ON THE QUENCHED FUNCTIONAL CLT IN RANDOM SCENERIES

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ON THE QUENCHED FUNCTIONAL CLT IN RANDOM SCENERIES

Guy Cohen, Jean-Pierre Conze

To cite this version:

Guy Cohen, Jean-Pierre Conze. ON THE QUENCHED FUNCTIONAL CLT IN RANDOM SCENER- IES. 2021. �hal-03205934�

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ON THE QUENCHED FUNCTIONAL CLT IN RANDOM SCENERIES

GUY COHEN AND JEAN-PIERRE CONZE

Abstract. We prove a quenched functional central limit theorem (quenched FCLT) for the sums of a random field (r.f.) along a Z^d-random walk in different frameworks: probabilistic (when the r.f. is i.i.d. or a moving average of i.i.d. random variables) and algebraic (when the r.f. is generated by commuting automorphisms of a torus or by commuting hyperbolic flows on homogeneous spaces).

Contents

Introduction 1

1. Summation along a r.w. and variance 3

1.1. Random walks and sums along random walks 3

1.2. Variance for quenched processes 4

2. Random walk in random scenery 12

3. Cumulants and CLT 15

3.1. Moments and cumulants 15

3.2. A sufficient condition for the CLT 17

4. Moving averages of i.i.d. random variables 20

5. Tightness and 4th-moment 21

6. Random walks and FCLT for automorphisms of a torus 24

7. Exponential mixing of all orders 27

7.1. FCLT and exponential mixing of all orders 27

7.2. Translations on homogeneous spaces 30

References 31

Date: April 22, 2021.

2010Mathematics Subject Classification. Primary: 60F05, 28D05, 22D40, 60G50; Secondary: 47B15, 37A25, 37A30.

Key words and phrases. quenched functional central limit theorem, Z^d-action, random walk in random scenery, self-intersections of a r.w., toral automorphisms, exponential mixing, flows on homogeneous spaces, S-unit, cumulant.

1

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Introduction

LetGbe a group acting on a probability space(E,B, µ)by a measure preserving action(g, x)∈ G×E → T^gx∈ E. A random walk (Zn) defined on a probability space (Ω,P) with values in the group Ginduces a random walk onE. Forf ∈L²(E,B, µ), we can consider the sums along the random walk: Pn−1

k=0f(T^Z^kx).

This general framework leads in practice to different situations and methods in the proof of a central limit theorem (CLT) and a functional central limit theorem (FCLT) along the paths of the random walk. In particular the proof of the tightness for the FCLT requires specific tools which it seems interesting to present in examples.

A first situation is that of a random walk in random sceneries (cf. [17], [5]). For d ≥ 1, let X = (Xℓ)_ℓ_∈Z^d be a strictly Z^d-stationary real random field (r.f.). One can assume that the r.v.s Xℓ are defined on a probability space (E,B, µ) on which commuting measure preserving maps T1, ..., Td act in such a way ¹ that Xℓ =T^ℓX0.

Conversely, given commuting measure preserving invertible maps T1, ..., Td and a measurable f on a probability space (E,B, µ), (T^ℓf)_ℓ_∈Z^d is a strictly Z^d-stationary random field. If (Zn) is a random walk in the group Z^d, then the sums along Z_n read Pn−1

k=0X_Z_k =Pn−1

k=0T^Z^kX₀, or Pn−1

k=0T^Z^kf. When (Xℓ) is a d-dimensional random field of i.i.d. random variables, we obtain the classical random walk in random sceneries.

Another types of examples in the algebraic case can be obtained as follows. Suppose that G = SL(ρ,Z) and that (E,B, µ) is the torus T^ρ, ρ ≥ 2, endowed with the Borel σ-algebra and the Lebesgue measure. The map x → Ax, where A is a matrix in SL(ρ,Z), defines an automorphism of T^ρ which preserves µ. When a spectral gap property is available for the transition operator associated to the random walk onG, the previous sums forf in a convenient class of observables satisfy a CLT (cf.[1], [11]) for P-a.e. ω ∈Ω.

The commutative case, which we will consider here, is different from the spectral point of view. For the action of commuting matrices in SL(ρ,Z) acting on T^ρ, we prove for P-a.e. ω a functional CLT, extending previous results in [8]. A second algebraic example comes from commuting flows on homogeneous spaces. Based on the exponential mixing of all orders proved in [3], a CLT has been shown in [4] for ergodic sums on Følner sets when the observables are smooth. Likewise we prove here a CLT and its functional version for the sums along a random walk.

The result, a functional CLT for the different models described above, is presented for a general aperiodic random walk in dimensiond >1with a moment of order 2, but the detailed proofs are given in the case of a centered 2-dimensional r.w. The proofs can be adapted easily to the case of transient random walks. We say also some words in the i.i.d. case, when the usual random walk is replaced by a plane Lorentz process generated by a periodic billiard with dispersive obstacles (cf. [23], [24]).

Beyond the CLT, tightness is a main step in the proof of a FCLT. To show it we use the method based on the maximal inequality for associated r.v.s due to Newman and Wright [22]

1Underlined letters represent elements of Z^d or T^d. We write ℓ for (ℓ1, ..., ℓd) and T^ℓ for T₁^ℓ¹...T_d^ℓ^d. The euclidean norm ofℓ∈Z^d is denoted by|ℓ|or kℓk.

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or, in the algebraic case, the method based on norm estimates for the maximum of partial sums (cf. Billingsley [2], Móricz [21]). A difficulty which occurs is that the estimates available for the random walk involve constants depending on the trajectory.

The content of the paper is the following. Section 1 contains results on the variance of sums along a random walk. The independent case is presented in Section 2. Some facts on cumulants are recalled in Section 3, then applied to moving averages in Section 4 and to the algebraic models in Sections 6 and 7. For the tightness in the latter cases, we use the method of maximum of partial sums in an adapted version presented in Section 5.

