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Asymptotic results on the length of coalescent trees. (English) Zbl 1141.60007

Summary: We give the asymptotic distribution of the length of partial coalescent trees for Beta and related coalescents. This allows us to give the asymptotic distribution of the number of (neutral) mutations in the partial tree. This is a first step to study the asymptotic distribution of a natural estimator of DNA mutation rate for species with large families.

MSC:

60F05 Central limit and other weak theorems
60G52 Stable stochastic processes
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
05C05 Trees
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[1] Basdevant, A.-L. and Goldschmidt, C. (2007). Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent. Available at http://fr.arXiv.org/abs/0706.2808. · Zbl 1190.60006
[2] Berestycki, J., Berestycki, N. and Schweinsberg, J. (2007). Beta-coalescents and continuous stable random trees. Ann. Probab. 35 1835-1887. · Zbl 1129.60067 · doi:10.1214/009117906000001114
[3] Berestycki, J., Berestycki, N. and Schweinsberg, J. (2007). Small time properties of Beta-coalescents. Ann. Inst. H. Poincaré Probab. Statist. · Zbl 1129.60067 · doi:10.1214/009117906000001114
[4] Bertoin, J. (1996). Lévy Processes . Cambridge Univ. Press. · Zbl 0861.60003
[5] Bertoin, J. and Le Gall, J.-F. (2006). Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 147-181. · Zbl 1110.60026
[6] Birkner, M., Blath, J., Capaldo, M., Etheridge, A., Möhle, M., Schweinsberg, J. and Wakolbinger, A. (2005). Alpha-stable branching and Beta-coalescents. Electron. J. Probab. 10 303-325. · Zbl 1066.60072
[7] Bolthausen, E. and Sznitman, A.-S. (1998). On Ruelle’s probability cascades and an abstract cavity method. Comm. Math. Phys. 197 247-276. · Zbl 0927.60071 · doi:10.1007/s002200050450
[8] Boom, J. D. G., Boulding, E. G. and Beckenbach, A. T. (1994). Mitochondrial DNA variation in introduced populations of pacific oyster, Crassostrea Gigas, in British Columbia. Can. J. Fish. Aquat. Sci. 51 1608-1614.
[9] Breiman, L. (1992). Probability . SIAM, Philadelphia. · Zbl 0753.60001
[10] Drmota, M., Iksanov, A., Möhle, M. and Rösler, U. (2007). Asymptotic results about the total branch length of the Bolthausen-Sznitman coalescent. Stoch. Process. Appl. 117 1404-1421. · Zbl 1129.60069 · doi:10.1016/j.spa.2007.01.011
[11] Eldon, B. and Wakeley, J. (2006). Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics 172 2621-2633.
[12] Feller, W. (1971). An Introduction to Probability Theory and Its Applications . II. Wiley, New York. · Zbl 0219.60003
[13] Gnedin, A. and Yakubovich, Y. (2007). On the number of collisions in \Lambda -coalescents. Available at http://arXiv.org/abs/0704.3902. · Zbl 1190.60061
[14] Iksanov, A. and Möhle, M. (2007). On a random recursion related to absorption times of death Markov chains. Available at http://arXiv.org/abs/0710.5826.
[15] Kallenberg, O. (2002). Foundations of Modern Probability , 2nd ed. Springer, New York. · Zbl 0996.60001
[16] Kimura, M. (1969). The number of heterozygous nucleotide sites maintained in a finite population due to steady flux of mutations. Genetics 61 893-903.
[17] Kingman, J. F. C. (1982). The coalescent. Stochastic Process. Appl. 13 235-248. · Zbl 0491.60076 · doi:10.1016/0304-4149(82)90011-4
[18] Kingman, J. F. C. (2000). Origins of the coalescent 1974-1982. Genetics 156 1461-1463.
[19] Möhle, M. (2006). On the number of segregating sites for populations with large family sizes. Adv. in Appl. Probab. 38 750-767. · Zbl 1112.92046 · doi:10.1239/aap/1158685000
[20] Mukherjea, A., Rao, M. and Suen, S. (2006). A note on moment generating functions. Statist. Probab. Lett. 76 1185-1189. · Zbl 1092.60008 · doi:10.1016/j.spl.2005.12.026
[21] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870-1902. · Zbl 0963.60079 · doi:10.1214/aop/1022677552
[22] Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 1116-1125. · Zbl 0962.92026 · doi:10.1239/jap/1032374759
[23] Schweinsberg, J. (2003). Coalescent processes obtained from super critical Galton-Watson processes. Stochastic Process. Appl. 106 107-139. · Zbl 1075.60571 · doi:10.1016/S0304-4149(03)00028-0
[24] Watterson, G. A. (1975). On the number of segregating sites in genetical models without recombination. Theoret. Population Biology 7 256-276. · Zbl 0294.92011 · doi:10.1016/0040-5809(75)90020-9
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