The quenched limiting distributions of a one-dimensional random walk in random scenery.

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The quenched limiting distributions of a

one-dimensional random walk in random scenery.

Nadine Guillotin-Plantard, Yueyun Hu, Bruno Schapira

To cite this version:

Nadine Guillotin-Plantard, Yueyun Hu, Bruno Schapira. The quenched limiting distributions of a one- dimensional random walk in random scenery.. Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2013, 18 (85), pp.1-7. �hal-00842602v2�

The quenched limiting distributions

of a one-dimensional random walk in random scenery

Nadine Guillotin-Plantard^∗, Yueyun Hu^†, and Bruno Schapira^‡ Université Lyon 1, Université Paris 13, and Université Aix-Marseille

Summary.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 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 approxima- tion.

AMS Subject Classification: 60F05, 60G52.

1. Introduction

Random walks in random sceneries were introduced independently by Kesten and Spitzer [9]

and by Borodin [3, 4]. LetS = (Sn)n≥0 be a random walk in Z^d starting at 0, i.e., S₀ = 0 and (S_n−S_n−1)_n≥1 is a sequence of i.i.d. Z^d-valued random variables.Letξ = (ξ_x)_x∈Z^d 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 := (K_n)_n≥0 is defined by setting K₀ := 0 and, for n∈N∗,

K_n:=

Xn

i=1

ξ_Si. (1)

We will denote by P the joint law of S and ξ. The law P is called the annealed law, while the conditional lawP(·|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 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].

∗Institut Camille Jordan, Université Lyon 1, 43, boulevard du 11 novembre 1918 69622 Villeurbanne. Research partially supported by ANR (MEMEMOII) 2010 BLAN 0125 Email: nadine.guillotin@univ-lyon1.fr

†Département de Mathématiques (LAGA CNRS-UMR 7539) Université Paris 13, 99 avenue J.B. Clément, 93430 Villetaneuse. Research partially supported by ANR (MEMEMOII) 2010 BLAN 0125 Email: yueyun@math.univ- paris13.fr

‡Centre de Mathématiques et Informatique, Aix-Marseille Université, 39, rue F. Joliot Curie, 13453 Marseille Cedex 13. Research partially supported by ANR (MEMEMOII) 2010 BLAN 0125. Email:

bruno.schapira@latp.univ-mrs.fr

1

2

In this note we consider the case of the simple symmetric random walk (Sn)n≥0 on Z, the random scenery (ξ_x)_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 byAC([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

K^∗ :=n

f ∈ AC(R→R) :f(0) = 0, Z _∞

−∞

( ˙f(x))²dx≤1o

. (2)

Theorem 1. For P-a.e. ξ, under the quenched probability P(.|ξ), the process K˜_n:= K_n

(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,asn→ ∞,under the topology of weak convergence of measures, are equal to the set of the laws of random variables inΘB, with

Θ_B :=n Z ∞

−∞

f(x)dL₁(x) :f ∈ K^∗o

, (3)

where(L₁(x), x∈R) denotes the family of local times at time1 of a one-dimensional Brownian motion B starting from 0.

The setΘ_B is closed for the topology of weak convergence of measures, and is a compact subset of L²((B_t)_t∈[0,1]).

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

−∞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(B˙ _s)ds, (4) 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 thatB := (B(t))_t≥0,W := (W(t))_t≥0,W˜ := ( ˜W(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 writeW(x) :=W(x)if x≥0andW˜(−x)if x <0and say thatW is a two-sided Brownian motion. We denote byP_B,P_W 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 calledBrownian motion in Brownian scenery, as

Z_t:=

Z _+∞

0

L_t(x)dW(x) + Z _+∞

0

L_t(−x)dW˜(x)≡ Z _+∞

−∞

L_t(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 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 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≤maxm≤n|K(m)−Z(m)|=o(n^{12+2(2+δ)1 +ε}), n→+∞. (5) Theorem 1 will follow from this strong approximation and the following result.

Theorem 2. P_W-almost surely, under the quenched probability P(·|W), the limit points of the law of

Z˜_t:= Z_t

(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 in Θ_B defined in Theorem 1. Consequently under P(·|W), as t → ∞, Z˜_t 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 the stochastic integral R

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 anyλ > e^e,

W_λ(t) := W(λt)

(2λlog logλ)^1/2, t∈R.

Lemma 3. (i) Almost surely, for anys <0< rrational numbers,(W_λ(t), s≤t≤r)is relatively compact in the uniform topology and the set of its limit points is Ks,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 variable AW only depending on (W(x), x∈R) such that for all λ≥e³⁶,

sup

t∈R,t6=0

|W_λ(t)|

q|t|log log(|t|+_|¹_t|+ 36) ≤ AW <∞.

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 (2), we remark that Ks,r coincides with the restriction ofK^∗ on [s, r]: for any s <0< r,

Ks,r =n f

[s,r]:f ∈ K^∗o .