The paper is an extension of a previous version. We have added the FCLT along a random walk for flows on homogeneous spaces, using the recent results in [3] and [4] on the multiple mixing and on the CLT for group actions which are exponentially mixing of all orders. We have also added some remarks about the non nullity of the variance, in particular, the observation that there is no degeneracy for the sums along a transient random walk.

Acknowledgements. This research started during visits of the first author to the IRMAR at the University of Rennes 1 and of the second author to the Center for Advanced Studies in Mathematics at Ben Gurion University. The authors are grateful to their hosts for their support. They thank Y. Guivarc’h and B. Bekka for helpful discussions.

1. Summation along a r.w. and variance 1.1. Random walks and sums along random walks.

First we recall some definitions and results about the random walks on Z^d (see [26], details on the results recalled here can also be found in [8]).

Let (ζi)i≥0 be a sequence of i.i.d. random vectors on a probability space (Ω, P) with values in Z^d and common probability distribution ν. The associated random walk(r.w.) Z = (Zn)inZ^d starting from 0 is defined by Z0 := 0, Zn:=ζ0+...+ζn−1,n ≥1.

The r.v.s ζ_i can be viewed as the coordinate maps on (Ω, P) obtained as (Z^d)^Z equipped with the product measure ν^⊗^Z and with the shift θ acting on the coordinates. We have ζi = ζ0◦θⁱ and the cocycle relation Zn+n^′ =Zn+Zn^′ ◦θⁿ,∀n, n^′ ≥0.

Let denote by S := {ℓ ∈ Z^d : P(ζ0 = ℓ) > 0} the support of ν and by L the sub-lattice of Z^d generated byS. Without loss of generality, we can assume that Z is reduced which means that L is cofinite in Z^d. Therefore the vector space generated by Lis R^d andd is the ‘genuine’

dimension of the random walk Z.

For simplicity, we will assume thatL=Z^d(the random walkZ is said to beaperiodic). Observe that one can replace a reduced r.w. Z by an aperiodic one, again without loss of generality.

Let D be the sub-lattice ofZ^d generated by{ℓ−ℓ^′, ℓ, ℓ^′ ∈ S}. We denote by Γ1 the annulator in T^d ofD, that is the closed subgroup of T^d defined by{t ∈T^d:e^2πi^h^r,tⁱ = 1,∀r ∈D} and by dγ1 the Haar probability measure of the group Γ1. The r.w. is said to be strictly (or strongly) aperiodic, if D=Z^d.

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Sums along random walks

Given a strictly Z^d-stationary random field X = (Xℓ)_ℓ_∈Z^d, where the real random variables Xℓ

are defined on a probability space (E,B, µ), the process of ‘ergodic sums’ along the random walk (Zn)is

S_n^ω,X(x) =S_n^ω(x) :=

n−1

X

k=0

X_Z_k_(ω)(x), n≥1, ω∈Ω.

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If the random field is represented as Xℓ = T^ℓf, where T1, ..., Td are commuting measure preserving maps and f ∈L²(E,B, µ), the sums read:

S_n^ωf =

n−1

X

k=0

T^Z^k^(ω)f =X

ℓ∈Z^d

wn(ω, ℓ)T^ℓf, (2)

where wn(ω, ℓ) (denoted also byw^ω_n(ℓ)) is the local time of the random walk at time n ≥1:

wn(ω, ℓ) = #{k < n:Zk(ω) =ℓ}=

n−1

X

k=0

1Zk(ω)=ℓ. (3)

Summing along the random walk amounts to take the ergodic sums for the skew product (ω, x) → T_ζ₀(ω, x) = (θω, T^ζ⁰^(ω)x) on Ω ×E. Putting F_f(ω, x) = F(ω, x) = f(x) for an observable f on E, we get that the ergodic sums of F for Tζ0 read:

SnF(ω, x) =

n−1

X

k=0

F(T_ζ^k₀(ω, x)) =

n−1

X

k=0

f(T^Z^k^(ω)x) = (S_n^ωf)(x).

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A limit theorem in distribution for the sumsS_n^ωf (with respect to the measureµonE) obtained for P-a.e. ω is sometimes called quenched. We will use this terminology ². If the random variables SnF(ω, x)are viewed as defined onΩ×E endowed with the probability P×µ, a limit theorem under P×µ for theses sums is called annealed.

1.2. Variance for quenched processes.

Let f be a function in L²(E,B, µ) with real values. Everywhere we assume (or prove) the absolute summability of the series of decorrelations

X

ℓ∈Z^d

| Z

X

T^ℓf f dµ|<∞, (5)

which implies existence and continuity of the spectral density, the even function given by ϕf(t) =X

ℓ∈Z^d

hT^ℓf fie^2πi^h^ℓ,tⁱ. (6)

The computation of the variance R

E|Pn−1

k=0 T^Z^k^(ω)f|²dµ, is related to the number of self- intersections of the random walk at time n≥1:

Vn(ω) := #{0≤u, v < n: Zu(ω) =Zv(ω)}= X

ℓ∈Z²

wn(ω, ℓ)² = Z

T^d

|X

ℓ∈Z^d

wn(ℓ)e^2πi^h^ℓ,tⁱ|²dt.

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2We follow here the terminology of [1] used in several papers. The term ‘quenched’ is also used in the random scenery when a limit theorem is shown for the distribution with respect to ω, conditionally to the scenery X.

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Let us consider the kernels (which are even functions) K(w_n^ω)(t) =|

n−1

X

k=0

e^2πi^h^Z^k^(ω),tⁱ|² =|X

ℓ∈Z^d

wn(ω, ℓ)e^2πi^h^ℓ,tⁱ|², K(w˜ _n^ω)(t) =Vn(ω)⁻¹K(w^ω_n)(t).

