Electron. Commun. Probab. 18 (2013), no. 85, 1–7.
DOI: 10.1214/ECP.v18-2918
ISSN: 1083-589X
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
The quenched limiting distributions of a
one-dimensional random walk in random scenery∗
Nadine Guillotin-Plantard†
Yueyun Hu‡
Bruno Schapira§
Abstract
For a one-dimensional random walk in random scenery (RWRS) on Z, we determine its quenched weak limits by applying Strassen [13]’s functional law of the iterated logarithm. As a consequence, conditioned on the random scenery, the onedimensional RWRS does not converge in law, in contrast with the multi-dimensional
case.
Keywords: Random walk in random scenery; Weak limit theorem; Law of the iterated logarithm; Brownian motion in Brownian Scenery; Strong approximation.
AMS MSC 2010: 60F05; 60G52.
Submitted to ECP on July 8, 2013, final version accepted on October 25, 2013.
1
Introduction
Random walks in random sceneries were introduced independently by Kesten and
Spitzer [9] and by Borodin [3, 4]. Let S = (Sn )n≥0 be a random walk in Zd starting at 0,
i.e., S0 = 0 and (Sn − Sn−1 )n≥1 is a sequence of i.i.d. Zd -valued random variables. Let
ξ = (ξx )x∈Zd be a field of i.i.d. real random variables independent of S . The field ξ is
called the random scenery. The random walk in random scenery (RWRS) K := (Kn )n≥0
is defined by setting K0 := 0 and, for n ∈ N∗ ,
Kn :=
n
X
ξSi .
(1.1)
i=1
We will denote by P the joint law of S and ξ . The law P is called the annealed law, while
the conditional law P(·|ξ) is called the quenched law.
Limit theorems for RWRS have a long history, we refer to [7] or [8] for a complete
review. Distributional limit theorems for quenched sceneries (i.e. under the quenched
law) are however quite recent. The first result in this direction that we are aware of was
obtained by Ben Arous and Černý [1], in the case of a heavy-tailed scenery and planar
√
random walk. In [7], quenched central limit theorems (with the usual n-scaling and
Gaussian law in the limit) were proved for a large class of transient random walks. More
∗ Research partially supported by ANR (MEMEMOII) 2010 BLAN 0125.
† Institut Camille Jordan, Université Lyon 1, France.
E-mail: nadine.guillotin@univ-lyon1.fr
‡ Département de Mathématiques (LAGA CNRS-UMR 7539) Université Paris 13, France.
E-mail: yueyun@math.univ-paris13.fr
§ Centre de Mathématiques et Informatique, Aix-Marseille Université, France.
E-mail: bruno.schapira@latp.univ-mrs.fr
Quenched CLT in dimension one
recently, in [8], the case of the planar random walk was studied, the authors proved a
quenched version of the annealed central limit theorem obtained by Bolthausen in [2].
In this note we consider the case of the simple symmetric random walk (Sn )n≥0 on
Z, the random scenery (ξx )x∈Z is assumed to be centered with finite variance equal to
one and there exists some δ > 0 such that E(|ξ0 |2+δ ) < ∞. We prove that under these
assumptions, there is no quenched distributional limit theorem for K . In the sequel, for
−∞ ≤ a < b ≤ ∞, we will denote by AC([a, b] → R) the set of absolutely continuous
functions defined on the interval [a, b] with values in R. Recall that if f ∈ AC([a, b] →
R), then the derivative of f (denoted by f˙) exists almost everywhere and is Lebesgue
integrable on [a, b]. Define
Z
n
K := f ∈ AC(R → R) : f (0) = 0,
∞
o
(f˙(x))2 dx ≤ 1 .
∗
(1.2)
−∞
Theorem 1. For P-a.e. ξ , under the quenched probability P (. | ξ), the process
K̃n :=
Kn
,
(2n3/2 log log n)1/2
n > ee ,
does not converge in law. More precisely, for P-a.e. ξ , under the quenched probability P (. | ξ), the limit points of the law of K̃n , as n → ∞, under the topology of weak
convergence of measures, are equal to the set of the laws of random variables in ΘB ,
with
nZ ∞
o
f (x)dL1 (x) : f ∈ K∗ ,
ΘB :=
(1.3)
−∞
where (L1 (x), x ∈ R) denotes the family of local times at time 1 of a one-dimensional
Brownian motion B starting from 0.
