2.6 LLN for Y
Similarly as in 2.29, define Z
L k
= r
L
Z
τ
L k
− Z
τ
L k−1
,
k ∈ N. 2.43
In this section we prove the LLN for these increments and this will imply the LLN in 2.8. Proof. By Lemma 2.1, we have
¯ P
µ,0,0
T
L k
, Z
L k
∈ · | H
k−1
− µ
L
·
tv
≤ ΦL a.s.
∀ k ∈ N, 2.44
where µ
L
A × B = ¯ P
µ,0,0
T
L 1
∈ A, Z
L 1
∈ B ∀ A ⊂ r
L
N , B ⊂ r
L
H .
2.45 Therefore, by Lemma 2.4, there exists an i.i.d. sequence of random variables
e T
L k
, e Z
L k
, ∆
L k
k∈N
2.46 on r
L
N × r
L
H × {0, 1}, where e
T
L k
, e Z
L k
is distributed according to µ
L
and ∆
L k
is Bernoulli distributed with parameter ΦL, and also a sequence of random variables
b T
L k
, b Z
L k
k∈N
, 2.47
such that ∆
L k
is independent of b T
L k
, b Z
L k
and T
L k
, Z
L k
= 1 − ∆
L k
e T
L k
, e Z
L k
+ ∆
L k
b T
L k
, b Z
L k
. 2.48
Let z
L
= ¯ E
µ,0,0
Z
L 1
, 2.49
which is finite by Lemma 2.2 because |Z
L 1
| ≤ T
L 1
.
Lemma 2.5. There exists a sequence of numbers δ
L L∈N
, satisfying lim
L→∞
δ
L
= 0, such that lim sup
n→∞
1 n
n
X
k=1
Z
L k
− z
L
δ
L
¯ P
µ,0,0
− a.s. 2.50
Proof. With the help of 2.48 we can write 1
n
n
X
k=1
Z
L k
= 1
n
n
X
k=1
e Z
L k
− 1
n
n
X
k=1
∆
L k
e Z
L k
+ 1
n
n
X
k=1
∆
L k
b Z
L k
. 2.51
By independence, the first term in the r.h.s. of 2.51 converges ¯ P
µ,0,0
-a.s. to z
L
as L → ∞. Hölder’s inequality applied to the second term gives, for
α, α
′
1 with α
−1
+ α
′−1
= 1, 1
n
n
X
k=1
∆
L k
e Z
L k
≤ 1
n
n
X
k=1
∆
L k
α
′ 1
α′
1 n
n
X
k=1
e Z
L k
α
1 α
. 2.52
599
Hence, by Lemma 2.2 and the inequality |e Z
L k
| ≤ e T
L k
compare 2.29 and 2.43, we have lim sup
n→∞
1 n
n
X
k=1
∆
L k
e Z
L k
≤ ΦL
1 α′
M α
1 α
¯ P
µ,0,0
− a.s. 2.53
It remains to analyze the third term in the r.h.s. of 2.51. Define the filtration
b G
k
= σ
∆
L i
, b Z
L i
: i k . Since |∆
L k
b Z
L k
| ≤ |Z
L k
|, it follows from Lemma 2.2 that M
α ≥ ¯ E
µ,0,0
|∆
L k
b Z
L k
|
α
| b G
k
= ΦL ¯ E
µ,0,0
|b Z
L k
|
α
| b G
k
a.s. 2.54
Next, put b Z
∗L k
= ¯ E
µ,0,0
b Z
L k
| b G
k
and note that M
n
=
n
X
k=1
∆
L k
k b
Z
L k
− b Z
∗L k
2.55 is a mean-zero martingale w.r.t. the filtration b
G
n
. By the Burkholder-Davis-Gundy inequality
Williams [23], 14.18, it follows that, for β = α ∧ 2,
¯ E
µ,0,0
sup
n∈N
M
n β
≤ Cβ ¯
E
µ,0,0
X
k∈N
[∆
L k
b Z
L k
− b Z
∗L k
]
2
k
2 β2
≤ Cβ X
k∈N
¯ E
µ,0,0
|∆
L k
b Z
L k
− b Z
∗L k
|
β
k
β
≤ C
′
β, 2.56
for some constants C β, C
′
β ∞. Hence M
n
a.s. converges to an integrable random variable as n → ∞, and by Kronecker’s lemma Williams [23], 12.7,
lim
n→∞
1 n
n
X
k=1
∆
L k
b Z
L k
− b Z
∗L k
= 0 a.s.
2.57 Moreover, if ΦL
0, then by Jensen’s inequality and 2.54 we have |b
Z
∗L k
| ≤ h
¯ E
µ,0,0
bZ
L k
α
| b G
k
i
1 α
≤ M
α ΦL
1 α
a.s. Hence
1 n
n
X
k=1
∆
L k
b Z
∗L k
≤ M
α ΦL
1 α
1 n
n
X
k=1
∆
L k
. 2.58
As n → ∞, the r.h.s. converges ¯ P
µ,0,0
-a.s. to M α
1 α
ΦL
1 α′
. Therefore, choosing
δ
L
= 2M
α
1 α
ΦL
1 α′
, we get the claim. Finally, since e
Z
L k
≥ r
L
and 1
n
n
X
k=1
T
L k
= t
L
= ¯ E
µ,0,0
T
L 1
¯ P
µ,0,0
− a.s., 2.59
600
Lemma 2.5 yields lim sup
n→∞ 1
n
P
n k=1
Z
L k
1 n
P
n k=1
T
L k
− z
L
t
L
C
1
δ
L
¯ P
µ,0,0
− a.s. 2.60
for some constant C
1
∞ and L large enough. By 2.29 and 2.43, the quotient of sums in the l.h.s. equals Z
τ
L n
τ
L n
. It therefore follows from a standard interpolation argument that lim sup
n→∞
Z
n
n −
z
L
t
L
C
2
δ
L
¯ P
µ,0,0
− a.s. 2.61
for some constant C
2
∞ and L large enough. This implies the existence of the limit lim
L→∞
z
L
t
L
, as well as the fact that lim
n→∞
Z
n
n = u ¯ P
µ,0,0
-a.s., which in view of 2.13 is equivalent to the statement in 2.8 with u = v, 1.
2.7 From discrete to continuous time
Kategori