be of independent interest and presents a “preview” of the main result as Theorem 1. Section 3 contains a “matured” statement of the main result Theorem 10, followed by a discussion of some
of its immediate consequences, and of the significance of a certain “regularity hypothesis”, while Section 4 is devoted to the proof of Theorem 10.
2 Definitions and preliminaries
2.1 Notation
In this section we recall some standard notation, as well as gather less standard notation that will be frequently used.
Denote the set of real numbers by R and set R
+
= 0, ∞. For a, b ∈ R, denote by a ∧ b resp. a ∨ b the minimum resp. maximum of the two numbers. Let
∆ := {x
1
, x
2
, . . . : x
1
≥ x
2
≥ . . . ≥ 0, X
i
x
i
≤ 1}, 2
be the infinite unit simplex. If x = x
1
, x
2
, . . . ∈ ∆ and c ∈ R, let c
x = c x
1
, c x
2
, . . .. Denote by 0 the zero 0, 0 . . . , in ∆.
Let N := {1, 2, . . .}, and P be the set of partitions of N. Furthermore, for n ∈ N denote by P
n
the set of partitions of [n] := {1, . . . , n}.
If f is a function, defined in a left-neighborhood s − ǫ, s of a point s, denote by f s− the left limit
of f at s. Given two functions f , g : R
+
→ R
+
, write f = Og if lim sup f x gx ∞, f = og
if lim sup f x gx = 0, and f ∼ g if lim f xgx = 1. Furthermore, write f = Θg if both
f = Og and g = O f . The point at which the limits are taken is determined from the context. If F = F
t
, t ≥ 0 is a filtration, and T is a stopping time relative to F , denote by F
T
the standard filtration generated by T , see for example [13], page 389.
For ν a finite or σ-finite measure on ∆ or on [0, 1], denote the support of ν by suppν.
2.2 Ξ-coalescents
Let Ξ be a finite measure on ∆, and write Ξ = Ξ
+ aδ ,
where a ≥ 0 and Ξ 0, 0, . . . = 0. As noted in [24], we may assume without loss of generality that
Ξ is a probability measure. The Ξ-coalescent driven by the above Ξ is a Markov process Π
t
, t ≥ 0 with values in P the set of partitions of N, characterized in the following way. If n ∈ N, then the
restriction Π
n t
, t ≥ 0 of Π
t
, t ≥ 0 to [n] is a Markov chain, taking values in P
n
, such that while Π
n t
consists of b blocks, any given k
1
-tuple, k
2
-tuple,. . ., and k
r
-tuple of its blocks here P
r i=1
k
i
≤ b and k
i
≥ 2, i = 1, . . . , r merge simultaneously each forming one new block at rate λ
b;k
1
,...,k
r
;¯ s
= Z
∆
P
¯ s
l=0
P
i
1
,...,i
r+l
¯ s
l
x
k
1
i
1
· · · x
k
r
i
r
x
i
r+1
· · · x
i
r+l
1 − P
∞ i=1
x
i ¯
s−l
P
∞ i=1
x
2 i
Ξdx,
220
where ¯ s := b −
P
r i=1
k
i
is the number of blocks that do not participate in the merger event, and where the sum
P
i
1
,...,i
r+l
in the above summation stands for the infinite sum P
∞ i
1
=1
P
∞ i
2
=1,i
2
6=i
1
. . . P
∞ i
r+l
=1,i
r+l
6∈{i
1
,...,i
r+l−1
}
over r + l different indices. It is easy to verify that each such coalescent process has the same rate of pairwise merging
λ
2;2;0
= Ξ∆ = 1. 3
2.3 Preview of the small-time asymptotics