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We say that the summation along the r.w. Z is ξ-regular, where ξ is a probability measure on T^d, if (for P-a.e. ω) the normalised kernel ( ˜K(w^ω_n))n≥1 converges weakly to ξ, i.e., limn→∞

R

T^dK˜(w_n^ω)ϕ dt=ξ(ϕ)for every continuous function ϕ onT^d. This property is equivalent to (for P-a.e. ω):

nlim→∞

Z

K(w˜ _n^ω)(t)e⁻^2πi^h^p,tⁱ dt= lim

n→∞

Z

K(w˜ ^ω_n)(t) cos(2πhp, ti) dt= ˆξ(p),∀p∈Z^d. (9)

Another equivalent formulation is

nlim→∞

Vn(ω, p)

Vn(ω) = ξ(p),ˆ ∀p∈Z^d, for a.e. ω,

with Vn(ω, p) := #{0≤u, v < n: Zu(ω)−Zv(ω) =p}, p∈Z^d. (10)

For f satisfying (5), it implies that the (asymptotic) normalised variance is, for a.e. ω, σ²(f) := lim

n

kP

ℓ∈Z^dwn(ℓ, ω)T^ℓfk²2

P

ℓ∈Z^d|wn(ℓ, ω)|² = lim

n

Z

T^d

K(w˜ _n^ω)(t)ϕf(t)dt=ξ(ϕf).

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It can be shown that every summation associated to a random walk in Z^d isξ-regular for some measure ξ (cf. [8]).

We summarize below the results on the asymptotic variance (see [8] for the proofs).

1.2.1. Recurrence/transience.

Recall that a r.w. Z = (Zn) is recurrent if and only if P_∞

n=1P(Zn= 0) = +∞. Let m1(Z) :=P

ℓ∈Z^d P(ζ0 =ℓ)kℓk, m2(Z) :=P

ℓ∈Z^d P(ζ0 =ℓ)kℓk². For d= 1, ifm1(Z)<∞, then Z is recurrent if and only if it is centered;

for d= 2, if m₂(Z)<∞, thenZ is recurrent if and only if it is centered;

for d≥3, if m2(Z)<∞, then it is always transient.

We denote by Ψ(t) =E[e^2πi^h^ζ⁰^,tⁱ], t∈T^d, the characteristic function of the r.w. and put Φ(t) := 1− |Ψ(t)|²

|1−Ψ(t)|². (12)

Remark 1.1. Fort6= 0inT^d,Φis well defined (sinceZ is aperiodic), nonnegative andΦ(t) = 0 only on Γ₁\ {0}. Hence it is positive for a.e. t, except when the r.w. is ‘deterministic’ (i.e., if P(ζ0 =ℓ) = 1 for some ℓ∈Z^d, so that |Ψ(t)| ≡1 in this case).

A r.w. of genuine dimensiondwhich is aperiodic is transient or recurrent depending on whether ℜe(₁₋¹_Ψ) is integrable or not on the d-dimensional unit cube ([26]).

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Transient case

In the transient case, one can show:

Theorem 1.2. ([26]) Let Z = (Zn) be a transient aperiodic random walk in Z^d. a) The function Φ is integrable on T^d and, with a nonnegative constant K, we have

I(ℓ) := 1ℓ=0+ X∞ k=1

[P(Zk=ℓ) +P(Zk=−ℓ)] = Z

T^d

cos(2πhℓ, ti) Φ(t)dt+K,∀ℓ∈Z^d. b1) Suppose d = 1. If m1(Z) = +∞, then K = 0. If m1(Z) < ∞, then Z is non centered (because it is transient) and K =|P

ℓ∈Z P(X₀ =ℓ)ℓ|⁻¹. b2) If d >1, then K = 0.

c) Denoting by dξ(t) the measure Φ(t)dt+Kδ0(t), we have, for a.e. ω, Z 1

nK_n^ω(t) cos(2πhℓ, ti)dt= 1ℓ=0+ 1 n

n−1

X

k=1 n−Xk−1

j=0

[1_Z_k_(θ^j_ω)=ℓ+ 1_Z_k_(θ^j_ω)=₋_ℓ]

n→→∞ I(ℓ) = Z

cos(2πhℓ, ti)dξ(t).

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It follows that the summation along a transient r.w. behaves for the normalisation like the iteration of a single transformation, is ξ-regular (up to a constant factor) and that

limn

1 nk

n−1

X

k=0

T^Z^k^(ω)fk²2 = Z

Φ(t)ϕf(t)dt+Kϕf(0).

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From (13) , (14) and the expression of ϕf, ϕf(t) =P

ℓ∈Z^dhT^ℓf, ficos(2πhℓ, ti), we deduce:

Z

Φ(t)ϕf(t)dt+Kϕf(0) = Z

ϕf(t)dξ(t) = X

ℓ∈Z^d

hT^ℓf, fi Z

cos(2πhℓ, ti)dξ(t) = X

ℓ∈Z^d

I(ℓ)hT^ℓf, fi=kfk²2+ 2X

k≥1

X

ℓ∈Z^d

P(Zk =ℓ)hT^ℓf, fi .

Remark 1.3. (about the variance in the non deterministic transient case)

Let f be in L²(E, µ) with real values and satisfying (5). By Remark 1.1, the (quenched) asymptotic variance is 6= 0, if f is not a.e. equal to 0. Let F_f(ω, x) =F(ω, x) =f(x). For the map Tζ0 acting on the product space Ω×E endowed with the product measureP×µ, we have

Z

T_ζⁿ₀F F dµ dP=X

ℓ

(E1Zn=ℓ)hT^ℓf fi=X

ℓ∈Z^d

P(Zn =ℓ)hT^ℓf, fi, n≥0.

