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General branching processes as Markov fields

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  • Jagers, Peter

Abstract

The natural Markov structure for population growth is that of genetics: newborns inherit types from their mothers, and given those they are independent of the history of their earlier ancestry. This leads to Markov fields on the space of sets of individuals, partially ordered by descent. The structure of such fields is investigated. It is proved that this Markov property implies branching, i.e. the conditional independence of disjoint daughter populations. The process also has the strong Markov property at certain optional sets of individuals. An intrinsic martingale (indexed by sets of individuals) is exhibited, that catches the stochastic element of population development. The deterministic part is analyzed by Markov renewal methods. Finally the strong Markov property found is used to divide the population into conditionally independent subpopulations. On those classical limit theory for sums of independent random variables can be used to catch the asymptotic population development, as real time passes.

Suggested Citation

  • Jagers, Peter, 1989. "General branching processes as Markov fields," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 183-212, August.
    • Handle: RePEc:eee:spapps:v:32:y:1989:i:2:p:183-212
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    Cited by:

    1. Bertoin, Jean, 2006. "Different aspects of a random fragmentation model," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 345-369, March.
    2. Kyprianou, Andreas E. & Pardo, Juan Carlos, 2012. "An optimal stopping problem for fragmentation processes," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1210-1225.
    3. Shankar Bhamidi & Steven N. Evans & Arnab Sen, 2012. "Spectra of Large Random Trees," Journal of Theoretical Probability, Springer, vol. 25(3), pages 613-654, September.
    4. Kyprianou, A. E., 1999. "A note on branching Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 82(1), pages 1-14, July.
    5. Iksanov, Alexander & Meiners, Matthias, 2015. "Rate of convergence in the law of large numbers for supercritical general multi-type branching processes," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 708-738.
    6. Kolesko, Konrad & Sava-Huss, Ecaterina, 2023. "Limit theorems for discrete multitype branching processes counted with a characteristic," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 49-75.
    7. Braunsteins, Peter & Decrouez, Geoffrey & Hautphenne, Sophie, 2019. "A pathwise approach to the extinction of branching processes with countably many types," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 713-739.
    8. Chen, Dayue & de Raphélis, Loïc & Hu, Yueyun, 2018. "Favorite sites of randomly biased walks on a supercritical Galton–Watson tree," Stochastic Processes and their Applications, Elsevier, vol. 128(5), pages 1525-1557.
    9. Komjáthy, Júlia & Lodewijks, Bas, 2020. "Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1309-1367.

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