~x y; {o}. In the remainder of this paper, we will specialise to the case where v = o and
where the superscript n of M
n
denotes the number of involved nested expectations, and, for n ≥ 0,
we abbreviate ˜ C
b
n+1
b
n
= ˜ C
n
, where we use the convention that b
= v , which is the initial vertex
in M
N +1
v
, ~x
J
; C
. Let
A
N
v , ~x
J
; C = M
N +1
v
, ~x
J
; C
1, B
N
v , y; C =
X
b= · ,y
M
N +1
v
,b; C
1 p
b
, 3.20
which are both nonnegative and agree with 3.16–3.17 when N = 0. We note that A
N
v , ~x
J
; C =
B
N
v , y; C = 0 for N
ǫ min
j ∈J
t
x
j
− t
v
, since, by the recursive definition 3.19, the operation M
N +1
eats up at least N time-units where one time-unit is ǫ.
We now resume the expansion of the right-hand side of 3.18. As we notice, we have Pv
C
−→ ~x
J
again in the right-hand side of 3.18, but now with v and C being replaced by b and ˜
C
b
v ,
respectively. Applying 3.18 to its own right-hand side, we obtain
P v
C
−→ ~x
J
= A
v , ~x
J
; C
− A
1
v , ~x
J
; C
+ X
y
B v , y; C − B
1
v , y; C τ~x
J
− y
+ X
b
2
p
b
2
M
2
v
,b
2
; C
P b
2
˜ C1
− → ~x
J
. 3.21
Define
Av , ~x
J
; C =
∞
X
N =0
−1
N
A
N
v , ~x
J
; C,
Bv , y; C =
∞
X
N =0
−1
N
B
N
v , y; C.
3.22 By repeated application of 3.18 to 3.21 until the remainder vanishes which happens after a
finite number of iterations, see below 3.20, we arrive at the following conclusion, which is the linear expansion for the generalised r-point function:
Proposition 3.6 Linear expansion. For any J
6= ∅, λ ≤ λ
c
and
~x
J
∈ Λ
|J|
,
P v
C
−→ ~x
J
= Av , ~x
J
; C +
X
y
Bv , y ; C
τ~x
J
− y. 3.23
Applying Proposition 3.6 to the r-point function in 3.3, we arrive at
τ~x
J
= A
~x
J
+ X
y
By τ~x
J
− y, 3.24
where we abbreviate A
~x
J
= Ao, ~x
J
; {o},
By = Bo, y; {o},
3.25 and similarly for A
N
~x
J
= A
N