In the transient case it holds, for everyℓ,P

k≥1P(Zk =ℓ)≤P

k≥1P(Zk = 0)<+∞. Therefore the density of the spectral measure for Ff and the map Tζ0 is

kfk²2+ 2X

k≥1

X

ℓ∈Z^d

P(Zk =ℓ)hT^ℓf, fi

cos 2πkt.

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The asymptotic variance limn 1

nkPn−1

k=0T_ζ^k₀Ffk²2 for the annealed model is the same as for the quenched model and is equal to kfk²2+ 2X

k≥1

X

ℓ∈Z^d

P(Zk =ℓ)hT^ℓf, fi .

It follows that the function Ff onΩ×E (which depends only on the second coordinate), withf as above and non a.e. null, is never a coboundary in L²(P×µ)forTζ0, because the asymptotic variance is non null. Observe also that Ff even is not a measurable coboundary, at least when the CLT holds, which is the case of the situations that we are going to considered here. This follows from the fact that, for a single measure preserving transformation, if an observable is a coboundary in the space of measurable functions, then the limiting distribution of the ergodic sums after normalisation by any sequence tending to infinity is the Dirac mass at 0, which is excluded here.

Recurrent case

Let us consider now the case d= 2 and a centered random walk Z with a moment of order 2.

By the local limit theorem (LLT), Z is recurrent.

A non standard normalization occurs in the CLT for sums along Zn as recalled below. There are C0, C finite positive constants ³ such that (cf. [5] Lemma 2.6, [20] Proposition 1.4 for (15) and (16), [8] Theorem 4.13 for (17)):

E(V_n)∼C₀nlnn, Var(V_n)≤Cn², (15)

ϕn(ω) := Vn(ω)

C₀nlnn →1, for a.e. ω, (16)

ϕ_n(ω, p) := Vn(ω, p)

C0nlnn →1,∀p∈Z^d, for a.e. ω.

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Therefore the summation along the r.w. Z is δ0-regular: the normalised kernel satisfies limn

R K(w˜ ^ω_n)(t)e⁻^2πi^h^p,tⁱ dt= 1,∀p∈Z^d and the asymptotic variance is σ²(f) = lim

n (C0nlnn)⁻¹k Xn k=0

T^Z^k^(ω)fk²2 = X

k∈Z^d

hT^kf fi=ϕf(0).

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The results presented below are valid for the cases covered above, hence excludes only the one-dimensional recurrent case.

We stress that, in the recurrent 2-dimensional case, the variance can be degenerate, while this does not occur in the transient case unless f = 0.

1.2.2. Number of self-intersections of a 2-dimensional centered r.w.

In this subsection, we study more precisely the case d = 2 and a centered random walk (Zn) with a moment of order 2.

If I, J are intervals, the quantity V(ω, I, J, p) :=

Z X

u∈I

e^2πi^h^Z^u^(ω),tⁱ X

v∈J

e⁻^2πi^h^Z^v^(ω),tⁱ

e⁻^2πi^h^p,tⁱdt = #{(u, v)∈I×J : Z_u(ω)−Z_v(ω) =p} (19)

3If the r.w. is strongly aperiodic,C0= (π√

det Σ)⁻¹.

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is non negative and increases when I orJ increases for the inclusion order.

We write simply V(ω, I, p) if I = J, V(ω, I) for V(ω, I,0), Vn(ω) and Vn(ω, p) as above for V(ω,[0, n[)and V(ω,[0, n[, p).

Observe that V(ω, J) = P

ℓ∈Z²w(ω, J, ℓ)², where w(ω, J, ℓ) = P

i∈J1Zi(ω)=ℓ. Notice also that V(ω,[b, b+k[) =V(θ^bω,[0, k[) =Vk(θ^bω), for b≥0, k≥1.

LetA, B be in[0,1]. For simplicity, in the formulas above and below, we write nA, nB instead of ⌊nA⌋or⌊nA⌋+ 1, ⌊nB⌋, θ^t instead of θ^⌊^t^⌋. The equalities are satisfied up to the addition of quantities which are bounded independently from A, B, n. We have:

V(ω,[nA, nB], p) = Z

( X

u∈[nA,nB]

e^2πi^h^Z^u^(ω),tⁱ) ( X

v∈[nA,nB]

e⁻^2πi^h^Z^v^(ω),tⁱ)e⁻^2πi^h^p,tⁱdt

= #{u, v ∈[0, n(B −A)] :

u−1

X

i=0

ζ₀(θ^i+nAω)−

v−1

X

i=0

ζ₀(θ^i+nAω) =p}=V(θ^nAω,[0, n(B−A)], p).

By (16) and (17) there a set Ω0 of full probability such that

Vn(ω)≤K(ω)nlnn, ∀n ≥2, where the function K ≥0 is finite on Ω0, (20)

for any fixed A∈]0,1], V(ω,[1, nA], p)∼C0nAlnn, for ω∈Ω0. (21)

By [5, Lemma 2.5] we have sup

ℓ∈Z²

w_n(ω, ℓ) =o(n^ε), for a.e. ω, for every ε >0.

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For a simple r.w. on Z², Erdös and Taylor ([14]) have shown: lim sup

n

sup

ℓ∈Z²

wn(ω, ℓ) (logn)² ≤ 1

π. The result has been extended by Dembo, Peres, Rosen and Zeitouni who proved for an aperiodic centered random walk on Z² with moments of all orders [13]:

limn sup

ℓ∈Z²

wn(ω, ℓ) (logn)² = 1

π. (23)

We will need also to bound, for ℓ₁, ℓ₂, ℓ₃ ∈Z^d,W_n(ω, ℓ₁, ℓ₂, ℓ₃) :=

(24) #{1≤i0, i1, i2, i3 < n: Zi1(ω)−Zi0(ω) =ℓ₁, Zi2(ω)−Zi0(ω) = ℓ₂, Zi3(ω)−Zi0(ω) =ℓ₃}. Lemma 1.4. There exists a positive integrable function C3 such that

Wn(ω, ℓ₁, ℓ₂, ℓ₃)≤C3(ω)n(lnn)⁵, ∀n ≥1.

