  The $\X$-coalescent processes were initially studied by M\"ohle 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 $\X$-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 a
deterministic "speed" function -- an almost sure small time asymptotic to the
number of blocks process, for a large class of $\X$-coalescents that come down
from infinity.
