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2 results

  • Total internal and external lengths of the Bolthausen-Sznitman coalescent

  • Part of
  • Götz Kersting, Juan Carlos Pardo, Arno Siri-Jégousse
  • Journal:
    Journal of Applied Probability / Volume 51 / Issue A / December 2014
    Published online by Cambridge University Press:
    30 March 2016, pp. 73-86
    Print publication:
    December 2014
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  • In this paper we study a weak law of large numbers for the total internal length of the Bolthausen-Sznitman coalescent, thereby obtaining the weak limit law of the centered and rescaled total external length; this extends results obtained in Dhersin and Möhle (2013). An application to population genetics dealing with the total number of mutations in the genealogical tree is also given.

  • Asymptotic hitting probabilities for the Bolthausen-Sznitman coalescent

  • Part of
  • Martin Möhle
  • Journal:
    Journal of Applied Probability / Volume 51 / Issue A / December 2014
    Published online by Cambridge University Press:
    30 March 2016, pp. 87-97
    Print publication:
    December 2014
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  • The probability h(n, m) that the block counting process of the Bolthausen-Sznitman n-coalescent ever visits the state m is analyzed. It is shown that the asymptotic hitting probabilities h(m) = limn→∞h(n, m), m ∈ N, exist and an integral formula for h(m) is provided. The proof is based on generating functions and exploits a certain convolution property of the Bolthausen-Sznitman coalescent. It follows that h(m) ∼ 1/log m as m → ∞. An application to linear recursions is indicated.