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1
)
N(v)
F(v)
V
F
((a, b
]
)
=
(
−
v
∈{
a
1
,b
1
}×···×{
a
d
,b
d
}
d
d
=
sgn
u
j
F(u
1
,...,u
d
)
−
sgn
u
j
F(a
1
,u
2
...,u
d
)
≥
0
.
j
=
2
j
=
2
Similarly for
u
1
<
0. This gives the lower bound with
a
1
=
0.
Let
I
={
i
}⊂{
1
,...,d
}
. Then,
d
sgn
u
j
F(u
1
,...,u
i
,...u
d
)
j
=
1
⎛
⎝
j
⎞
⎠
F(n
sgn
u
1
,...,u
i
,...,n
sgn
u
d
)
≤
sgn
u
i
lim
n
→∞
sgn
u
j
c
∈
I
⎛
⎝
j
⎞
⎠
F(v
1
,...,u
i
,...,v
d
)
≤
sgn
u
i
lim
n
→∞
sgn
v
j
c
c
∞}
|
I
|
∈
I
(v
j
)
j
∈
I
∈{−
n,
c
sgn
u
i
F
i
(u
i
)
= |
u
i
|
.
=
Since
i
∈{
is arbitrary, we obtain the upper bound.
To simplify notation, we suppose 0
1
,...,d
}
u
i
≤
=
≤
v
i
,
i
1
,...,d
. The general case can
be treated similarly. We have
|
F(v
1
,...,v
d
)
−
F(u
1
,...,u
d
)
|
= |
V
F
((
0
,v
1
]×···×
(
0
,v
d
]
)
−
V
F
((
0
,u
1
]×···×
(
0
,u
d
]
)
|
d
V
F
(
i
i
−
1
d
−
≤
lim
a
−
a,
∞]
×
(u
i
,v
i
]×
(
−
a,
∞]
→∞
i
=
1
d
F
i
(v
i
)
F
i
(u
i
)
=
−
i
=
1
d
=
1
|
v
i
−
u
i
|
.
i
=
We also need tail integrals of Lévy processes.
d
Definition 14.2.5
Let
X
be a Lévy process with state space
R
and Lévy measure
ν
.
d
The
tail integral
of
X
is the function
U
: R
\{
0
}→R
given by
sgn
(x
j
)ν
I(x
j
)
,
d
d
U(x
1
,...,x
d
)
=
i
=
1
j
=
1
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