The set ΘB is closed for the topology of weak convergence of measures, and is a
compact subset of L2 ((Bt )t∈[0,1] ).
Let us mention that the set K∗ directly comes from Strassen [13]’s limiting set. The
R∞
precise meaning of −∞ f (x)dL1 (x) can be given by the integration by parts and the
occupation times formula:
Z
∞
Z
∞
L1 (x)f˙(x)dx = −
f (x)dL1 (x) = −
−∞
1
Z
−∞
f˙(Bs )ds,
(1.4)
0
where as before, f˙ denotes the almost everywhere derivative of f .
Instead of Theorem 1, we shall prove that there is no quenched limit theorem for the
continuous analogue of K introduced by Kesten and Spitzer [9] and deduce Theorem
1 by using a strong approximation for the one-dimensional RWRS. Let us define this
continuous analogue: Assume that B := (B(t))t≥0 , W := (W (t))t≥0 , W̃ := (W̃ (t))t≥0 are
three real Brownian motions starting from 0, defined on the same probability space and
independent of each other. For brevity, we shall write W (x) := W (x) if x ≥ 0 and W̃ (−x)
if x < 0 and say that W is a two-sided Brownian motion. We denote by PB , PW the law
of these processes. We will also denote by (Lt (x))t≥0,x∈R a continuous version with
compact support of the local time of the process B . We define the continuous version
of the RWRS, also called Brownian motion in Brownian scenery, as
Z
+∞
Zt :=
Z
0
+∞
Z
+∞
Lt (−x)dW̃ (x) ≡
Lt (x)dW (x) +
Lt (x)dW (x).
−∞
0
In dimension one, under the annealed measure, Kesten and Spitzer [9] proved that the
process (n−3/4 K([nt]))t≥0 weakly converges in the space of continuous functions to the
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Quenched CLT in dimension one
continuous process Z = (Z(t))t≥0 . Zhang [14] (see also [6, 10]) gave a stronger version
of this result in the special case when the scenery has a finite moment of order 2 + δ
for some δ > 0, more precisely, there is a coupling of ξ , S , B and W such that (ξ, W ) is
independent of (S, B) and for any ε > 0, almost surely,
1
1
max |K(m) − Z(m)| = o(n 2 + 2(2+δ) +ε ), n → +∞.
(1.5)
0≤m≤n
Theorem 1 will follow from this strong approximation and the following result.
Theorem 2. PW -almost surely, under the quenched probability P(·|W ), the limit points
of the law of
Z̃t :=
(2t3/2
Zt
,
log log t)1/2
t → ∞,
under the topology of weak convergence of measures, are equal to the set of the laws of
random variables in ΘB defined in Theorem 1. Consequently under P(·|W ), as t → ∞,
Z̃t does not converge in law.
To prove Theorem 2, we shall apply Strassen [13]’s functional law of the iterated
logarithm applied to the two-sided Brownian motion W ; we shall also need to estimate
R
the stochastic integral g(x)dL1 (x) for a Borel function g , see Section 2 for the details.
2
Proofs
For a two-sided one-dimensional Brownian motion (W (t), t ∈ R) starting from 0, let
us define for any λ > ee ,
Wλ (t) :=
W (λt)
,
(2λ log log λ)1/2
t ∈ R.
Lemma 3. (i) Almost surely, for any s < 0 < r rational numbers, (Wλ (t), s ≤ t ≤ r) is
relatively compact in the uniform topology and the set of its limit points is Ks,r , with
Z r
n
o
Ks,r := f ∈ AC([s, r] → R) : f (0) = 0,
(f˙(x))2 dx ≤ 1 .
s
(ii) There exists some finite random variable AW only depending on (W (x), x ∈ R)
such that for all λ ≥ e36 ,
sup
t∈R,t6=0
|Wλ (t)|
q
|t| log log(|t| +
1
|t|
≤ AW < ∞.
+ 36)
Remark 4. The statement (i) is a reformulation of Strassen’s theorem and holds in fact
for all real numbers s and r . Moreover, using the notation K∗ in (1.2), we remark that
Ks,r coincides with the restriction of K∗ on [s, r]: for any s < 0 < r,
n
o
Ks,r = f
: f ∈ K∗ .
