PERSISTENCE EXPONENT FOR RANDOM WALK ON DIRECTED VERSIONS OF Z 2

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PERSISTENCE EXPONENT FOR RANDOM WALK ON DIRECTED VERSIONS OF Z 2

Nadine Guillotin-Plantard, Françoise Pene

To cite this version:

Nadine Guillotin-Plantard, Françoise Pene. PERSISTENCE EXPONENT FOR RANDOM WALK ON DIRECTED VERSIONS OF Z 2. 2015. �hal-01187968�

VERSIONS OF Z²

NADINE GUILLOTIN-PLANTARD AND FRANC¸ OISE P`ENE

Abstract. We study the persistence exponent for random walks in random sceneries (RWRS) with integer values and for some special random walks in random environment inZ² including random walks inZ² with random orientations of the horizontal layers.

1. Introduction and main results

Random walks in random sceneries were introduced independently by H. Kesten and F. Spitzer [15] and by A. N. Borodin [6]. LetS = (S_n)_n≥0 be a random walk inZstarting at 0, i.e.,S₀ = 0 andX_n:=S_n−S_n−1, n≥1 is a sequence of i.i.d. (independent identically distributed)Z-valued random variables. Let ξ = (ξ_x)_x∈Z be a field of i.i.d. Z-valued random variables independent of S. The field ξ is called the random scenery. The random walk in random scenery (RWRS) Z:= (Z_n)_n≥0 is defined by settingZ₀ := 0 and, for n∈N∗,

Z_n:=

n

X

i=1

ξ_Si. (1)

We will denote byPthe joint law ofS andξ. Limit theorems for RWRS have a long history, we refer to [14] for a complete review.

In the following, we consider the case when the common distribution of the scenery ξ_x is assumed to be symmetric with a third moment and with positive varianceσ²_ξ. Concerning the random walk (S_n)_n≥1, the distribution of X₁ is assumed to be centered and square integrable with positive variance σ²_X. We assume without any loss of generality that neither the support of the distribution ofX₁ nor the one of ξ₀ are contained in a proper subgroup ofZ.¹

Under the previous assumptions, the following weak convergence holds in the space of c`adl`ag real-valued functions defined on [0,∞), endowed with the Skorokhod topology (with respect to the classical J₁-metric):

n⁻¹²S_⌊nt⌋

t≥0

=L⇒

n→∞(σ_XY(t))_t≥0,

where Y is a standard real Brownian motion. We will denote by (Lt(x))x∈R,t≥0 a continuous version with compact support of the local time of the process (σ_XY(t))_t≥0 (see [19]). In [15], Kesten and Spitzer proved the convergence in distribution of ((n^−3/4Z_[nt])_t≥0)_n, to a process

2000Mathematics Subject Classification. 60F05; 60G52.

Key words and phrases. Random walk in random scenery; local limit theorem; local time; persistence; random walk in random environment

This research was supported by the french ANR project MEMEMO2.

1If the subgroup ofZgenerated by the support of the distribution ofX1ismZfor somem >1, then we replace (Sn)nby (Sn/m)nand (ξx)x by (ξmx)x and we observe that these changes do not affect the RWRSZ.

If the greatest common divisor of the support of the distribution ofξ0is ˜m >1, we divide the RWRSZ by ˜m.

1

∆ = (∆t)t≥0 defined by

∆_t:=σ_ξ Z

R

L_t(x) dW(x),

where (W(x))_x≥0 and (W(−x))_x≥0are independent standard Brownian motions independent of Y. The process ∆ is called Kesten-Spitzer process in the literature. We are interested in the persistence properties of the sum Zn, n≥1. Our main result in this setup is the following one.

Theorem 1. There exists a constant c >0 such that for large enough T Ph

k=1,...,Tmax Z_k≤1i

≤T^−1/4(logT)^c. (2) If moreover E[e^ξ1]<∞, then there exist positive constants c^′, c^′′ andT₀ such that

T^−1/4(logT)^−c′h H⁻¹

c^′′T⁻¹⁴i₋1

≤P h

k=1,...,Tmax Z_k≤1i

(3) for everyT > T₀, where H is given by H(t) :=E[e^ξ11_{eξ1>t}] and where H⁻¹(x) := inf{t > 0 : H(t)< x}, for every x >0.

