a
1
y
1
, ~x
I
; 3
+
:
1
y
o
2
b
1
b
a
1
y
1
, ~x
I
; 3
−
:
1
y
o
1
b
2
b
+
1
y
o
1
b
2
b
Figure 7: Schematic representations of a
1
y
1
,
~x
I
; 3
±
. The random variable B
δ
b
N +1
, y
1
; Cb
N
in 4.51 for N = 1 becomes B
b
2
, y
1
; Cb
1
in bold dashed lines. The expansion for ˜
B
N
y
1
,
~x
I
is completed by using 4.25, 4.30 and 4.46 as follows. For convenience, we let
˜ M
1
b
1
X = ˜ M
b
1
o
,b
1
; {o}
X . 4.47
Moreover, for a measurable function X v that depends explicitly on v ∈ Λ, we abuse notation to
write ˜
M
N +1
b
N +1
X b
N
= X
b
N
p
b
N
M
N
b
N
˜ M
b
N +1
b
N
,b
N +1
;˜ C
N −1
X b
N
N ≥ 1, 4.48
where b
N
in the left-hand side is a dummy variable that has already been summed over, as in the right-hand side. Using this notation, as well as the abbreviations
C
N
= Cb
N
, ˜
C
e
N
= ˜ C
e
b
N
, C
+
= {b
N
} and
C
−
= ˜ C
N −1
, 4.49
we define, for N ≥ 0,
φ
N
y
1
, y
2
±
= X
b
N +1
,e b
N +1
6=e
p
b
N +1
p
e
˜ M
N +1
b
N +1
1
{H
t y1
b
N
,e; C
±
in ˜ C
e N
}
B
δ
b
N +1
, y
1
; C
N
B
δ
e, y
2
; ˜ C
e
N
, 4.50
and, for ℓ = 3, 4,
a
N
y
1
, ~x
I
; ℓ = a
N
y
1
, ~x
I
; ℓ
+
−
1
{ j
I
1}
a
N
y
1
, ~x
I
; ℓ
−
, 4.51
where a
N
y
1
,
~x
I
; 3
±
= X
b
N +1
p
b
N +1
˜ M
N +1
b
N +1
1
H
t y1
b
N
,
~x
I
; C
±
B
δ
b
N +1
, y
1
; C
N
, 4.52
a
N
y
1
, ~x
I
; 4
±
= − X
b
N +1
,e b
N +1
6=e
p
b
N +1
p
e
˜ M
N +1
b
N +1
1
{H
t y1
b
N
,e; C
±
in ˜ C
e N
}
B
δ
b
N +1
, y
1
; C
N
Ae, ~x
I
; ˜ C
e
N
. 4.53
837
φ
1
y
1
, y
2
+
:
2
y
e
o
1
y
b
1
b
2
φ
1
y
1
, y
2
−
:
2
y
e
o
1
b
1
y
b
2
+
2
y
e
o
1
b
1
y
b
2
a
1
y
1
, ~x
I
; 4
+
:
1
y
e
o
b
1
b
2
a
1
y
1
, ~x
I
; 4
−
:
1
y
e
o
b
2
b
1
+
e
o
1
b
1
y
b
2
Figure 8: Schematic representations of
φ
1
y
1
, y
2
±
and a
1
y
1
, ~x
I
; 4
±
. The random
variables B
δ
b
N +1
, y
1
; Cb
N
, B
δ
e, y
2
; ˜ C
e
b
N
and Ae, ~x
I
; ˜ C
e
b
N
in 4.50–4.53 become B
b
2
, y
1
;
Cb
1
, B e, y
2
; ˜ C
e
b
1
and A e,
~x
I
; ˜ C
e
b
1
, respectively depicted in bold dashed lines, when N = 1.
These functions correspond to the second term in the left-hand side of 4.46 and the first and second terms in the right-hand side of 4.46, respectively, when 4.46 is substituted into 4.25.
We note that the functions 4.51 depend on I via the indicator
1
{ j
I
1}
, which is due to the fact that both terms in the right-hand side of 4.30 contribute to the case of j
I
1, while for the case of j
I
= 1, the contribution is only from the first term that has been treated as the case of A = {b
N
}. Now we arrive at
˜ B
N
y
1
, ~x
I
− a
N
y
1
, ~x
I
; 3 = X
y
2
φ
N
y
1
, y
2 +
−
1
{ j
I
1}
φ
N
y
1
, y
2 −
τ~x
I
− y
2
+ a
N
y
1
, ~x
I
; 4, 4.54
where a
N
y
1
,
~x
I
;
ℓ for ℓ = 3, 4 turn out to be error terms. This extracts the factor τ~x
I
− y
2
from ˜
B
N
y, ~x
I
.
838
4.3 Summary of the expansion for A
