On the Speed of Coming Down from Infinity for $\Xi$-Coalescent Processes

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.