In particular, if there exist η >1, A₁>0 and A₂ >0 such that

∀x >0, P(ξ₀> x)≤A₁e^−A2xη, then there exist c^′ >0 and T0 >0 such that

T^−1/4e^{−c′(logT)}

η1

≤P h

k=1,...,Tmax Z_k≤1i

(4) for everyT > T₀.

If the distribution ofξ₁ has compact support, then there exist positive constantsc^′ andT₀ such that

T^−1/4(logT)^−c′ ≤P h

k=1,...,Tmax Z_k ≤1i

(5) for everyT > T0.

The corresponding results for the continuous-time Kesten-Spitzer process ∆ were obtained in [10], also cf. [18, 20, 11]. The case of random walk in random gaussian scenery was treated in [3] with a lower bound in T^−1/4e^{−c′√logT}, coherent with our result. We would particularly like to stress that in all these results, the scenery was supposed to be gaussian.

Now we will state an analogous result for particular models of random walks (M_n)_nin random environment on Z² including random walks on Z² with random orientation of the horizontal layers.

To they-th horizontal line, we associate theZ-valued random variableξ_y, corresponding to the only authorized horizontal displacement of the walk (Mn)n on this horizontal line. We assume that (ξ_y)_y∈Z is a sequence of i.i.d. random variables the distribution of which is symmetric, has a moment of order 3 and a positive varianceσ²_ξ. We consider a distributionν onZadmiting a vari- ance and with null expectation (corresponding to the distribution of the vertical displacements when vertical displacement occur). We fix a parameter δ ∈(0,1). We consider a random walk in random environment M = (M_n)_n on Z² starting from the origin (i.e. M₀ := (0,0)), moving horizontally (with respect to (ξ_y)_y) with probabilityδ and moving vertically (with respect to ν) with probability 1−δ as follows:

P(Mn+1= (x+ξy, y)|Mn= (x, y)) =δ (horizontal displacement) P(Mn+1 = (x, y+z)|Mn = (x, y)) = (1−δ)ν({z}) (vertical displacement).

Observe that if theξy’s have Rademacher distribution (i.e. takes their values in{−1,1}), thenM is a walk onZ² with random orientations of the horizontal layers, they-th horizontal layer being oriented to the left if ξ_y = −1 and to the right if ξ_y = 1). Such models have been considered by Matheron and de Marsilly in [20], their transience has been established by Campanino and P´etritis in [8], see also [13] for their asymptotic behaviour and [9] for local limit theorem in this context.

The processM is strongly related to RWRS. Indeed it can be represented as follows M_n= ( ˜Z_n, S_n) =

n

X

k=1

ξ_Skε_k, S_n

!

, S_n:=

n

X

k=1

X˜_k(1−ε_k),

where ( ˜X_k)_k is a sequence of i.i.d. random variables with distributionν and (ε_k)_k is a sequence of i.i.d. Bernoulli random variables with parameter δ (i.e. P(ε_k= 1) =δ= 1−P(ε_k = 0)). We assume that (ξ_y)_y, ( ˜X_k)_k and (ε_k)_k are independent. We then set X_k := ˜X_k(1−ε_k). As for RWRS, we assume without any loss of generality that neither the support ofνnor the one of the distribution ofξ₀ are contained in a proper subgroup of Z. Observe that the second coordinate S of M is a random walk. Hence we focus our study on the first coordinate ˜Z of M, which is very similar to RWRS. Our second main result states that the conclusion of Theorem 1 is still valid for ˜Z.

Theorem 2 (Persistence of M on the leftside). There exists a constant c > 0 such that for large enough T

P h

k=1,...,Tmax

Z˜_k≤1i

≤T^−1/4(logT)^+c. (6) If moreover E[e^ξ1]<∞, then there exist positive constants c^′, c^′′ andT₀ such that

T^−1/4(logT)^−c′h H⁻¹

c^′′T⁻¹⁴i₋1

≤P h

k=1,...,Tmax

Z˜_k≤1i

, (7)

for everyT > T₀. The function H is defined as in Theorem 1.

Let us recall that Z and ˜Z are stationary but non-markovian processes with respect to the annealed distributionP and that they are markovian but non-stationary given the sceneryξ.

