one-dimensional random walk in random scenery

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ISSN:1083-589X 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 deter- mine its quenched weak limits by applying Strassen [13]’s functional law of the iterated logarithm. As a consequence, conditioned on the random scenery, the one- dimensional 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]. LetS= (Sn)_n≥0be a random walk inZ^dstarting at0, i.e., S0 = 0and (Sn−S_n−1)_n≥1 is a sequence of i.i.d.Z^d-valued random variables. Let ξ = (ξx)_x∈_Zd be a field of i.i.d. real random variables independent ofS. The fieldξ is called the random scenery. The random walk in random scenery (RWRS)K:= (Kn)n≥0

is defined by settingK₀:= 0and, forn∈N^∗^,

K_n:=

n

X

i=1

ξ_S_i. (1.1)

We will denote byPthe joint law ofSandξ. The lawPis called theannealed law, while the conditional lawP(·|ξ)is called thequenchedlaw.

Limit theorems for RWRS have a long history, we refer to [7] or [8] for a complete review. Distributional limit theorems forquenchedsceneries (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 ˇCerný [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:[email protected]

‡Département de Mathématiques (LAGA CNRS-UMR 7539) Université Paris 13, France.

§Centre de Mathématiques et Informatique, Aix-Marseille Université, France.

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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≥0on Z, the random scenery(ξx)_x∈Z is assumed to be centered with finite variance equal to one and there exists someδ >0 such thatE(|ξ0|^2+δ)<∞. We prove that under these assumptions, there is no quenched distributional limit theorem forK. 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 byf˙) exists almost everywhere and is Lebesgue integrable on[a, b]. Define

K^∗:=n

f ∈ AC(R→R) :f(0) = 0, Z ∞

−∞

( ˙f(x))²dx≤1o

. (1.2)

Theorem 1. ForP^-a.e.ξ, under the quenched probabilityP(.|ξ), the process

K˜n:= Kn

(2n^3/2log logn)^1/2, n > e^e,

does not converge in law. More precisely, for P^-a.e.ξ, under the quenched probability P(.|ξ), the limit points of the law ofK˜_n, as n → ∞, under the topology of weak convergence of measures, are equal to the set of the laws of random variables inΘB, with

ΘB :=nZ ∞

−∞

f(x)dL1(x) :f ∈ K^∗o

, (1.3)

where(L1(x), x ∈ R)denotes the family of local times at time1 of a one-dimensional Brownian motionB starting from0.

The set ΘB is closed for the topology of weak convergence of measures, and is a compact subset ofL²((Bt)_t∈[0,1]).

Let us mention that the setK^∗ directly comes from Strassen [13]’s limiting set. The precise meaning ofR∞

−∞f(x)dL1(x) can be given by the integration by parts and the occupation times formula:

Z ∞

−∞

f(x)dL1(x) =− Z ∞

−∞

L1(x) ˙f(x)dx=− Z 1

0

f˙(Bs)ds, (1.4)

where as before,f˙denotes the almost everywhere derivative off.

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 thatB:= (B(t))_t≥0,W := (W(t))_t≥0,W˜ := ( ˜W(t))_t≥0are three real Brownian motions starting from0, defined on the same probability space and independent of each other. For brevity, we shall writeW(x) :=W(x)ifx≥0andW˜(−x) ifx <0and say thatW is a two-sided Brownian motion. We denote byPB,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 processB. We define the continuous version of the RWRS, also calledBrownian motion in Brownian scenery, as

Zt:=

Z +∞

0

Lt(x)dW(x) + Z +∞

0

Lt(−x)dW˜(x)≡ Z +∞

−∞

Lt(x)dW(x).

In dimension one, under the annealed measure, Kesten and Spitzer [9] proved that the process(n^−3/4K([nt]))t≥0weakly converges in the space of continuous functions to the

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continuous processZ= (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 order2 +δ for someδ >0, more precisely, there is a coupling ofξ,S,B andW such that(ξ, W)is independent of(S, B)and for anyε >0, almost surely,

0≤m≤nmax |K(m)−Z(m)|=o(n¹²⁺^2(2+δ)¹ ^+ε), n→+∞. (1.5) Theorem 1 will follow from this strong approximation and the following result.

