IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v128y2018i3p939-962.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414917301679
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2017.06.012?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. Christoph, Gerd & Schreiber, Karina, 1998. "Discrete stable random variables," Statistics & Probability Letters, Elsevier, vol. 37(3), pages 243-247, March.
    3. 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.
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Christoph, Gerd & Schreiber, Karina, 2000. "Scaled Sibuya distribution and discrete self-decomposability," Statistics & Probability Letters, Elsevier, vol. 48(2), pages 181-187, June.
    2. Wojdyła, Tomasz & Kimmel, Marek & Bobrowski, Adam, 2011. "Time to the MRCA of a sample in a Wright–Fisher model with variable population size," Theoretical Population Biology, Elsevier, vol. 80(4), pages 265-271.
    3. Klebanov, Lev B. & Slámová, Lenka, 2013. "Integer valued stable random variables," Statistics & Probability Letters, Elsevier, vol. 83(6), pages 1513-1519.
    4. Buddana Amrutha & Kozubowski Tomasz J., 2014. "Discrete Pareto Distributions," Stochastics and Quality Control, De Gruyter, vol. 29(2), pages 143-156, December.
    5. Vladimir V. Vinogradov & Richard B. Paris, 2017. "On Poisson–Tweedie mixtures," Journal of Statistical Distributions and Applications, Springer, vol. 4(1), pages 1-23, December.
    6. Michael Grabchak, 2022. "Discrete Tempered Stable Distributions," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1877-1890, September.
    7. Wirtz, Johannes & Wiehe, Thomas, 2019. "The Evolving Moran Genealogy," Theoretical Population Biology, Elsevier, vol. 130(C), pages 94-105.
    8. Rémillard Bruno & Theodorescu Radu, 2000. "Inference Based On The Empirical Probability Generating Function For Mixtures Of Poisson Distributions," Statistics & Risk Modeling, De Gruyter, vol. 18(4), pages 349-366, April.
    9. Di Noia, Antonio & Marcheselli, Marzia & Pisani, Caterina & Pratelli, Luca, 2023. "Censoring heavy-tail count distributions for parameter estimation with an application to stable distributions," Statistics & Probability Letters, Elsevier, vol. 202(C).
    10. Tomasz J. Kozubowski & Krzysztof Podgórski, 2018. "A generalized Sibuya distribution," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(4), pages 855-887, August.
    11. Sagitov, Serik, 2013. "Linear-fractional branching processes with countably many types," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 2940-2956.
    12. Zhu, Rong & Joe, Harry, 2009. "Modelling heavy-tailed count data using a generalised Poisson-inverse Gaussian family," Statistics & Probability Letters, Elsevier, vol. 79(15), pages 1695-1703, August.
    13. Amaury Lambert & Florian Simatos, 2015. "Asymptotic Behavior of Local Times of Compound Poisson Processes with Drift in the Infinite Variance Case," Journal of Theoretical Probability, Springer, vol. 28(1), pages 41-91, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:128:y:2018:i:3:p:939-962. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.