432 Electronic Communications in Probability
i N
n d
→ N ; ii We have λM
c
= 0 and the following two conditions hold: a
′
lim
m →m
lim sup
n →∞
|n[PM
m,n
x − PM
m −1,n
x] − λM
x
| = 0, for any x 0, x
6∈ D
′
, b
′
lim
m →m
lim sup
n →∞
|n[PN
m,n
∈ M, M
m,n
x − PN
m −1,n
∈ M, M
m −1,n
x] −λM ∩ M
x
| = 0, for any x 0, x 6∈ D
′
and for any set M ∈ J
x,λ
.
Remark 2.6. Note that
PM
m,n
x − PM
m −1,n
x = P max
1 ≤ j≤m−1
|X
j,n
| ≤ x, |X
m,n
| x.
Remark 2.7. Suppose that m = 1 in Theorem 2.5. One can prove that in this case, the limit N is
a Poisson process of intensity ν given by: νB = λ{µ ∈ M
p
E\{o}; µB = 1}, ∀B ∈ B.
3 The Proofs
3.1 Proof of Theorem 2.4
Before giving the proof, we need some preliminary results.
Lemma 3.1. Let E be a LCCB space and M = ∩
d i=1
{µ ∈ M
p
E; µB
i
≥ l
i
} for some B
i
∈ B, l
i
≥ 1 integers and d
≥ 1. Then: i M is closed with respect to the vague topology;
ii ∂ M ⊂ ∪
d i=1
{µ ∈ M
p
E; µ∂ B
i
0}.
Proof: Note that ∂ M
⊂ ∪
d i=1
∂ M
i
, where M
i
= {µ ∈ M
p
E; µB
i
≥ l
i
}. Since the finite intersec- tion of closed sets is a closed set, it suffices to consider the case d = 1, i.e. M =
{µ ∈ M
p
E; µB ≥ l
} for some B ∈ B and l ≥ 1. i Let µ
n n
⊂ M be such that µ
n v
→ µ. If µ∂ B = 0, then µ
n
B → µB, and since µ
n
B ≥ l for all n, it follows that µB
≥ l. If not, we proceed as in the proof of Lemma 3.15 of [ 21
]. Let B
δ
be a δ-swelling of B. Then S = {δ ∈ 0, δ
]; µ∂ B
δ
0} is a countable set. By the previous argument, µB
δ
≥ l for all δ ∈ 0, δ ]\S. Let δ
n n
∈ 0, δ ]\S be such that δ
n
↓ 0. Since µB
δ
n
≥ l for all n, and µB
δ
n
↓ µB, it follows that µB ≥ l, i.e. µ ∈ M. ii By part i, ∂ M = ¯
M \M
o
= M ∩ M
o c
. We will prove that ∂ M ⊂ {µ ∈ M; µ∂ B 0}, or
equivalently A :=
{µ ∈ M; µ∂ B = 0} ⊂ M
o
. Since M
o
is the largest open set included in M and A ⊂ M, it suffices to show that A is open. Let
µ ∈ A and µ
n n
⊂ M
p
E be such that µ
n v
→ µ. Then µ
n
B → µB, and since µB ≥ l and {µ
n
B}
n
are integers, it follows that µ
n
B ≥ l for all n ≥ n
1
, for some n
1
. On the other hand, µ
n
∂ B → µ∂ B, since ∂ B ∈ B and µ∂ B = 0 note that ∂ ∂ B = ∂ B. Since µ∂ B = 0 and
{µ
n
∂ B}
n
are integers, it follows that µ
n
∂ B = 0 for all n ≥ n
2
, for some n
2
. Hence µ
n
∈ A for all n ≥ max{n
1
, n
2
}.