Theorem 2. PW-almost surely, under the quenched probabilityP(·|W), the limit points of the law of

Z˜_t:= Zt

(2t^3/2log logt)^1/2, t→ ∞,

under the topology of weak convergence of measures, are equal to the set of the laws of random variables inΘ_B defined in Theorem 1. Consequently underP(·|W), ast→ ∞, Z˜_tdoes 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 motionW; we shall also need to estimate the stochastic integralR

g(x)dL₁(x)for a Borel functiong, see Section 2 for the details.

2 Proofs

For a two-sided one-dimensional Brownian motion(W(t), t∈R)starting from0, let us define for anyλ > e^e,

Wλ(t) := W(λt)

(2λlog logλ)^1/2, t∈R.

Lemma 3. (i) Almost surely, for any s <0 < rrational numbers,(Wλ(t), s≤t ≤r)is relatively compact in the uniform topology and the set of its limit points isKs,r, with

Ks,r:=n

f ∈ AC([s, r]→R) :f(0) = 0, Z r

s

( ˙f(x))²dx≤1o .

(ii) There exists some finite random variableAW only depending on(W(x), x ∈R) such that for allλ≥e³⁶,

sup

t∈R,t6=0

|Wλ(t)|

q|t|log log(|t|+_|t|¹ + 36)

≤ AW <∞.

Remark 4. The statement (i) is a reformulation of Strassen’s theorem and holds in fact for all real numberssandr. Moreover, using the notationK^∗ in (1.2), we remark that Ks,rcoincides with the restriction ofK^∗on[s, r]: for anys <0< r,

Ks,r=n f

[s,r]:f ∈ K^∗o .

Proof: (i) For any fixed s < 0 < r, by applying Strassen’s theorem ([13]) to the two- dimensional rescaled Brownian motion: (^√_2λr^W^(λru)_{log log}_λ,√ ^W^(λsu)

2λ|s|log logλ)0≤u≤1, we get that a.s.,(Wλ(t), s≤t≤r)is relatively compact in the uniform topology withKs,ras the set of limit points. By inverting a.s. ands, r, we obtain (i).

(ii) By the classical law of the iterated logarithm for the Brownian motionW (both at0and at∞), we get that

Ae_W := sup

x∈R,x6=0

|W(x)|

q|x|log log(|x|+_|x|¹ + 36)

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is a finite variable. Observe that for anyt >0andλ > e³⁶, log log

λt+ 1 λt + 36

= log

logλ+ log t+ 1

λ²t +36 λ

≤ log

logλ+ log t+1

t + 36

≤ log logλ+ log log t+1

t + 36

using that for everya, b≥2,log(a+b)≤log(a) + log(b). The Lemma follows if we take for e.g.AW := 2AeW.2

Next, we recall some properties of Brownian local times: The processx7→L₁(x)is a (continuous) semimartingale (by Perkins [11]), moreover, the quadratic variation of x7→ L1(x)equals 4Rx

−∞L1(z)dz. By Revuz and Yor ([12], Exercise VI (1.28)), for any locally bounded Borel functionf,

1 2

Z ∞

−∞

f(x)dL₁(x) =− Z B1

0

f(u)du+ Z 1

0

f(B_u)dB_u. (2.1) Let us define for allλ > e^eandn≥0,

Hλ:=

Z ∞

−∞

Wλ(x)dL1(x), H_λ⁽ⁿ⁾:=

Z n

−n

Wλ(x)dL1(x),

withH_λ⁽⁰⁾= 0. Denote byE^B the expectation with respect to the law ofB.

Lemma 5. There exists some positive constantc1such that for anyλ > e³⁶andn≥0, we have

E^B

Hλ−H_λ⁽ⁿ⁾

≤ c1e⁻ⁿ

2

4 AW, (2.2)

E^BZ ∞

−∞

f(x)dL1(x)²

≤ 16s(f), (2.3)

E^B

Z ∞

−∞

f(x)dL1(x)− Z n

−n

f(x)dL1(x)

≤ 4p

2s(f)e⁻ⁿ

2

4 , (2.4)

for any Borel functionf :R→R^{such that}s(f) := sup_0≤u≤1E^Bh

f²(Bu)i

<∞.

Remark that iff is bounded, thens(f)≤sup_x∈_Rf²(x).

