Scaling limit of senile RRW 111
where the convergence is in
D[0, 1], R
d
, U
× D[0, 1], R, J
1
, and where U and J
1
denote the uniform and Skorokhod J
1
topologies respectively. Proof of Theorem 2.1 assuming Proposition 4.1. Since ⌊g
−1 α
n⌋ is a sequence of positive integers such that ⌊g
−1 α
n⌋ → ∞ and n g
α
⌊g
−1 α
n⌋ → 1 as n → ∞, it follows from 4.1 that as n → ∞,
S
τ
⌊⌊g−1 α n⌋t⌋
Æ
p d−p
⌊g
−1 α
n⌋ ,
τ
⌊⌊g
−1 α
n⌋t⌋
n
=⇒ B
d
t, V
α
t ,
4.2 in
D[0, 1], R
d
, U
× D[0, 1], R, J
1
. Let
Y
n
t = S
τ
⌊⌊g−1 α n⌋t⌋
Æ
p d−p
⌊g
−1 α
n⌋ ,
and T
n
t = τ
⌊⌊g
−1 α
n⌋t⌋
n ,
4.3 and let T
−1 n
t := inf{s ≥ 0 : T
n
s t} = inf{s ≥ 0 : τ
⌊⌊g
−1 α
n⌋s⌋
nt}. It follows e.g. see the proof of Theorem 1.3 in [1] that Y
n
T
−1 n
t =⇒ B
d
V
−1 α
t in
D[0, 1], R
d
, U
. Thus, S
τ
⌊⌊g−1 α n⌋T
−1 n
t⌋
Æ
p d−p
g
−1 α
n =⇒ B
d
V
−1 α
t. 4.4
By definition of T
−1 n
, we have τ
⌊⌊g
−1 α
n⌋T
−1 n
t⌋−1
≤ nt ≤ τ
⌊⌊g
−1 α
n⌋T
−1 n
t⌋
and hence |S
⌊nt⌋
−S
τ
⌊⌊g−1 α n⌋T
−1 n
t⌋
| ≤ 3. Together with 4.4 and the fact that g
−1 α
n ⌊g
−1 α
n⌋ → 1, this proves Theorem 2.1.
5 Proof of Proposition 4.1
The proof of Proposition 4.1 is broken into two parts. The first part is the observation that the marginal processes converge, i.e. that the time-changed walk and the time-change converge to
B
d
t and V
α
t respectively, while the second is to show that these two processes are asymptoti- cally independent.
5.1 Convergence of the time-changed walk and the time-change.
Lemma 5.1. Suppose that assumptions A1 and A2 hold for some
α 0, then as n → ∞, S
τ
⌊nt⌋
Æ
p d−p
n =⇒ B
d
t in D[0, 1], R
d
, U , and
τ
⌊nt⌋
g
α
n =⇒ V
α
t in D[0, 1], R, J
1
. 5.1 Proof. The first claim is the conclusion of Proposition 3.1, so we need only prove the second claim.
Recall that τ
n
= 1 + P
n i=1
T
i
where the T
i
are i.i.d. with distribution T . Since g
α
n → ∞, it is enough to show convergence of
τ
∗ ⌊nt⌋
= τ
⌊nt⌋
− 1 g
α
n. For processes with independent and identically distributed increments, a standard result of Sko-
rokhod essentially extends the convergence of the one-dimensional distributions to a functional central limit theorem. When E[T ] exists, convergence of the one-dimensional marginals
τ
∗ ⌊nt⌋
nE[T ] =⇒ t is immediate from the law of large numbers. The case
α 1 is well known, see for example [6, Section XIII.6] and [16, Section 4.5.3]. The case where
α = 1 but 2.1 is not summable is perhaps less well known. Here the result is immediate from the following lemma.
112 Electronic Communications in Probability
Lemma 5.2. Let T
k
≥ 0 be independent and identically distributed random variables satisfying 2.1 and 2.2 with
α = 1. Then for each t ≥ 0, τ
∗ ⌊nt⌋
n ℓn
P
−→ t. 5.2
Lemma 5.2 is a corollary of the following weak law of large numbers due to Gut [7].
Theorem 5.3 [7], Theorem 1.3. Let X
k
be i.i.d. random variables and S
n
= P
n k=1
X
k
. Let g
n
= n
1 α
ℓn for n ≥ 1, where α ∈ 0, 1] and ℓn is slowly varying at infinity. Then S
n
− nE
X I
{|X |≤g
n
}
g
n
P
−→ 0, as n → ∞,
5.3 if and only if nP|X |
g
n
→ 0. Proof of Lemma 5.2. Note that
E
T I
{T ≤n ℓn}
=
⌊n ℓn⌋
X
j=1
P n
ℓn ≥ T ≥ j =
⌊n ℓn⌋
X
j=1
P T ≥ j − ⌊n
ℓn⌋PT ≥ nℓn. 5.4
Now by assumption A2b, n
n ℓn
E
T I
{|T |≤n ℓn}
=
P
⌊n ℓn⌋
j=1
P T ≥ j
ℓn −
⌊n ℓn⌋
ℓn P
T ≥ n ℓn
∼ P
⌊n ℓn⌋
j=1
j
−1
L j ℓn
− ⌊n
ℓn⌋ ℓn
n ℓn
−1
Ln ℓn → 1.
5.5
Theorem 5.3 then implies that n ℓn
−1
τ
n
P
−→ 1, from which it follows immediately that n
ℓn
−1
τ
⌊nt⌋
= n ℓn
−1
⌊nt⌋ ℓ⌊nt⌋⌊nt⌋ℓ⌊nt⌋
−1
τ
⌊nt⌋
P
−→ t. 5.6
This completes the proof of Lemma 5.2, and hence Lemma 5.1.
5.2 Asymptotic Independence