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Proof. It suffices to bound the sum with strict inequality between indices W_n^′(ω) = X

1≤i0<i1<i2<i3≤n

1_Z_i₁₋_Z_i₀₌_ℓ₁.1_Z_i₂₋_Z_i₁₌_ℓ₂₋_ℓ₁.1_Z_i₃₋_Z_i₂₌_ℓ₃₋_ℓ₂.

Using independence and the local limit theorem for the random walk, we find the bound Z

W_n^′(ω)dP(ω)≤C1

X

i0,i1,i2,i3∈[1,n]

(i1i2i3)⁻¹ ≤C2n(lnn)³.

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Therefore X∞

p=1

Z

2⁻^p(ln(2^p))⁻⁵W₂^′^pdP < ∞. The function C(ω) := P_∞

p=12⁻^p(ln(2^p))⁻⁵W₂^′^p is integrable and we have: W₂^′^p(ω)≤C(ω) 2^p(ln(2^p))⁵, ∀p≥1.

Let pn be such that: 2^pⁿ⁻¹ ≤n <2^pⁿ. Since W_n^′ is increasing with n, we obtain:

W_n^′(ω)≤W₂^′pn(ω)≤C(ω) 2^pⁿ(ln(2^pⁿ))⁵ ≤C(ω) 2n(ln(2n))⁵ ≤C^′(ω)n(lnn)⁵.

Variance for the finite dimensional distributions

The following lemma will be applied to the successive return times of a point ωinto a set under the iteration of the shift θ.

Lemma 1.5. Let(y(j), j ≥1) be a sequence with values in{0,1}such that limn 1 n

Pn

j=1y(j) = a > 0. If (kr) is the sequence of successive times such that y(kr) = 1, then, for every δ > 0, there is n(δ) such that, for n ≥n(δ), kr+1−kr ≤δn, for all r ∈[1, n].

Proof. Since r = Pkr

j=1 y(j), we have: kr/r = kr/Pkr

j=1 y(j) → a⁻¹. Hence, for every δ > 0, there is n1(δ) such that 0 < kr+1 −kr ≤ δr, for r ≥ n1(δ). Therefore, if n ≥ n1(δ), then 0< kr+1−kr ≤δr ≤δn, forr∈[n1(δ), n].

Ifn(δ)≥n₁(δ)is such thatk_r+1−k_r ≤δn(δ)forr ≤n₁(δ), we get the result of the lemma.

Lemma 1.6. Let Λ be a measurable set in Ω of positive measure. Let kr = kr(ω) be the successive times such that θ^k^rω ∈ Λ. For a.e. ω, for every positive small enough δ, there is n(δ) such that for n ≥n(δ)

1) kr+1−kr ≤δn, for all r∈[1, n]; moreover, kn∼cn, with c=P(Λ)⁻¹, when n → ∞; 2) there are integers v < 2/δ and 0 =ρ⁽ⁿ⁾₁ < ρ⁽ⁿ⁾₂ < ... < ρ⁽ⁿ⁾v ≤ n < ρ⁽ⁿ⁾_v+1, such that θ^ρ⁽ⁿ⁾ⁱ ω ∈Λ and ¹₂δn≤ρ⁽ⁿ⁾_i+1−ρ⁽ⁿ⁾_i ≤ ³₂δn, for i= 1, ..., v.

Proof. Since θ is ergodic on (Ω,P), Birkhoff ergodic theorem implies lim_n_n¹ Pn−1

0 1_Λ(θ^kω) = P(Λ) > 0, for a.e. ω and kn/n → P(Λ)⁻¹. Hence Lemma 1.5 implies 1). For 2), we select in the sequence (kr) an increasing sequence of visit times to the set Λ satisfying the prescribed conditions by eliminating successive times which are at a distance < ¹₂δn.

Asymptotic orthogonality of the cross terms

Proposition 1.7. For 0< A < B < C < D <1, p∈Z, Z (

XnB v=nA

e^2πi^h^Z^v^(ω),uⁱ) ( XnD w=nC

e⁻^2πi^h^Z^w^(ω),uⁱ)e⁻^2πi^h^p,uⁱ

du=εn(ω)nlogn, with εn(ω)→0.

(26)

The above integral is the non negative self-intersection quantity: V(ω,[nA, nB],[nC, nD], p).

By (19), V(ω, I, J, p) increases when I or J increases. Hence, it suffices to show (26) for the intervals [1, nA],[nA, n], for 0< A <1. The proof below is based on (17) and (21).

Lemma 1.8. There is a set Ωˆ ⊂Ω such that P( ˆΩ) = 1 and for all ω ∈Ω, the following holds:ˆ limn ϕnB(θ^nAω, p) = lim

n

V(ω,[nA, n], p)

C₀nBlnn = 1, for A∈]0,1[, B = 1−A;

(27)

V(ω,[1, nA],[nA, n], p) +V(ω,[nA, n],[1, nA], p) =εn(ω)nlogn, with εn(ω)→0.

(28)

(11)

Proof. 1) The setΩ.ˆ For everyL≥1andδ >0, let Λ(L, δ) :={ω :ϕn(ω, p)−1∈[−δ, δ],∀n≥ L}. We have limL↑∞P(Λ(L, δ)) = 1. There is L(δ) such thatP(Λ(L(δ), δ))≥ ¹₂.

