## Electronic Communications in Probability

### A spectral decomposition for the block counting process of the Bolthausen-Sznitman coalescent

#### Abstract

A spectral decomposition for the generator and the transition probabilities of the block counting process of the Bolthausen-Sznitman coalescent is derived. This decomposition is closely related to the Stirling numbers of the first and second kind. The proof is based on generating functions and exploits a certain factorization property of the Bolthausen-Sznitman coalescent. As an application we derive a formula for the hitting probability $h(i,j)$ that the block counting process of the Bolthausen-Sznitman coalescent ever visits state $j$ when started from state $i\ge j$. Moreover, explicit formulas are derived for the moments and the distribution function of the absorption time $\tau_n$ of the Bolthausen-Sznitman coalescent started in a partition with $n$ blocks. We provide an elementary proof for the well known convergence of $\tau_n-\log\log n$ in distribution to the standard Gumbel distribution. It is shown that the speed of this convergence is of order $1/\log n$.

#### Article information

Source
Electron. Commun. Probab., Volume 19 (2014), paper no. 47, 11 pp.

Dates
Accepted: 23 July 2014
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465316749

Digital Object Identifier
doi:10.1214/ECP.v19-3464

Mathematical Reviews number (MathSciNet)
MR3246966

Zentralblatt MATH identifier
1334.60157

Rights

#### Citation

Möhle, Martin; Pitters, Helmut. A spectral decomposition for the block counting process of the Bolthausen-Sznitman coalescent. Electron. Commun. Probab. 19 (2014), paper no. 47, 11 pp. doi:10.1214/ECP.v19-3464. https://projecteuclid.org/euclid.ecp/1465316749

#### References

• Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. 9th printing. Dover, New York.
• Bolthausen, E.; Sznitman, A.-S. On Ruelle's probability cascades and an abstract cavity method. Comm. Math. Phys. 197 (1998), no. 2, 247-276.
• Drmota, Michael; Iksanov, Alex; Moehle, Martin; Roesler, Uwe. Asymptotic results concerning the total branch length of the Bolthausen-Sznitman coalescent. Stochastic Process. Appl. 117 (2007), no. 10, 1404-1421.
• Freund, F.; MÃ¶hle, M. On the time back to the most recent common ancestor and the external branch length of the Bolthausen-Sznitman coalescent. Markov Process. Related Fields 15 (2009), no. 3, 387-416.
• Gladstien, K. The characteristic values and vectors for a class of stochastic matrices arising in genetics. SIAM J. Appl. Math. 34 (1978), no. 4, 630-642.
• Goldschmidt, Christina; Martin, James B. Random recursive trees and the Bolthausen-Sznitman coalescent. Electron. J. Probab. 10 (2005), no. 21, 718-745 (electronic).
• Henard, O. The fixation line. Preprint. (2013) arXiv:1307.0784
• Kingman, J. F. C. On the genealogy of large populations. Essays in statistical science. J. Appl. Probab. 1982, Special Vol. 19A, 27-43.
• Kukla, Jonas; Miller, Luke; Pitters, Helmut. A spectral decomposition for the Kingman and the Bolthausen-Sznitman coalescent. In preparation.
• MÃ¶hle, Martin. On hitting probabilities of beta coalescents and absorption times of coalescents that come down from infinity, ALEA, Lat. Am. J. Probab. Stat. 11 (2014), 141-159. number not yet available
• MÃ¶hle, Martin. Asymptotic hitting probabilities for the Bolthausen-Sznitman coalescent, J. Appl. Probab. 51A (2014), to appear.
• MÃ¶hle, Martin and Pitters, Helmut. Absorption time and tree length of the Kingman coalescent and the Gumbel distribution. Preprint (2014).
• Norris, J. R. Markov chains. Reprint of 1997 original. Cambridge Series in Statistical and Probabilistic Mathematics, 2. Cambridge University Press, Cambridge, 1998. xvi+237 pp. ISBN: 0-521-48181-3
• Pitman, Jim. Coalescents with multiple collisions. Ann. Probab. 27 (1999), no. 4, 1870-1902.
• Sagitov, Serik. The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 (1999), no. 4, 1116-1125.
• Tavare, Simon. Line of descent and genealogical processes, and their applications in population genetics models. Theor. Popul. Biol. 26 (1984), no. 2, 119-164.