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Continuum random trees and branching processes with immigration

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  • Duquesne, Thomas

Abstract

We study a genealogical model for continuous-state branching processes with immigration with a (sub)critical branching mechanism. This model allows the immigrants to be on the same line of descent. The corresponding family tree is an ordered rooted continuum random tree with a single infinite end defined by two continuous processes denoted by and that code the parts at resp. the left and the right hand side of the infinite line of descent of the tree. These processes are called the left and the right height processes. We define their local time processes via an approximation procedure and we prove that they enjoy a Ray-Knight property. We also discuss the important special case corresponding to the size-biased Galton-Watson tree in a continuous setting. In the last part of the paper we give a convergence result under general assumptions for rescaled discrete left and right contour processes of sequences of Galton-Watson trees with immigration. We also provide a strong invariance principle for a sequence of rescaled Galton-Watson processes with immigration that also holds in the supercritical case.

Suggested Citation

  • Duquesne, Thomas, 2009. "Continuum random trees and branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 119(1), pages 99-129, January.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:1:p:99-129
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    References listed on IDEAS

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    1. Grimvall, Anders, 1973. "On the transition from a Markov chain to a continuous time process," Stochastic Processes and their Applications, Elsevier, vol. 1(4), pages 335-368, October.
    2. Bingham, N. H., 1976. "Continuous branching processes and spectral positivity," Stochastic Processes and their Applications, Elsevier, vol. 4(3), pages 217-242, August.
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    Cited by:

    1. David Clancy, 2021. "The Gorin–Shkolnikov Identity and Its Random Tree Generalization," Journal of Theoretical Probability, Springer, vol. 34(4), pages 2386-2420, December.
    2. Abraham, Romain & Delmas, Jean-François & He, Hui, 2015. "Pruning of CRT-sub-trees," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1569-1604.
    3. Archer, Eleanor & Pein, Anne, 2023. "Parabolic Anderson model on critical Galton–Watson trees in a Pareto environment," Stochastic Processes and their Applications, Elsevier, vol. 159(C), pages 34-100.

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