Conditions for Point Process Convergence 433
Lemma 3.2. Let E be a LCCB space and Q
n n
, Q be probability measures on M
p
E. Let B
Q
be the class of all sets B
∈ B which satisfy: Q
{µ ∈ M
p
E; µ∂ B 0} = 0, and
J
Q
be the class of sets M = ∩
d i=1
{µ ∈ M
p
E; µB
i
≥ l
i
} for some B
i
∈ B
Q
, l
i
≥ 1 integers and d
≥ 1. Then Q
n w
→ Q if and only if Q
n
M → QM for all M ∈ J
Q
.
Proof: Let N
n n
, N be point processes on E, defined on a probability space Ω, F , P, such that
P ◦ N
−1 n
= Q
n
for all n, and P ◦ N
−1
= Q. Note that B
Q
= B
N
:= {B ∈ B; N ∂ B = 0 a.s.}.
By definition, N
n d
→ N if and only if Q
n w
→ Q. By Theorem 4.2 of [ 15
], N
n d
→ N if and only if N
n
B
1
, . . . , N
n
B
d d
→ N B
1
, . . . , N B
d
for any B
1
, . . . , B
d
∈ B
N
and for any d ≥ 1. Since these random vectors have values in Z
d +
, the previous convergence in distribution is equivalent to:
PN
n
B
1
= l
1
, . . . , N
n
B
d
= l
d
→ PN B
1
= l
1
, . . . , N B
d
= l
d
for any l
1
, . . . , l
d
∈ Z
+
, which is in turn equivalent to PN
n
B
1
≥ l
1
, . . . , N
n
B
d
≥ l
d
→ PN B
1
≥ l
1
, . . . , N B
d
≥ l
d
for any l
1
, . . . , l
d
∈ Z
+
. Finally, it suffices to consider only integers l
i
≥ 1 since, if there exists a set I
⊂ {1, . . . , d} such that l
i
= 0 for all i ∈ I and l
i
≥ 1 for i 6∈ I, then PN
n
B
1
≥ l
1
, . . . , N
n
B
d
≥ l
d
= PN
n
B
i
≥ l
i
, i 6∈ I → PN B
i
≥ l
i
, i 6∈ I = PN B
1
≥ l
1
, . . . , N B
d
≥ l
d
.
Proof of Theorem 2.4: Note that {max
j ≤r
n
|X
j,n
| x} = {N
r
n
,n
∈ M
x
}. Suppose that i holds. As in the proof of Theorem 3.6 of [
2 ], it follows that λM
c
= 0 and a holds. Moreover, we have P
n,x w
→ P
x
where P
n,x
and P
x
are probability measures on M
p
E defined by:
P
n,x
M = k
n
PN
r
n
,n
∈ M ∩ M
x
k
n
PN
r
n
,n
∈ M
x
and P
x
M = λM ∩ M
x
λM
x
. Therefore, P
n,x
M → P
x
M for any M ∈ M
p
E with P
x
∂ M = 0. Since k
n
PN
r
n
,n
∈ M
x
→ λM
x
by a, it follows that k
n
PN
r
n
,n
∈ M ∩ M
x
→ λM ∩ M
x
, 3
for any M ∈ M
p
E with λ∂ M ∩ M
x
= 0. In particular, 3 holds for a set M =
∩
d i=1
{µ ∈ M
p
E; µB
i
≥ l
i
}, with B
i
∈ B
x,λ
, l
i
≥ 1 integers and d
≥ 1. To see this, note that by Lemma 3.1, ∂ M ∩ M
x
⊂ ∪
d i=1
{µ ∈ M
x
; µ∂ B
i
0}, and hence λ∂ M ∩ M
x
≤
d
X
i=1
λ{µ ∈ M
x
; µ∂ B
i
0} = 0. Suppose that ii holds. As in the proof of Theorem 3.6 of [
2 ], it suffices to show that P
n,x w
→ P
x
. This follows by Lemma 3.2, since the class of sets B
∈ B which satisfy: P
x
{µ ∈ M
p
E; µ∂ B 0} = 0 coincides with
B
x,λ
.
434 Electronic Communications in Probability
3.2 Proof of Theorem 2.5