In Section 2, we prove some useful technical lemmas concerning the random walkS as well as the random walk in random sceneryZ and the analogous process ˜Z. Section 3 is devoted to the proof of Theorems 1 and 2.

2. Preliminary results

For everyy∈Zand every integer n≥1, we writeN_n(y) for the number of visits of the walk S to sitey before time n, i.e.

N_n(y) := #{k= 1, ..., n : S_k=y}. Using this notation, we observe thatZ can be rewritten as follows:

Zn=X

y∈Z

ξyNn(y).

Analogously

Z˜_n=X

y∈Z

ξ_yN˜_n(y),

with ˜N_n(y) := #{k= 1, ..., n : S_k =y and ε_k = 1}. The behaviour of ( ˜N_n(y))_y will appear to be very similar to the behaviour of (Nn(y))y, at least for our purpose.

2.1. Preliminary results on the random walk. We set N_n^∗ = sup_yNn(y) andRn:= #{y∈ Z : N_n(y)>0} for the number of sites that have been visited by the walkS before time n.

Lemma 3. Let γ ∈(0,¹₂). We set

Ω⁽¹⁾_n (γ) :={N_n^∗ ≤n^12+γ, R_n≤n^12+γ} There exists C_γ >0 such that

P[Ω⁽¹⁾_n (γ)] = 1−O(exp(−C_γn^γ)).

Proof. Due to Lemma 34 of [9], we know that there exists c_γ > 0 such that P[R_n ≤ n^21+γ] = 1−O(exp(−cγn^γ)). Let us prove that the argument therein can be adapted to prove the same result forN_n^∗ instead ofR_n. Observe first that a7→P[N_n^∗≥a] is sub-multiplicative. Indeed, let a, bbe two positive integers. Let us writeτ_a:= min{k≥1, N_k^∗ =a}.

P[N_n^∗ ≥a+b] =

n

X

j=1

P[τa=j, N_n^∗−N_j^∗≥b]

≤

n

X

j=1

P[τ_a=j, sup

y (N_n(y)−N_j(y))≥b]

≤

n

X

j=1

P[τ_a=j]P[N_n^∗_−j ≥b]

≤ P[N_n^∗ ≥a]P[N_n^∗≥b].

HenceP[N_n^∗≥a b]≤(P[N_n^∗ ≥a])^b and so

P[N_n^∗≥E[N_n^∗]n^γ] ≤ P[N_n^∗ ≥ ⌊3E[N_n^∗]⌋]^⌊nγ/3⌋

≤

E[N_n^∗]

⌊3E[N_n^∗]⌋

_⌊n^γ/3⌋

≤2^{−⌊nγ/3⌋}. We conclude by using the fact that E[N_n^∗]∼c^′√

n.

Lemma 4. Let µ ∈ (0,1] and γ ∈ (0,1/2) and ϑ > 0 such that γ > 2(1−µ)ϑ. For any δ∈(0,^γ₂ −(1−µ)ϑ), we have

P

"

sup

y,z∈Z, 0<|y−z|<n^ϑ

|N_n(y)−N_n(z)|

|y−z|^µ > n^41+γ

#

=O

e⁻ⁿ

δ2

. (8)

Under the assumptions of Theorem 2, we also have

P

"

sup

y,z∈Z, 0<|y−z|<n^ϑ

|N˜_n(y)−N˜_n(z)|

|y−z|^µ > n^41+γ

#

=O

e⁻ⁿ

δ2

. (9)

Proof. Due to Lemma 3, it is enough to prove that P

h sup

k=1,...,n|S_k|> eⁿ

δ2i

=O

e⁻ⁿ

δ2

(10) and that

P

"

sup

y,z∈En,0<|y−z|<n^ϑ

N_n(y)−N_n(z)

|y−z|^µ > n^14+γ

#

=O e^−nδ

, (11)

where E_n := {y ∈ Z : |y| ≤ e^nδ2, N_n(y) ≤ n^12+γ} (and the analogous estimate with N_n(·) replaced by ˜Nn(·) under the assumptions of Theorem 2). We start with the proof of the first estimate. From Doob’s inequality, there exists some constant C >0 such that

