On the other hand, if T
◦ 3
= t
◦ 3
, we have to add the new leaf to the first branch point to get t
◦ 4
. Thus P
T
◦ 3
, T
◦ 4
= t
◦ 3
, t
◦ 4
= α − γ
3 − α
P T
◦ 3
= t
◦ 3
= α − γ2 − 2α + γ
2 − α3 − α .
It is easy to check that PT
◦ 4,−1
, T
◦ 4
= t
◦ 3
, t
◦ 4
6= PT
◦ 3
, T
◦ 4
= t
◦ 3
, t
◦ 4
if γ 6= 1 − α, which means that the alpha-gamma model is then not strongly sampling consistent.
3 Dislocation measures and asymptotics of alpha-gamma trees
3.1 Dislocation measures associated with the alpha-gamma-splitting rules
Theorem 2 claims that the alpha-gamma trees are sampling consistent, which we proved in Section 2.5, and identifies the integral representation of the splitting rule in terms of a dislocation measure,
which we will now establish.
Proof of Theorem 2. In the binary case γ = α, the expression simplifies and the result follows from
Ford [12], see also [16, Section 5.2]. In the multifurcating case
γ α, we first make some rearrangement for the coefficient of the sam- pling consistent splitting rules of alpha-gamma trees identified in Proposition 10:
γ + 1 − α − γ 1
nn − 1 X
i6= j
n
i
n
j
= n + 1 − α − γn − α − γ
nn − 1
γ + 1 − α − γ
X
i6= j
A
i j
+ 2
k
X
i=1
B
i
+ C
, where
A
i j
= n
i
− αn
j
− α n + 1 − α − γn − α − γ
, B
i
= n
i
− αk − 1α − γ n + 1 − α − γn − α − γ
, C
= k − 1α − γkα − γ
n + 1 − α − γn − α − γ .
Notice that B
i
p
PD
∗
α,−α−γ
n
1
, . . . , n
k
simplifies to n
i
− αk − 1α − γ n + 1 − α − γn − α − γ
α
k−2
Γk − 1 − γα Z
n
Γ1 − γα Γ
α
n
1
. . . Γ
α
n
k
= Z
n+2
Z
n
n + 1 − α − γn − α − γ α
k−1
Γk − γα Z
n+2
Γ1 − γα Γ
α
n
1
. . . Γ
α
n
i
+ 1 . . . Γ
α
n
k
= e
Z
n+2
e Z
n
p
PD
∗
α,−α−γ
n
1
, . . . , n
i
+ 1, . . . , n
k
, 1,
416
where Γ
α
n = Γn−αΓ1−α and e Z
n
= Z
n
αΓ1−γαΓn−α−γ is the normalisation constant in 4 for
ν = PD
∗ α,−γ−α
. The latter can be seen from [17, Formula 17], which yields e
Z
n
= X
{A
1
,...,A
k
}∈P
[n]
\{[n]}
α
k−1
Γk − 1 − γα Γn − α − γ
k
Y
i=1
ΓA
i
− α Γ1 − α
, whereas Z
n
is the normalisation constant in 2 and hence satisfies Z
n
= X
{A
1
,...,A
k
}∈P
[n]
\{[n]}
α
k−2
Γk − 1 − γα Γ1 − γα
k
Y
i=1
ΓA
i
− α Γ1 − α
. According to 4,
p
PD
∗
α,−α−γ
n
1
, . . . , n
k
= 1
e Z
n
Z
S
↓
X
i
1
,...,i
k
≥1 distinct
k
Y
l=1
s
n
l
i
l
PD
∗ α,−α−γ
ds.
Thus,
k
X
i=1
B
i
p
PD
∗
α,−α−γ
n
1
, . . . , n
k
= 1
e Z
n
Z
S
↓
X
i
1
,...,i
k
≥1 distinct
k
Y
l=1
s
n
l
i
l
X
u∈{i
1
,...,i
k
},v6∈{i
1
,...,i
k
}
s
u
s
v
PD
∗ α,−α−γ
ds
Similarly, X
i6= j
A
i j
p
PD
∗
α,−α−γ
n
1
, . . . , n
k
= 1
e Z
n
Z
S
↓
X
i
1
,...,i
k
≥1 distinct
k
Y
l=1
s
n
l
i
l
X
u,v∈{i
1
,...,i
k
}:u6=v
s
u
s
v
PD
∗ α,−α−γ
ds
C p
PD
∗
α,−α−γ
n
1
, . . . , n
k
= 1
e Z
n
Z
S
↓
X
i
1
,...,i
k
≥1 distinct
k
Y
l=1
s
n
l
i
l
X
u,v6∈{i
1
,...,i
k
}:u6=v
s
u
s
v
PD
∗ α,−α−γ
ds,
Hence, the EPPF p
seq α,γ
n
1
, . . . , n
k
of the sampling consistent splitting rule takes the following form: n + 1 − α − γn − α − γZ
n
nn − 1Γ
α
n
γ + 1 − α − γ
X
i6= j
A
i j
+ 2
k
X
i=1
B
i
+ C
p
PD
∗
α,γ
n
1
, . . . , n
k
= 1
Y
n
Z
S
↓
X
i
1
,...,i
k
≥1 distinct
k
Y
l=1
s
n
l
i
l
γ + 1 − α − γ X
i6= j
s
i
s
j
PD
∗ α,−α−γ
ds, 11
where Y
n
= nn − 1Γ
α
nαΓ1 − γαΓn + 2 − α − γ is the normalisation constant. Hence, we have
ν
α,γ
ds = γ + 1 − α − γ
P
i6= j
s
i
s
j
PD
∗ α,−α−γ
ds.
417
3.2 The alpha-gamma model when
