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On the speed of coming down from infinity for \(\Xi \)-coalescent processes. (English) Zbl 1203.60111
Summary: The \(\Xi \)-coalescent processes were initially studied by Möhle and Sagitov [Ann. Probab. 29, No. 4, 1547–1562 (2001; Zbl 1013.92029)], and introduced by Schweinsberg [Electron. J. Probab. 5, Paper No. 12, 50 p., electronic only (2000; Zbl 0959.60065)] in their full generality. They arise in the mathematical population genetics as the complete class of scaling limits for genealogies of Cannings’ models. The \(\Xi \)-coalescents generalize \(\Lambda \)-coalescents, where now simultaneous multiple collisions of blocks are possible. The standard version starts with infinitely many blocks at time 0, and it is said to come down from infinity if its number of blocks becomes immediately finite, almost surely. This work builds on the technique introduced recently by J. Berstycki, N. Berestycki and V. Limic [Ann. Probab. 38, No. 1, 207–233 (2010; Zbl 1247.60110)], and exhibits deterministic “speed” function – an almost sure small time asymptotic to the number of blocks process, for a large class of \(\Xi \)-coalescents that come down from infinity.

MSC:
60J25 Continuous-time Markov processes on general state spaces
60F99 Limit theorems in probability theory
92D25 Population dynamics (general)
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