P[ sup

k=1,...,n|S_k|> eⁿ

δ2

] ≤ C E[S²_n]e⁻²ⁿ

δ2

=O

ne⁻²ⁿ

δ2

so (10). Let us prove now the second estimate. Let τ_j(y) be the j-th visit time of (S_n)_n to y, that is

τ₀(y) := 0 and ∀j≥0, τ_j+1(y) = inf{k > τ_j(y) : S_k=y}

(resp. ˜τ₀(y) := 0, ˜τ_j+1(y) = inf{k > τ˜_j(y) : S_k = y, ε_k = 1}). Let y, z ∈ E_n be such that Nn(y)−Nn(z)>0, then there existsj∈ {1, ...,⌊n^12+γ⌋}such that τj(y)≤n < τj+1(y) (observe thatτ

⌊n^{2 +1 γ}⌋+1(y)> n). For this choice of j, we have

N_n(y)−N_n(z)≤N_τj(y)(y)−N_τj(y)(z).

Therefore P

"

sup

y,z∈En,0<|y−z|<n^ϑ

N_n(y)−N_n(z)

|y−z|^µ > n^14+γ

#

≤ X

y,z∈En,0<|y−z|<n^ϑ

⌊n^{12 +γ}⌋

X

j=1

Ph

N_τj(y)(y)−N_τj(y)(z)≥ |y−z|^µn^14+γi ,

≤ X

y,z∈En,0<|y−z|<n^ϑ

⌊n^{12 +γ}⌋

X

j=1

P

"

1 +

j−1

X

k=1

(1−M_k(y, z))≥ |y−z|^µn^14+γ

#

, (12) sinceN_τj(y)(y)−N_τj(y)(z) =j−Pj−1

k=0M_k(y, z)≤j−Pj−1

k=1M_k(y, z), where, following [15], we write M_k(y, z) for the number of visits of (S_n)_n to z between its k-th and (k+ 1)-th visit to y, i.e.

M_k(y, z) := X

τk(y)<n≤τk+1(y)

1_{Sn=z}.

Under assumptions of Theorem 2, (12) still holds for ˜N_n(·) instead of N_n(·) if we replace τ_j(y) by ˜τ_j(y) and M_k(y, z) by ˜M_k(y, z) :=P

˜

τk(y)<n≤τ˜k+1(y)1_{Sn=z,εn=1}.

Due to the strong Markov property, (M_k(y, z))_k≥1 is a sequence of i.i.d. random variables.

Let us recall (see pages 13-14 in [15] for more details) that its common law is given by

P[M_k(y, z) = 0] = 1−p(|y−z|), ∀ℓ≥1, P[M_k(y, z) =ℓ] = (1−p(|y−z|))^ℓ−1(p(|y−z|))², withp(x) =p(−x)∼˜c|x|⁻¹. Observe also that, under the assumptions of Theorem 2, ( ˜M_k(y, z))_k≥1 is a sequence of i.i.d. random variables with P[ ˜M_k(y, z) = 0] = 1−p(z˜ −y) and P[ ˜M_k(y, z) = ℓ] = ˜p(z−y)(1−p(y˜ −z))^ℓ−1p(y˜ −z) ifℓ≥1 where ˜p(x) denotes the probability that (Sn, εn)n≥1

visits (x,1) before (0,1). Observe that ²

˜

p(x) =X

k≥0

(1−δ−p(x))^kp(x)(1−p(˜−x)) = p(x)

δ+p(x)(1−p(˜−x)).

2The fact that (Sn, εn)n≥1 visits (x,1) before (0,1) means that S visits 0 several times (let us say k times, withk≥0) before its first visit atxbut thatǫn= 0 at each of these visits to 0 (this happens with probability (1−δ−p(x))^k), thatS goes toxbefore coming back to 0 (this happens with probabilityp(x)) and finally that, starting fromS=x, (Sn, εn)n≥1 visits (x,1) before (0,1) (this happens with probability 1−p(−x)).˜

Iterating this formula we obtain that ˜p(x) = _δ+p(x)^p(x)

1−_δ+p(x)^p(x) (1−p(x))˜

= p(x)(δ+p(x)˜p(x)) (δ+p(x))²

which leads to

p(x)

2 +δ ≤p(x) =˜ p(x)

δ+ 2p(x) ≤ p(x) δ . There existsC₀ >1 such that

∀x6= 0, C₀⁻¹|x|⁻¹ ≤p(x)≤C₀|x|⁻¹ (13) and

∀x6= 0, C₀⁻¹|x|⁻¹ ≤p(x)˜ ≤C₀|x|⁻¹. (14) Observe thatM₁(y, z) has expectation 1 and admits exponential moment of every order:

∀t >0, G_|y−z|(t) :=E h

e^{t(1−M1(0,|y−z]))}i

= (1−p(|y−z|))e^t−1 + 2p(|y−z|) 1−(1−p(|y−z|))e^−t .