Proof:We first prove that there exists some positive constantc2such that for alln≥0 andλ > e³⁶,

E^Bh

(Hλ−H_λ⁽ⁿ⁾)²i

≤c2A²_W. (2.5)

In fact, by applying (2.1) and using the Brownian isometry forf(x) =Wλ(x)1_(|x|>n), we get that

E^Bh

(Hλ−H_λ⁽ⁿ⁾)²i

≤8E^Bh

Fn,λ(B1)²i

+ 8E^BhZ 1 0

(Wλ(Bu))²1_(|B_u_|>n)dui , withFn,λ(x) :=Rx

0 Wλ(y)1_(|y|>n)dyfor anyx∈R. By Lemma 3 (ii),

|Fn,λ(x)| ≤ AW

Z x

0

(|y|log log(|y|+ 1

|y|+ 36))^1/2dy

≤c3AW(1 +x²), ∀x∈R, with some constantc3 > 0. (Here we used that logx < xforx > 0 and that for any a, b > 0, √

a+b ≤ √ a+√

b). Hence EB

hF_n,λ(B₁)²i

≤ 6c²₃A²_W. In the same way,

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EB

(W_λ(B_u))²

≤ A²_W EB

|Bu|log log(|Bu|+_|B¹

u|+ 36)

which is integrable foru∈(0,1]. Then (2.5) follows.

To check (2.2), we remark thatHλ−H_λ⁽ⁿ⁾= 0ifsup_0≤u≤1|Bu| ≤n. Then by Cauchy- Schwarz’ inequality and (2.5), we have that

EB

H_λ−H_λ⁽ⁿ⁾

= EB

h

H_λ−H_λ⁽ⁿ⁾

1_(sup_0≤u≤1_|B_u_|>n)i

≤ r

E^Bh

(Hλ−H_λ⁽ⁿ⁾)²is P^B

sup

0≤u≤1

|Bu|> n

≤ √

c₂A_W√ 2e⁻ⁿ

2 4 , by the standard Gaussian tail:P^B sup_0≤u≤1|Bu|> x

≤2e^−x²^/2for anyx >0. Then we get (2.2).

To prove (2.3), we use again (2.1) and the Brownian isometry to arrive at

E^BZ ∞

−∞

f(x)dL1(x)²

≤8E^Bh

G²(B1)i + 8

Z 1

0

E^Bh

f²(Bu)i

du≤8E^Bh

G²(B1)i

+ 8s(f),

with G(x) := Rx

0 f(y)dy for any x ∈ R^. By Cauchy-Schwarz’ inequality, (G(x))² ≤

xRx

0 f²(y)dy

^{for any}x∈R, from which we use the integration by parts for the density ofB1and deduce thatE^Bh

G²(B1)i

≤E^Bh

f²(B1)i

. Then (2.3) follows.

Finally for (2.4), we use (2.3) to see that EB

Z ∞

−∞

f(x)dL₁(x)− Z n

−n

f(x)dL₁(x)²

=EB

Z ∞

−∞

f(x)1_(|x|>n)dL₁(x)²

≤16s(f), for anyn. 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 ofΘB. For anyp > 0, it is easy to see that ΘB ⊂ L^p(B), since from Cauchy-Schwarz’ inequality, using the relation (1.4), we deduce that

Z ∞

−∞

f(x)dL₁(x)²

≤ Z ∞

−∞

(L₁(x))²dx Z ∞

−∞

( ˙f(x))²dx

≤sup

x

L₁(x)∈L^p(B),

see Csáki [5], Lemma 1 for the tail ofsup_xL₁(x). Writed_L1(B)(ξ, η)for the distance in L¹(B)for anyξ, η∈L¹(B).

Lemma 6. P^W-almost surely,

d_L1(B)(H_λ,Θ_B)→0, asλ→ ∞,

whereΘ_B is defined in (1.3). Moreover,PW-almost surely for anyξ∈Θ_B, lim inf

λ→∞ d_L1(B)(Hλ, ξ) = 0.

Proof: Let ε > 0. Choose a large n = n(ε)such that c1e⁻ⁿ²^/4 ≤ ε. By Lemma 3 (i), for all largeλ≥λ0(W, ε, n), there exists some functiong =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 byε, we get that

EB

H_λ⁽ⁿ⁾−

Z n

−n

g(x)dL₁(x) ≤4p

s(f)≤4ε.

