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On the block counting process and the fixation line of the Bolthausen–Sznitman coalescent

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  • Kukla, Jonas
  • Möhle, Martin

Abstract

The block counting process and the fixation line of the Bolthausen–Sznitman coalescent are analyzed. It is shown that these processes, properly scaled, converge in the Skorohod topology to the Mittag-Leffler process and to Neveu’s continuous-state branching process respectively as the initial state tends to infinity. Strong relations to Siegmund duality, Mehler semigroups and self-decomposability are pointed out. Furthermore, spectral decompositions for the generators and transition probabilities of the block counting process and the fixation line of the Bolthausen–Sznitman coalescent are provided leading to explicit expressions for functionals such as hitting probabilities and absorption times.

Suggested Citation

  • Kukla, Jonas & Möhle, Martin, 2018. "On the block counting process and the fixation line of the Bolthausen–Sznitman coalescent," Stochastic Processes and their Applications, Elsevier, vol. 128(3), pages 939-962.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:3:p:939-962
    DOI: 10.1016/j.spa.2017.06.012
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    References listed on IDEAS

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    1. Christoph, Gerd & Schreiber, Karina, 1998. "Discrete stable random variables," Statistics & Probability Letters, Elsevier, vol. 37(3), pages 243-247, March.
    2. Huillet, Thierry & Möhle, Martin, 2013. "On the extended Moran model and its relation to coalescents with multiple collisions," Theoretical Population Biology, Elsevier, vol. 87(C), pages 5-14.
    3. Pfaffelhuber, P. & Wakolbinger, A., 2006. "The process of most recent common ancestors in an evolving coalescent," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1836-1859, December.
    4. Sagitov, Serik, 1995. "A key limit theorem for critical branching processes," Stochastic Processes and their Applications, Elsevier, vol. 56(1), pages 87-100, March.
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