[s,r]
Proof: (i) For any fixed s < 0 < r , by applying Strassen’s theorem ([13]) to the twoW (λru)
W (λsu)
dimensional rescaled Brownian motion: ( √2λr log log λ , √
)0≤u≤1 , we get that
2λ|s| log log λ
a.s., (Wλ (t), s ≤ t ≤ r) is relatively compact in the uniform topology with Ks,r as the set
of limit points. By inverting a.s. and s, r , we obtain (i).
(ii) By the classical law of the iterated logarithm for the Brownian motion W (both
at 0 and at ∞), we get that
AeW :=
sup
x∈R,x6=0
|W (x)|
q
|x| log log(|x| +
ECP 18 (2013), paper 85.
1
|x|
+ 36)
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Quenched CLT in dimension one
is a finite variable. Observe that for any t > 0 and λ > e36 ,
1
log log λt +
+ 36
=
λt
1
36 log log λ + log t + 2 +
λ t
λ
1
≤ log log λ + log t + + 36
t
1
≤ log log λ + log log t + + 36
t
using that for every a, b ≥ 2, log(a + b) ≤ log(a) + log(b). The Lemma follows if we take
eW . 2
for e.g. AW := 2A
Next, we recall some properties of Brownian local times: The process x 7→ L1 (x) is
a (continuous) semimartingale (by Perkins [11]), moreover, the quadratic variation of
Rx
x 7→ L1 (x) equals 4 −∞ L1 (z)dz . By Revuz and Yor ([12], Exercise VI (1.28)), for any
locally bounded Borel function f ,
1
2
Z
∞
B1
Z
f (x)dL1 (x) = −
−∞
Z
1
f (u)du +
f (Bu )dBu .
0
(2.1)
0
Let us define for all λ > ee and n ≥ 0,
Z
∞
Hλ :=
(n)
Wλ (x)dL1 (x),
Hλ
−∞
(0)
with Hλ
Z
n
:=
Wλ (x)dL1 (x),
−n
= 0. Denote by EB the expectation with respect to the law of B .
Lemma 5. There exists some positive constant c1 such that for any λ > e36 and n ≥ 0,
we have
Z
EB ∞
−∞
(n) EB Hλ − Hλ Z ∞
2
EB
f (x)dL1 (x)
−∞
Z n
f (x)dL1 (x) −
f (x)dL1 (x)
≤ c1 e −
n2
4
AW ,
(2.2)
≤ 16 s(f ),
≤ 4
(2.3)
p
n2
2s(f ) e− 4 ,
(2.4)
−n
h
i
for any Borel function f : R → R such that s(f ) := sup0≤u≤1 EB f 2 (Bu ) < ∞.
Remark that if f is bounded, then s(f ) ≤ supx∈R f 2 (x).
Proof: We first prove that there exists some positive constant c2 such that for all n ≥ 0
and λ > e36 ,
h
i
(n)
EB (Hλ − Hλ )2 ≤ c2 A2W .
(2.5)
In fact, by applying (2.1) and using the Brownian isometry for f (x) = Wλ (x)1(|x|>n) , we
get that
h
i
h
i
hZ 1
i
(n) 2
2
EB (Hλ − Hλ ) ≤ 8EB Fn,λ (B1 ) + 8EB
(Wλ (Bu ))2 1(|Bu |>n) du ,
0
Rx
Wλ (y)1(|y|>n) dy for any x ∈ R. By Lemma 3 (ii),
Z x
1
+ 36))1/2 dy ≤ c3 AW (1 + x2 ),
|Fn,λ (x)| ≤ AW (|y| log log(|y| +
|y|
0
with Fn,λ (x) :=
0
∀x ∈ R,
with some constant c3 > 0. (Here we usedh that log xi < x for x > 0 and that for any
√
√
√
a, b > 0, a + b ≤ a + b). Hence EB Fn,λ (B1 )2 ≤ 6 c23 A2W . In the same way,
ECP 18 (2013), paper 85.
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Quenched CLT in dimension one
EB (Wλ (Bu ))2 ≤ A2W EB |Bu | log log(|Bu | + |B1u | + 36) which is integrable for u ∈ (0, 1].
Then (2.5) follows.