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We extendg toR^{by letting}g(x) =g(n)ifx≥n andg(x) = g(−n)ifx≤ −n, then g∈ K^∗andR∞

−∞g(x)dL₁(x) =Rn

−ng(x)dL₁(x). By the triangular inequality and (2.2), E^B

Hλ−

Z ∞

−∞

g(x)dL1(x)

≤4ε+E^B

Hλ−H_λ⁽ⁿ⁾

≤(4 +c1AW)ε.

It follows thatd_L1(B)(H_λ,Θ_B)≤(4 +c₁A_W)ε. HencePW-a.s., lim sup

λ→∞

dL¹(B)(Hλ,ΘB)≤(4 +c1AW)ε, showing the first part in the Lemma.

For the other part of the Lemma, leth∈ K^∗ such thatξ=R∞

−∞h(x)dL1(x). Observe that |h(x)| ≤

r xRx

0( ˙h(y))²dy ≤ p

|x| for all x ∈ R^, s(h) = sup_0≤u≤1E^B[h²(Bu)] ≤ E^B[|B1|], then for anyε >0, we may use (2.4) and choose an integern=n(ε)such that (c1+ 4√

2)e⁻ⁿ²^/4≤εand

dL¹(B)(ξ, ξn)≤ε, whereξn :=Rn

−nh(x)dL1(x). Applying Lemma 3 (i) to the restriction ofhon[−n, n], we may find a sequenceλ_j = λ_j(ε, W, n) → ∞such thatsup_|x|≤n|Wλj(x)−h(x)| ≤ ε. By applying (2.3) tof(x) = (W_λ_j(x)−h(x))1_(|x|≤n), we have that

d_L1(B)(H_λ⁽ⁿ⁾

j , ξn)≤4ε.

By (2.2) and the choice ofn,d_L1(B)(H_λ⁽ⁿ⁾

j , H_λ_j)≤εAW for all largeλ_j, it follows from the triangular inequality that

dL¹(B)(ξ, Hλ_j)≤(5 +AW)ε,

implying thatP^W^-a.s.,lim inf_λ→∞d_L1(B)(Hλ, ξ)≤(5 +AW)ε→0asε→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., Zt

t^3/4

(d)= − Z M1

m1

1 t^1/4W(√

ty)dL1(y). (2.6)

In fact, by the change of variablesx=y√

t, we get Z +∞

−∞

L_t(x)dW(x) =√ t

Z +∞

−∞

L_t(y√

√ t) t

dW(y√ t)

which has the same distribution as

√ t

Z +∞

−∞

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 processW, from the formula of integration by parts, we get thatPW -a.s.,

√ t

Z +∞

−∞

L₁(y)dW(y√

t) =−t^3/4 Z M1

m1

W(√ ty) t^1/4

dL₁(y),

yielding (2.6). Theorem 2 follows from Lemma 6.2

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Proof of Theorem 1.We use the strong approximation of Zhang [14] : there exists on a suitably enlarged probability space, a coupling ofξ,S, B and W such that (ξ, W)is independent of(S, B)and for anyε >0, almost surely,

0≤m≤nmax |K(m)−Z(m)|=o(n¹²⁺^2(2+δ)¹ ^+ε), 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 ofKñ and Zñ are the same ones. Now, by adapting the proof of Theorem 2, we have that forP^-a.e. (ξ, W), under the quenched probabilityP(.|ξ, W), the limit points of the laws ofZñ, asn→ ∞, under the topology of weak convergence of measures, are equal to the set of the laws of random variables inΘ_B. It gives that forP^-a.e.(ξ, W), under the quenched probability P(.|ξ, W), the limit points of the laws of K˜_n, as n → ∞, under the topology of weak convergence of measures, are equal to the set of the laws of random variables inΘB

and the first part of Theorem 1 follows.

Let (ζn)n be a sequence of random variables in ΘB, eachζn being associated to a functionfn ∈ K^∗. The sequence of the (almost everywhere) derivatives offn is then a bounded sequence in the Hilbert spaceL²(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 ofΘBfollows. Since the sequence(ζn)n

is bounded inL^p(B)for anyp≥1, the convergence also holds inL²(B). ThereforeΘB

is a compact set ofL²(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.

one-dimensional random walk in random scenery

The quenched limiting distributions of a