Proof: (i) For any fixeds <0< r, by applying Strassen’s theorem ([13]) to the two-dimensional rescaled Brownian motion: (√ ^W(λru)

2λrlog logλ,√ ^W(λsu)

2λ|s|log logλ)_0≤u≤1, we get that a.s.,(W_λ(t), s≤t≤r) is relatively compact in the uniform topology withKs,r as 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 motion W (both at0 and at ∞), we get that

AeW := sup

x∈R,x6=0

|W(x)|

q|x|log log(|x|+_|x¹_|+ 36)

4

is a finite variable. Observe that for anyt > 0 and λ > e³⁶,log log(λt+ _λt¹ + 36) ≤log logλ+ log log(t+¹_t+ 36). The Lemma follows if we take for e.g. AW := 2AeW.

Next, we recall some properties of Brownian local times: The processx7→L₁(x) is a (contin- uous) semimartingale (by Perkins [11]), moreover, the quadratic variation ofx 7→ 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. (6)

Let us define for allλ > e^e and n≥0, H_λ :=

Z _∞

−∞

W_λ(x)dL1(x), H_λ⁽ⁿ⁾:=

Z n

−n

W_λ(x)dL1(x),

withH_λ⁽⁰⁾= 0. Denote byE_B the expectation with respect to the law ofB.

Lemma 5. There exists some positive constant c₁ such that for any λ > e³⁶ andn≥0, we have

E_B

H_λ−H_λ⁽ⁿ⁾

≤ c₁e⁻ⁿ

2

4 AW, (7)

E_B Z ∞

−∞

f(x)dL₁(x)2

≤ 16s(f), (8)

E_B

Z _∞

−∞

f(x)dL₁(x)− Z n

−n

f(x)dL₁(x)

≤ 4p

2s(f)e⁻ⁿ

2

4 , (9)

for any Borel function f :R→R such that s(f) := sup_0≤u≤1E_B h

f²(B_u)i

<∞.

Remark that iff is bounded, then s(f)≤sup_x∈Rf²(x).

Proof: We first prove that there exists some positive constant c₂ such that for all n ≥0 and λ > e³⁶,

E_B h

(H_λ−H_λ⁽ⁿ⁾)²i

≤c₂A²W. (10)

In fact, by applying (6) and using the Brownian isometry forf(x) =W_λ(x)1_(|x|>n), we get that

E_B h

(H_λ−H_λ⁽ⁿ⁾)²i

≤8E_B h

F_n,λ(B1)²i + 8E_B

h Z ¹

0

(W_λ(Bu))²1_(|Bu|>n)dui ,

withF_n,λ(x) :=Rx

0 W_λ(y)1_(|y|>n)dy for any x∈R. By Lemma 3 (ii),

|F_n,λ(x)| ≤ AW

Z _x

0

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

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

≤c₃AW(1 +x²), ∀x∈R,

with some constantc₃ >0. HenceE_B h

F_n,λ(B1)²i

≤6c²₃A²W. In the same way,E_B

(W_λ(Bu))²

≤ A²WE_B

|B_u|log log(|B_u|+_|B¹

u|+ 36)

which is integrable for u∈(0,1]. Then (10) follows.

To check (7), we remark thatH_λ−H_λ⁽ⁿ⁾= 0if sup_0≤u≤1|B_u| ≤n. Then by Cauchy-Schwarz’

inequality and (10), we have that E_B

H_λ−H_λ⁽ⁿ⁾

= E_BhH_λ−H_λ⁽ⁿ⁾

1_{(sup0≤u≤1|Bu|>n)}i

≤ r

E_Bh

(H_λ−H_λ⁽ⁿ⁾)²i s P_B

sup

0≤u≤1|B_u|> n

≤ √c₂AW

√2e⁻ⁿ

2 4 , by the standard Gaussian tail: P_B sup_0≤u≤1|B_u|> x

≤2e^−x2/2 for any x >0. Then we get (7).

To prove (8), we use again (6) and the Brownian isometry to arrive at E_B

Z ∞

−∞

f(x)dL1(x)2

≤8E_B h

G²(B1)i + 8

Z 1

0

E_B h

f²(Bu)i

du≤8E_B h

G²(B1)i

+ 8s(f),

withG(x) :=R_x

0 f(y)dyfor anyx∈R. By Cauchy-Schwarz’ inequality,(G(x))² ≤ xR_x

0 f²(y)dy for anyx ∈R, from which we use the integration by parts for the density ofB₁ and deduce that E_B

h

G²(B₁)i

≤E_B h

f²(B₁)i

. Then (8) follows.

Finally for (9), we use (8) to see that E_B

Z ∞

−∞

f(x)dL₁(x)− Z n

−n

f(x)dL₁(x)2

=E_B Z ∞

−∞

f(x)1_(|x|>n)dL₁(x)2

≤16s(f), for any n. Then (9) follows from the Cauchy-Schwarz inequality and the Gaussian tail, exactly in the same way as (7).