Let(δj)be a sequence tending to 0. We apply Lemma 1.6 toΛ(L(δj), δj)for eachj. By taking the intersection of the corresponding sets, we get a set of ω’s of full P-measure. The set Ωˆ is the intersection of this set with the set Ω0 (of full measure) for which the law of large numbers holds for (Vn(ω)). Let ω ∈Ω.ˆ

2) Proof of (27). We have V(ω,[nA, n[, p) =V(θ^nAω,[0, n(1−A)[, p)and V(ω,[1, n], p)−V(ω,[1, nA[, p)−V(ω,[nA, n], p)

=V(ω,[1, nA[,[nA, n[, p) +V(ω,[nA, n],[1, nA[, p)≥0.

(29)

Claim: for an absolute constant C1 depending on A and p, for every δ, forn big enough, ϕ_nB(θ^nAω, p) = V(ω,[nA, n], p)

C0n(1−A) lnn ∈[1−C₁δ,1 +C₁δ].

(30)

Upper bound: The law of large numbers forVn(ω, p)implies, with|εn|,|ε^′_n| ≤δfornbig enough, C₀⁻¹V(ω,[1, n], p) = (1 +εn)nlnn, C₀⁻¹V(ω,[1, nA], p) = (1 +ε^′_n)nAlnn.

With B = 1−A, this implies by (29) V(ω,[nA, n], p)

C0nBlnn ≤ (1 +εn)nlnn−(1 +ε^′_n)nAlnn

nBlnn ≤1 + |εn|

B +|ε^′_n|A

B ≤1 + 1 +A B δ.

Lower bound: We apply Lemma 1.6 to Λ(L(δ), δ). Let nA, n^′_A be two consecutive visit times

≤n such thatn_A≤nA < n^′_A. Forn big enough, we have 0< n^′_A−n_A≤δn and nA=nA(1−ρn), n^′_A =nA(1 +ρ^′_n), with 0≤Aρn, Aρ^′_n≤δ.

Since ω ∈Ω, we have forˆ n big enough, with|δ_n^′| ≤δ,

C₀⁻¹V(ω,[n^′_A, n], p)≥(1−δ_n^′)(n−n^′_A) ln(n−n^′_A) = (1−δ^′_n)(nB−nAρ^′_n) ln(nB−nAρ^′_n).

It follows, for δ (hence ρ^′_n) small:

V(ω,[n^′_A, n], p)

C0(1−δ_n^′)nBln(nB) ≥ (nB−nAρ^′_n) ln(nB−nAρ^′_n)

nBln(nB) = (B −Aρ^′_n) [ln(nB) + ln(1−_B^Aρ^′_n)]

Bln(nB)

≥(1− A

Bρ^′_n)−2(1− A Bρ^′_n)

A Bρ^′_n

ln(nB) ≥1− A

Bρ^′_n−2

A Bρ^′_n

ln(nB) ≥1−B⁻¹δ(1 + 2 ln(nB)).

As V(ω, J, p) increases when the setJ increases, we have by the choice of nA and n^′_A: V(ω,[n^′_A, n], p)≤V(ω,[nA, n], p).

Therefore, for n such that ln(nB)≥2, we have V(ω,[nA, n], p)

C0nBln(nB) ≥(1−δ) (1− 2

Bδ)≥1−δ(1 + 2 B).

This shows the lower bound. Altogether with the upper bound, this proves the claim (30).

(12)

3) Proof of (28). Let δ > 0. According to (29) and (30), for n big enough, we have with

|ε^′′_n| ≤C₁δ:

V(ω,[1, nA],[nA, n], p) +V(ω,[nA, n],[1, nA], p) =V(ω,[1, n], p)−V(ω,[1, nA], p)−V(ω,[nA, n], p)

=C0[(1 +εn)nlnn−(1 +ε^′_n)nAlnn−(1 +ε^′′_n)n(1−A) lnn≤(2 +C1)C0δ nlnn.

Let a1, ..., as be real numbers and 0 = t0 < t1 < ... < ts−1 < ts = 1 a subdivision of [0,1]. For the asymptotic variance of Ps

j=0 ajPntj

k=ntj−1T^Z^k^(ω)f, which is used later, we need the following lemma. Recall that f has a continuous spectral density ϕf.

Lemma 1.9. For a.e. ω and for every partition (tj), we have

(C0nlnn)⁻¹k Xs

j=1

aj ntj

X

k=ntj−1

T^Z^k^(ω)fk²2 →ϕf(0) Xs

j=1

a²_j(tj −tj−1).

(31)

Proof. 1) Recall that proving (31) amounts to prove

(C0nlnn)⁻¹ Z

| Xs

j=1

aj ntj

X

k=ntj−1

e^2πi^h^Z^k^(ω),uⁱ|²ϕf(u)du→ϕf(0) Xs

j=1

a²_j(tj −tj−1).

1) First suppose that ϕf is a trigonometric polynomial ρ, which allows to use (26) for a finite set of characters e⁻^2πi^h^p,uⁱ. Using (18) for the asymptotic variance starting from 0, we have (C0nlnn)⁻¹kP_⌊tn⌋

k=0T^Z^k^(ω)fk²2 →tρ(0), for t∈]0,1[. By Lemma 1.8, (C0nlnn)⁻¹k

⌊tn⌋

X

k=⌊sn⌋

T^Z^k^(ω)fk²2→(t−s)ρ(0), for 0< s < t < 1.

Expanding the square and using that the cross terms are asymptotically negligible, we have

(C0nlnn)⁻¹ Z

| Xs

j=1

aj ntj

X

k=ntj−1

e^2πi^h^Z^k^(ω),uⁱ|²ρ(u)du

∼ (C0nlnn)⁻¹ Xs

j=1

a²_j Z

|

ntj

X

k=ntj−1

e^2πi^h^Z^k^(ω),uⁱ|²ρ(u)du

→ρ(0) Xs j=1

a²_j(tj−tj−1).

