Abstract
The $\Xi$-coalescent processes were initially studied by Möhle and Sagitov (2001), and introduced by Schweinsberg (2000) 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 Berstycki, Berestycki and Limic (2009), 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.
Highlights
Kingman’s coalescent [15; 16] is one of the central models of mathematical population genetics
The Kingman coalescent emerges in the scaling limit of genealogies of all evolutionary models that are asymptotically linked to Fisher-Wright diffusions
One can identify each of the active ancestral lineages, at any particular time, with a unique equivalence class of {1, 2, . . . , m} that consists of all the individuals that descend from this lineage
Summary
Kingman’s coalescent [15; 16] is one of the central models of mathematical population genetics. The Ξ-coalescent processes were initially studied by Möhle and Sagitov [19], and introduced by Schweinsberg [24] in their full generality It is shown in [19] that any limit of genealogies arising from a population genetics model with exchangeable reproduction mechanism must be a Ξ-coalescent. In the Λ-coalescent setting, weaker asymptotic results (than (1)) on N Ξ/v = N Λ/v can be deduced by an entirely different approach, based on the theory of Lévy processes and superprocesses This link was initially discovered in [6; 7] in the special case of so-called Beta-coalescents, and recently understood in the context of general Λ-coalescents in [5].
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