Recalling (3) for the definition ofΘ_B. For any p >0, it is easy to see thatΘ_B⊂L^p(B), since from Cauchy-Schwarz’ inequality, using the relation (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 of sup_xL₁(x). Write d_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, as λ→ ∞,

whereΘ_Bis defined in(3). Moreover,P_W-almost surely for anyξ∈Θ_B,lim inf_λ→∞d_L1(B)(H_λ, ξ) = 0.

Proof: Letε >0. Choose a largen=n(ε)such thatc₁e^−n2/4≤ε. By Lemma 3 (i), for all large λ ≥ λ₀(W, ε, n), there exists some function g = g_λ,W,ε,n ∈ K−n,n such that sup_|x|≤n|W_λ(x)− g(x)| ≤ε. Applying (8) to f(x) = (W_λ(x)−g(x))1_(|x|≤n) which is a bounded by ε, we get that

E_B H_λ⁽ⁿ⁾−

Z n

−n

g(x)dL1(x) ≤4p

s(f)≤4ε.

6

We extendg to Rby letting g(x) =g(n) if x≥n and g(x) = g(−n) if x ≤ −n, then g∈ K^∗ andR_∞

−∞g(x)dL₁(x) =R_n

−ng(x)dL₁(x). By the triangular inequality and (7), E_B

H_λ−

Z _∞

−∞

g(x)dL₁(x)

≤4ε+E_B

H_λ−H_λ⁽ⁿ⁾

≤(4 +c₁AW)ε.

It follows thatd_L1(B)(H_λ,Θ_B)≤(4 +c₁AW)ε. HenceP_W-a.s., lim sup_λ→∞d_L1(B)(H_λ,Θ_B)≤ (4 +c₁AW)ε, showing the first part in the Lemma.

For the other part of the Lemma, let h ∈ K^∗ such that ξ = R_∞

−∞h(x)dL1(x). Observe that

|h(x)| ≤r xR_x

0( ˙h(y))²dy ≤p

|x|for all x ∈R,s(h) = sup_0≤u≤1E_B[h²(B_u)]≤E_B[|B₁|], then for anyε > 0, we may use (9) and choose an integer n=n(ε) such that (c₁+ 4√

2)e^−n2/4 ≤ε and

d_L1(B)(ξ, ξ_n)≤ε, where ξ_n := R_n

−nh(x)dL₁(x). Applying Lemma 3 (i) to the restriction of h on [−n, n], we may find a sequenceλ_j =λ_j(ε, W, n)→ ∞ such that sup_|x|≤n|W_λj(x)−h(x)| ≤ ε. By applying (8) to f(x) = (W_λj(x)−h(x))1_(|x|≤n), we have that

d_L1(B)(H_λ⁽ⁿ⁾

j , ξ_n)≤4ε.

By (7) and the choice of n, d_L1(B)(H_λ⁽ⁿ⁾

j , H_λj) ≤ εAW for all large λ_j, it follows from the triangular inequality that

d_L1(B)(ξ, H_λj)≤(5 +AW)ε,

implying thatP_W-a.s., lim inf_λ→∞d_L1(B)(H_λ, ξ)≤(5 +AW)ε→0asε→0.

We now are ready to give the proof of Theorems 2 and 1.

Proof of Theorem 2. Firstly, we remark that by Brownian scaling,P_W-a.s., Z_t

t^3/4

(d)= − Z _M1

m1

1 t^1/4W(√

ty)dL₁(y). (11)

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 process W, from the formula of integration by parts, we get thatP_W -a.s.,

√t Z _+∞

−∞

L₁(y)dW(y√

t) =−t^3/4 Z _M1

m1

W(√ ty) t^1/4

dL₁(y),

yielding (11). The first part of Theorem 2 follows from Lemma 6.

Let (ζ_n)_n be a sequence of random variables in Θ_B, each ζ_n being associated to a function f_n∈ K^∗. The sequence of the (almost everywhere) derivatives off_n 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 (4), (ζn)n converges almost surely and

the closure of ΘB follows. Since the sequence (ζn)n is bounded in L^p(B) for any p ≥ 1, the convergence also holds inL²(B). ThereforeΘ_B is a compact set ofL²(B)as closed and bounded subset.

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

0≤maxm≤n|K(m)−Z(m)|=o(n^{12+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 of K˜_n and Z˜_n are the same ones.

Now, by adapting the proof of Theorem 2, we have that forP-a.e. (ξ, W), under the quenched probability P(.|ξ, W), the limit points of the laws ofZ˜_n, as n→ ∞, under the topology of weak convergence of measures, are equal to the set of the laws of random variables in Θ_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 inΘ_B and Theorem 1 follows.

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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