(n)
To check (2.2), we remark that Hλ − Hλ = 0 if sup0≤u≤1 |Bu | ≤ n. Then by CauchySchwarz’ inequality and (2.5), we have that
(n) EB Hλ − Hλ =
i
h
(n) EB Hλ − Hλ 1(sup0≤u≤1 |Bu |>n)
r h
is (n) 2
EB (Hλ − Hλ )
PB sup |Bu | > n
≤
0≤u≤1
√
≤
c2 AW
√
2 e−
n2
4
,
2
by the standard Gaussian tail: PB sup0≤u≤1 |Bu | > x ≤ 2e−x /2 for any x > 0. Then we
get (2.2).
To prove (2.3), we use again (2.1) and the Brownian isometry to arrive at
EB
Z
∞
f (x)dL1 (x)
2
1
Z
h
i
≤ 8EB G2 (B1 ) + 8
−∞
h
i
h
i
EB f 2 (Bu ) du ≤ 8EB G2 (B1 ) + 8s(f ),
0
Rx
2
R
0 f (y)dy for any x ∈ R. By Cauchy-Schwarz’ inequality, (G(x)) ≤
x 2
x 0 f (y)dy for any x ∈ R, from which we use the integration by parts for the density
h
i
h
i
of B1 and deduce that EB G2 (B1 ) ≤ EB f 2 (B1 ) . Then (2.3) follows.
with G(x) :=
Finally for (2.4), we use (2.3) to see that
EB
Z
∞
Z
n
f (x)dL1 (x) −
−∞
f (x)dL1 (x)
2
= EB
Z
−n
∞
2
f (x)1(|x|>n) dL1 (x) ≤ 16s(f ),
−∞
for any n. Then (2.4) follows from the Cauchy-Schwarz inequality and the Gaussian tail,
exactly in the same way as (2.2). 2
Recalling (1.3) for the definition of ΘB . For any p > 0, it is easy to see that ΘB ⊂
Lp (B), since from Cauchy-Schwarz’ inequality, using the relation (1.4), we deduce that
Z
∞
2
f (x)dL1 (x) ≤
−∞
Z
∞
(L1 (x))2 dx
Z
−∞
∞
(f˙(x))2 dx ≤ sup L1 (x) ∈ Lp (B),
x
−∞
see Csáki [5], Lemma 1 for the tail of supx L1 (x). Write dL1 (B) (ξ, η) for the distance in
L1 (B) for any ξ, η ∈ L1 (B).
Lemma 6. PW -almost surely,
dL1 (B) (Hλ , ΘB ) → 0,
as λ → ∞,
where ΘB is defined in (1.3). Moreover, PW -almost surely for any ξ ∈ ΘB ,
lim inf dL1 (B) (Hλ , ξ) = 0.
λ→∞
2
Proof: Let ε > 0. Choose a large n = n(ε) such that c1 e−n /4 ≤ ε. By Lemma 3 (i),
for all large λ ≥ λ0 (W, ε, n), there exists some function g = gλ,W,ε,n ∈ K−n,n such that
sup|x|≤n |Wλ (x) − g(x)| ≤ ε. Applying (2.3) to f (x) = (Wλ (x) − g(x))1(|x|≤n) which is
bounded by ε, we get that
Z
(n)
EB Hλ −
n
−n
p
g(x)dL1 (x) ≤ 4 s(f ) ≤ 4ε.
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Quenched CLT in dimension one
We extend g to R by letting g(x) = g(n) if x ≥ n and g(x) = g(−n) if x ≤ −n, then
R∞
Rn
g(x)dL1 (x) = −n g(x)dL1 (x). By the triangular inequality and (2.2),
−∞
g ∈ K∗ and
Z
EB Hλ −
∞
−∞
(n) g(x)dL1 (x) ≤ 4ε + EB Hλ − Hλ ≤ (4 + c1 AW )ε.
It follows that dL1 (B) (Hλ , ΘB ) ≤ (4 + c1 AW )ε. Hence PW -a.s.,
lim sup dL1 (B) (Hλ , ΘB ) ≤ (4 + c1 AW )ε,
λ→∞
showing the first part in the Lemma.
