Google Scholar