This shows (31) for trigonometric polynomials.

2) For a general continuous spectral density ϕf, for ε >0, let ρbe a trigonometric polynomial, such that kϕf −ρk∞< ε. Remark that

Z

| Xs

j=1

aj ntj

X

k=ntj−1

e^2πi^h^Z^k^(ω),uⁱ|²du≤ Xs j,j^′=1

ajaj^′V(ω,[ntj−1, ntj],[ntj^′−1, ntj^′],0)≤( Xs

j=1

|aj|)²Vn(ω).

(13)

Therefore we have:

(C0nlnn)⁻¹ Z

| Xs

j=1

aj ntj

X

k=ntj−1

e^2πi^h^Z^k^(ω),uⁱ|²ϕf(u)du−ϕf(0) Xs

j=1

a²_j(tj−tj−1)

≤(C0nlnn)⁻¹ Z

| Xs

j=1

aj ntj

X

k=ntj−1

e^2πi^h^Z^k^(ω),uⁱ|²ρ(u)du−ρ(0) Xs

j=1

a²_j(tj −tj−1)

+ε[(C₀nlnn)⁻¹ Z

| Xs

j=1

a_j

ntj

X

k=ntj−1

e^2πi^h^Z^k^(ω),uⁱ|²du+ Xs

j=1

a²_j(t_j−t_j₋₁)].

By the remark, the above quantity inside [ ] is less than (Ps

j=1 |aj|)²(C0nlnn)⁻¹Vn(ω) + Ps

j=1a²_j(tj−tj−1), which is bounded uniformly with respect to n. Therefore we can conclude

for a general continuous spectral density by step 1).

Remarks 1.10. 1) In Lemma 1.6, the dynamical system (Ω, θ,P) can be replaced by any ergodic dynamical system.

2) If the spectral density is constant (i.e., when the Xk’s are pairwise orthogonal), (26) and (31) are a consequence of the law of large numbers for the number of self-intersections, that is

Vn(ω)

C0nlnn →1. The law of large numbers for Vn(ω, p), p6= 0, is not needed.

3) A result analogous to Proposition 1.7 is valid if the r.w. Z is transient: for 0< A < B <

C < D < 1, p∈Z, Z (

XnB v=nA

e^2πi^h^Z^v^(ω),uⁱ) ( XnD w=nC

e⁻^2πi^h^Z^w^(ω),uⁱ)e⁻^2πi^h^p,uⁱ

du=εn(ω)n, with εn(ω)→0.

(32)

4) The quenched FCLT shown in the different examples below is valid for a set of ω’s of P- measure 1 given by the results of this Section 1. This set does not depend on the Z^d-dynamical systems considered in the further sections. The joint distribution on Ω×E is used only when the annealed model is mentionned, like for Bolthausen’s result recalled in Section 2.

1.2.3. Formulation of the quenched FCLT.

Let (Yn(t), t ∈ [0,1]) be a process on (E, µ) with values in the space C[0,1) of real valued continuous functions on [0,1] or in the space D[0,1] of right continuous real valued functions on [0,1]with left limits, endowed with the uniform norm.

Let (W(t), t ∈ [0,1]) be the Wiener process on [0,1]. To show a functional limit theorem (FCLT) for (Y_n(t), t∈[0,1]), i.e., weak convergence to the Wiener process, it suffices to prove the two following properties (where “=⇒” denotes the convergence in distribution):

1) Convergence of the finite dimensional distributions:

∀0 =t0 < t1 < ... < tr = 1, (Yn(t1), ..., Yn(tr)) =⇒

n→∞(Wt1, ..., Wtr), a property which follows (by the Cramér-Wold device) from

Pr

j=1aj(Yn(tj)−Yn(tj−1)) =⇒ N(0,Pr

j=1a²_j(tj −tj−1)), ∀(aj)1≤j≤r∈R. (33)

(14)

2) Tightness of the process. The condition of tightness reads:

∀ε >0, lim

δ→0⁺lim sup

n

µ(x∈E : sup

|t^′−t|≤δ|Yn(ω, x, t^′)−Yn(ω, x, t)| ≥ε) = 0.

(34)

Let (Zn) be a random walk on (Ω,P) with values in Z^d, d ≥ 1, and let X = (Xℓ(x))_ℓ_∈Z^d = (T^ℓf(x))_ℓ_∈Z^d be a d-dimensional random field defined on a probability space (E,B, µ). A quenched FCLT is satisfied by the sums along Zn if, for P-a.e. ω, the functional central limit theorem holds for the process (cf. Notation (1))

(Yn(ω, x, t))t∈[0,1] := S_[nt]^ω,X(x)

√C0nlogn

t∈[0,1]. (35)

2. Random walk in random scenery

We consider in this section d= 2 and the random walk in random scenery S_n^ω,X(x), that is the process (Y_n)defined by (35) whenXis a 2-dimensional random field of i.i.d. real variables with E(X₀²) = 1 and mean 0 on a probability space (E,B, µ).

It was shown by E. Bolthausen [5] that this process satisfies an annealed FCLT: with respect to the probability P×µ, the law of Yn converges weakly to the Wiener measure.

We show a quenched FCLT for the r.f. X (and when X is a r.f. of moving averages of i.i.d.

random variables in Section 4). As for the annealed FCLT in [5] and for Theorem 2.2 in [12] for a 1-dimensional stable r.w., the proof of Theorem 2.2 below is based on the maximal inequality shown by Newman and Wright [22] for associated r.v.s.

Definition 2.1. (cf. [15]) Recall that real random variables X1, . . . , Xn are associated if, for every n ≥ 1, for all non-decreasing (in each coordinate) functions f, g : Rⁿ 7→ R, we have Cov(f(X1, . . . , Xn), g(X1, . . . , Xn))≥0(if the covariance exists). Non-decreasing functions of a family of associated random variables are associated [15]. Independent variables are associated.

