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Size-biased and conditioned random splitting trees

Author

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  • Geiger, Jochen

Abstract

Random splitting trees share the striking independence properties of the continuous time binary Galton-Watson tree. They can be represented by Poisson point processes and their contour processes are strong Markov processes. Here we study splitting trees conditioned on extinction, respectively non-extinction as well as size-biased splitting trees. We give explicit probabilistic constructions of those trees by decomposing them into independent parts along a distinguished line of descent. The size-biased trees are shown to have stationary contour processes. Splitting trees are related to M/G/1-queuing systems which allows to translate the results on the trees into statements on the queues.

Suggested Citation

  • Geiger, Jochen, 1996. "Size-biased and conditioned random splitting trees," Stochastic Processes and their Applications, Elsevier, vol. 65(2), pages 187-207, December.
    • Handle: RePEc:eee:spapps:v:65:y:1996:i:2:p:187-207
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    Cited by:

    1. Champagnat, Nicolas & Lambert, Amaury, 2012. "Splitting trees with neutral Poissonian mutations I: Small families," Stochastic Processes and their Applications, Elsevier, vol. 122(3), pages 1003-1033.
    2. Cécile Delaporte, 2015. "Lévy Processes with Marked Jumps I: Limit Theorems," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1468-1499, December.
    3. Champagnat, Nicolas & Lambert, Amaury, 2013. "Splitting trees with neutral Poissonian mutations II: Largest and oldest families," Stochastic Processes and their Applications, Elsevier, vol. 123(4), pages 1368-1414.

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