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Minimal clade size in the Bolthausen-Sznitman coalescent. (English) Zbl 1320.60132

Summary: In this article we show the asymptotics of distribution and moments of the size \(X_n\) of the minimal clade of a randomly chosen individual in a Bolthausen-Sznitman \(n\)-coalescent for \(n\to\infty\). The Bolthausen-Sznitman \(n\)-coalescent is a Markov process taking states in the set of partitions of \(\{1,\dots, n\}\), where \(1,\dots, n\) are referred to as individuals. The minimal clade of an individual is the equivalence class the individual is in at the time of the first coalescence event this individual participates in. We also provide exact formulae for the distribution of \(X_n\). The main tool used is the connection of the Bolthausen-Sznitman \(n\)-coalescent with random recursive trees introduced by C. Goldschmidt and J. B. Martin [Electron. J. Probab. 10, Paper No. 21, 718–745 (2005; Zbl 1109.60060)]. With it, we show that \(X_n-1\) is distributed as the size of a uniformly chosen table in a standard Chinese restaurant process with \(n-1\) customers.

MSC:

60J27 Continuous-time Markov processes on discrete state spaces
92D25 Population dynamics (general)
60C05 Combinatorial probability
05C80 Random graphs (graph-theoretic aspects)
60G09 Exchangeability for stochastic processes
60F05 Central limit and other weak theorems

Citations:

Zbl 1109.60060
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References:

[1] Arratia, R., Barbour, A. D. and Tavaré, S. (2003). Logarithmic Combinatorial Structures: A Probabilistic Approach . European Mathematical Society, Zürich. · Zbl 1040.60001
[2] Berestycki, J., Berestycki, N. and Schweinsberg, J. (2013). The genealogy of branching Brownian motion with absorption. Ann. Prob. 41 , 527-618. · Zbl 1304.60088 · doi:10.1214/11-AOP728
[3] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopedia Math. Appl. 27 ). Cambridge University Press. · Zbl 0617.26001
[4] Blum, M. G. B. and François, O. (2005). Minimal clade size and external branch length under the neutral coalescent. Adv. Appl. Prob. 37 , 647-662. · Zbl 1071.92020 · doi:10.1239/aap/1127483740
[5] Bolthausen, E. and Sznitman, A.-S. (1998). On Ruelle’s probability cascades and an abstract cavity method. Commun. Math. Phys. 197 , 247-276. · Zbl 0927.60071 · doi:10.1007/s002200050450
[6] Bovier, A. and Kurkova, I. (2007). Much ado about Derrida’s GREM. In Spin Glasses (Lecture Notes Math. 1900 ), Springer, Berlin, pp. 81-115. · Zbl 1116.82018 · doi:10.1007/978-3-540-40908-3_4
[7] Brunet, É. and Derrida, B. (2013). Genealogies in simple models of evolution. J. Statist. Mech. Theory Exp. 2013 , P01006.
[8] Brunet, É., Derrida, B., Mueller, A. H. and Munier, S. (2006). Noisy traveling waves: effect of selection on genealogies. Europhys. Lett. 76 , 1-7. · doi:10.1209/epl/i2006-10224-4
[9] Brunet, É., Derrida, B., Mueller, A. H. and Munier, S. (2007). Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization. Phys. Rev. E 76 , 041104.
[10] Caliebe, A., Neininger, R., Krawczak, M. and Rösler, U. (2007). On the length distribution of external branches in coalescence trees: genetic diversity within species. Theoret. Pop. Biol. 72 , 245-252. · Zbl 1123.92024 · doi:10.1016/j.tpb.2007.05.003
[11] DeLaurentis, J. M. and Pittel, B. G. (1985). Random permutations and Brownian motion. Pacific J. Math. 119 , 287-301. · Zbl 0578.60033 · doi:10.2140/pjm.1985.119.287
[12] Desai, M. M., Walczak, A. M. and Fisher, D. S. (2013). Genetic diversity and the structure of genealogies in rapidly adapting populations. Genetics 193 , 565-585.
[13] Dhersin, J.-S., Freund, F., Siri-Jégousse, A. and Yuan, L. (2013). On the length of an external branch in the beta-coalescent. Stoch. Process. Appl 123 , 1691-1715. · Zbl 1281.60069 · doi:10.1016/j.spa.2012.12.010
[14] Freund, F. and Möhle, M. (2009). On the time back to the most recent common ancestor and the external branch length of the Bolthausen-Sznitman coalescent. Markov Process. Relat. Fields 15 , 387-416. · Zbl 1203.60110
[15] Fu, Y. X. and Li, W. H. (1993). Statistical tests of neutrality of mutations. Genetics 133 , 693-709.
[16] Gnedin, A., Iksanov, A. and Möhle, M. (2008). On asymptotics of exchangeable coalescents with multiple collisions. J. Appl. Prob. 45 , 1186-1195. · Zbl 1159.60016 · doi:10.1239/jap/1231340242
[17] Goldschmidt, C. and Martin, J. B. (2005). Random recursive trees and the Bolthausen-Sznitman coalescent. Electron. J. Prob. 10 , 718-745. · Zbl 1109.60060 · doi:10.1214/EJP.v10-265
[18] Hansen, J. C. (1990). A functional central limit theorem for the Ewens sampling formula. J. Appl. Prob. 27 , 28-43. · Zbl 0704.92011 · doi:10.2307/3214593
[19] Hwang, H.-K. (1995). Asymptotic expansions for the Stirling numbers of the first kind. J. Combin. Theory A 71 , 343-351. · Zbl 0833.05005 · doi:10.1016/0097-3165(95)90010-1
[20] Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13 , 235-248. · Zbl 0491.60076 · doi:10.1016/0304-4149(82)90011-4
[21] Neher, R. and Hallatschek, O. (2013). Genealogies of rapidly adapting populations. Proc. Nat. Acad. Sci. USA 110 , 437-442.
[22] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27 , 1870-1902. · Zbl 0963.60079 · doi:10.1214/aop/1022677552
[23] Pitman, J. (2005). Combinatorial Stochastic Processes (Lecture Notes Math. 1875 ). Springer, Berlin.
[24] Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36 , 1116-1125. · Zbl 0962.92026 · doi:10.1239/jap/1032374759
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