It follows that, if (Xℓ, ℓ ∈ Z²), are associated r.v.s, in particular independent, then the r.v.s (XZk(ω), k ≥0), are associated for every ω ∈Ω.

Theorem 2.2. If E(X₀²) = 1, for P-a.e. ω, the process Yn(ω, x, t

t∈[0,1] = S_⌊^ω,X_nt_⌋(x)

√nlogn

t∈[0,1]

satisfies a FCLT with asymptotic variance σ² = (π√

det Σ)⁻¹.

Proof. 1) For the convergence of the finite dimensional distributions, the proof, relying on Cramér-Wold’s theorem and Lindeberg’s CLT, is as in [5]. Another proof, based on truncation and cumulants, can be given, like for the more general case of moving averages in Section 4.

2) Tightness of the process(Y_n). The following is shown in [22], p. 673:

Let U1, U2, . . . be centered associated random variables with finite second order moment. Put Sk =Pk

j=1Uj, fork ≥1. Then, for every λ >0 and n≥1, we have µ( max

1≤k≤n|S_k| ≥λkS_nk2)≤2µ |S_n| ≥(λ−√

2)kS_nk2

. (36)

Inequality (36) can be applied for every fixed ω to Uj = XZj(ω) and to the sums SJ = Pb+k

j=bXZj(ω) for any interval J = [b, b+k]⊂ [0, n]. We also note that E(S_J²) =kX0k²2V(ω, J).

(15)

a) First, let us assume that E(X₀⁴)<∞. With K given by (20), we have kX

i∈J

XZi(ω)k⁴4,µ = 3E(X₀²)² X

ℓ₁6=ℓ₂

w(ω, J, ℓ₁)²w(ω, J, ℓ₂)²+ E(X₀⁴) X

ℓ

w(ω, J, ℓ)⁴

≤4E(X₀⁴)V(ω, J)² ≤4E(X₀⁴) (K(θ^bω))²(klnk)². (37)

Let C1 be a constant > 0 such that P{ω : K(ω) ≤ C1} > 0. Using Lemma 1.6, for n big enough and δ ∈]0,1[, there are times 0 = ρ1 < ρ2 < ... < ρv ≤ n < ρv+1, with v < 2/δ, such that K(θ^ρⁱω)≤C1 and ¹₂δn≤ρi+1−ρi ≤ ³₂δn, for i= 1, . . . , v.

Let ti = ^ρ_nⁱ, λ = ^√^ε_δ, Ji = [ρi−1, . . . , ρi[, mi = ²₃(ρi+1 −ρi)≤δn. There is C such that, by (20) and (37),

k

ρi

X

j=ρi−1

XZj(ω)k² ≤CkX0k²(n δ log(nδ))¹²,k

ρi

X

j=ρi−1

XZj(ω)k⁴ ≤CkX0k⁴(n δ log(nδ))¹²,∀i.

(38)

Using (36), we get, with σ⁽ⁱ⁾ := kPρi

j=ρi−1X_Z_j_(ω)k², λi := ε√

nlogn/σ⁽ⁱ⁾, by Chebyshev’s inequality (for a moment of order 4):

µ( sup

ti−1≤s≤ti

|

⌊sn⌋

X

j=[ti−1n⌋

XZj(ω)| ≥εp

nlogn) =µ( max

ρi−1≤k≤ρi| Xk j=ρi−1

XZj(ω)| ≥λiσ⁽ⁱ⁾)

≤2µ(|

ρi

X

j=ρi−1

XZj(ω)| ≥(λi−√

2)σ⁽ⁱ⁾)≤2µ(|

ρi

X

j=ρi−1

XZj(ω)| ≥ 1

2λiσ⁽ⁱ⁾)

≤2µ(|

ρi

X

j=ρi−1

XZj(ω)| ≥ 1 2εp

nlogn)≤2C⁴kX0k⁴4

(n δ log(nδ))²

1

16ε⁴(nlogn)² ≤32C⁴kX0k⁴4

δ² ε⁴. (39)

We have used that λi is big if δ is small. Observe now that (cf. [2]) µ( sup

|t^′−t|≤δ|Yn(t)−Yn(s)| ≥3ε)≤ Xv

i=1

µ( sup

ti−1≤s≤ti

|

⌊sn⌋

X

j=[ti−1n⌋

XZj(ω)| ≥εp

nlogn).

Hence, by (39) we get µ( sup

|t^′−t|≤δ|Yn(t)−Yn(s)| ≥3ε)≤32C⁴kX0k⁴4

2 δ

δ²

ε⁴ = 64C⁴kX0k⁴4

δ ε⁴. b) Now we use a truncation. For L >0, let

Xˆ_k^L:=Xk1{Xk≤L}−E(Xk1{Xk≤L}), X˜_k^L :=Xk−Xˆ_k^L =Xk1{Xk>L}−E(Xk1{Xk>L}), Yˆ_n^L(t) = 1

√C₀nlogn

⌊tn⌋

X

j=0

Xˆ_Z^L_j_(ω) and Y˜_n^L(t) := Y_n(t)−Yˆ_n^L(t) = 1

√C₀nlogn

⌊tn⌋

X

j=0

X˜_Z^L_j_(ω). Since we have still sums of associated random variables, all what we have done above (including (36) holds for both sums, except that for the unbounded part of the truncation we only have a moment of order 2. We use Chebyshev’s inequality (for a moment of order 2) to control the unbounded truncated part:

µ(|

ρi

X

j=ρi−1

X˜_Z^L_j_(ω)| ≥ 1 2εp

nlogn)≤C²kX˜₀^Lk²2

n δ log(nδ)

1

4ε²nlogn ≤4C²kX˜₀^Lk²2

δ ε².