R∞
For the other part of the Lemma, let h ∈ K∗ such that ξ = −∞ h(x)dL1 (x). Observe
r
p
Rx
that |h(x)| ≤
|x| for all x ∈ R, s(h) = sup0≤u≤1 EB [h2 (Bu )] ≤
x 0 (ḣ(y))2 dy ≤
EB [|B1√
|], then for any ε > 0, we may use (2.4) and choose an integer n = n(ε) such that
2
(c1 + 4 2)e−n /4 ≤ ε and
dL1 (B) (ξ, ξn ) ≤ ε,
Rn
where ξn := −n h(x)dL1 (x). Applying Lemma 3 (i) to the restriction of h on [−n, n], we
may find a sequence λj = λj (ε, W, n) → ∞ such that sup|x|≤n |Wλj (x) − h(x)| ≤ ε. By
applying (2.3) to f (x) = (Wλj (x) − h(x))1(|x|≤n) , we have that
(n)
dL1 (B) (Hλj , ξn ) ≤ 4ε.
(n)
By (2.2) and the choice of n, dL1 (B) (Hλj , Hλj ) ≤ εAW for all large λj , it follows from
the triangular inequality that
dL1 (B) (ξ, Hλj ) ≤ (5 + AW )ε,
implying that PW -a.s., lim inf λ→∞ dL1 (B) (Hλ , ξ) ≤ (5 + AW )ε → 0 as ε → 0. 2
We now are ready to give the proof of Theorems 2 and 1.
Proof of Theorem 2. Firstly, we remark that by Brownian scaling, PW -a.s.,
Z
Zt (d)
= −
t3/4
M1
m1
√
1
W ( ty)dL1 (y).
t1/4
(2.6)
√
In fact, by the change of variables x = y t, we get
+∞
Z
√ Z
Lt (x)dW (x) = t
+∞
−∞
−∞
√ √
Lt (y t)
√
dW (y t)
t
which has the same distribution as
√ Z
t
+∞
√
L1 (y)dW (y t)
−∞
from the scaling property of the local time of the Brownian motion. Since (L1 (x))x∈R
is a continuous semi-martingale, independent from the process W , from the formula of
integration by parts, we get that PW -a.s.,
√ Z
t
+∞
−∞
√
L1 (y)dW (y t) = −t
3/4
Z
M1
m1
√ W ( ty)
dL1 (y),
t1/4
yielding (2.6). Theorem 2 follows from Lemma 6. 2
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Quenched CLT in dimension one
Proof of Theorem 1. We use the strong approximation of Zhang [14] : there exists on
a suitably enlarged probability space, a coupling of ξ , S , B and W such that (ξ, W ) is
independent of (S, B) and for any ε > 0, almost surely,
1
1
max |K(m) − Z(m)| = o(n 2 + 2(2+δ) +ε ), n → +∞.
0≤m≤n
From the independence of (ξ, W ) and (S, B), we deduce that for P-a.e. (ξ, W ), under
the quenched probability P(.|ξ, W ), the limit points of the laws of K̃n and Z̃n are the
same ones. Now, by adapting the proof of Theorem 2, we have that for P-a.e. (ξ, W ),
under the quenched probability P(.|ξ, W ), the limit points of the laws of Z̃n , as n → ∞,
under the topology of weak convergence of measures, are equal to the set of the laws of
random variables in ΘB . It gives that for P-a.e. (ξ, W ), under the quenched probability
P(.|ξ, W ), the limit points of the laws of K̃n , as n → ∞, under the topology of weak
convergence of measures, are equal to the set of the laws of random variables in ΘB
and the first part of Theorem 1 follows.
Let (ζn )n be a sequence of random variables in ΘB , each ζn being associated to a
function fn ∈ K∗ . The sequence of the (almost everywhere) derivatives of fn is then a
bounded sequence in the Hilbert space L2 (R), so we can extract a subsequence which
weakly converges. Using the definition of the weak convergence and the relation (1.4),
(ζn )n converges almost surely and the closure of ΘB follows. Since the sequence (ζn )n
is bounded in Lp (B) for any p ≥ 1, the convergence also holds in L2 (B). Therefore ΘB
is a compact set of L2 (B) as closed and bounded subset. 2
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Acknowledgments. We are grateful to Mikhail Lifshits for interesting discussions. The
authors thank the referee for recommending various improvements in exposition.
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