B, t = G B, t, G B, t} B, t} is relatively compact in D B, t. It follows that B, t → B, [[B]], G B, t in law in D B, t have the same law in D B, t have the same law. This can be B, t
Theorem 3.8. If g ∈ C
∞
R is bounded with bounded derivatives, then the sequence B, G
+ n
1, B, G
+ n
g, B converges to B, [[B]], 18 R
g
′′′
B ds + R
gBd[[B]] in the sense of finite- dimensional distributions on [0,
∞, where G
+ n
g, B, t := 1
p n
⌊nt⌋
X
j=1
gBt
j
h
3
n
1 6
∆B
j
, t
≥ 0, n
≥ 1.
Proof. The proof is exactly the same as the proof of Theorem 3.7, except
ǫ
j −1
must be everywhere replaced
ǫ
j
, and Lemma 3.2 v must used instead of Lemma 3.2 i v.
Proof of Theorem 3.1. We begin by observing the following general fact. Suppose U and V are càdlàg processes adapted to a filtration under which V is a semimartingale. Similarly, suppose
e U and e
V are càdlàg processes adapted to a filtration under which e V is a semimartingale. If the
processes U, V and e U, e
V have the same law, then U0V 0 + R
·
Us − dV s and e
U0e V 0 +
R
·
e Us
− d e V s have the same law. This is easily seen by observing that these integrals are the limit
in probability of left-endpoint Riemann sums. Now, let G
− n
and G
+ n
be as defined previously in this section. Define G
−
g, B, t = − 1
8 Z
t
g
′′′
Bs ds + Z
t
gBs d[[B]]
s
, G
+
g, B, t = 1
8 Z
t
g
′′′
Bs ds + Z
t
gBs d[[B]]
s
. Let
t = t
1
, . . . , t
d
, where 0 ≤ t
1
· · · t
d
. Let G
− n