Hence, for every positive integer J ≤n^12+γ, due to the Markov inequality, we obtain that for everyt >0,

P

"

1 +

J

X

k=1

(1−M_k(y, z))≥ |y−z|^µn^14+γ

#

=P

"

exp t+t

J

X

k=1

(1−M_k(y, z))

!

≥exp

t|y−z|^µn^14+γ

#

≤ exp

−t|y−z|^µn^14+γ E

"

exp t+t

J

X

k=1

(1−M_k(y, z))

!#

≤ exp

−t|y−z|^µn^14+γ

(G_|y−z|(t))^Je^t

≤ exp

−t|y−z|^µn^14+γ

(1−C₀⁻¹|y−z|⁻¹)e^t−1 + 2C₀⁻¹|y−z|⁻¹ 1−(1−C₀⁻¹|y−z|⁻¹)e^−t

J

e^t

≤ exp

−t|y−z|^µn^14+γ

(1−C₀⁻¹|y−z|⁻¹)e^t−1 + 2C₀⁻¹|y−z|⁻¹ 1−(1−C₀⁻¹|y−z|⁻¹)e^−t

n^{12 +γ}

e^t

since the functionf :p7→ ⁽¹₁⁻₋^p)e₍₁₋^t−_p)e^1+2p−t is decreasing on (0,1) such thatf(0) =e^t and f(1) = 1.

Now using the Taylor expansion ofe^t at 0, we observe that (1−p)e^t−1 + 2p

1−(1−p)e^−t = 1 +_p^qt+^q_p^t₂² + ^q_pO(t³) 1 +_p^qt−^q_p^t₂² + ^q_pO(t³)

with p = C₀⁻¹|y−z|⁻¹ and q = 1−p where O(t³) is uniform in p. Taking t = p n^−14−γ2, we obtain

(1−p)e^t−1 + 2p

1−(1−p)e^−t = 1 +q n^−14−γ2 +qpⁿ⁻

12−γ

2 +p²O(n^−34−3γ2 ) 1 +q n^−14−γ2 −qpⁿ⁻

12−γ

2 +p²O(n^−34−3γ2 )

= 1 +qp n^−12−γ+O(n^−34−3γ2 ).

and so P

"

1 +

J

X

k=1

(1−M_k(y, z))≥ |y−z|^µn^14+γ

#

=O

e^{−C0−1|y−z|µ−1n}

γ 2

=O

e^{−C−10 n−(1−µ)ϑ+}

γ 2

.

Taking δ ∈ (0,^γ₂ −(1−µ)ϑ) and combining this with (12), we deduce (11) and the analogous estimate for ˜N_n(·) instead of N_n(·) under the assumptions of Theorem 2 (replacing M_k by ˜M_k

andp(·) by ˜p(·) in the above argument).

2.2. A conditional local limit Theorem for the RWRS. Let ϕ_ξ be the characteristic function of ξ₁. Since ξ₁ takes integer values, e^2iπξ1 = 1 a.s. and so ϕ_ξ(u) = 1 for every u∈2πZ. Let us consider the positive integer dsuch that d{u : |ϕ_ξ(u)|= 1}= 2πZ. Another characterization of dis that it is the positive generator of the subgroup of Z generated by the b−c, withbandcin the support of the distribution ofξ1(i.e. by the support of the distribution of ξ₀ −ξ₁). Since the support of ξ₁ is not contained in a proper subgroup of Z, we also have d = inf{n ≥ 1 : e^2iπnξ1/d = 1 a.s.}. Observe that e^{2iπd ξ1} is almost surely constant and so (e^{2iπd ξ1})² = ϕ_ξ ^2π_d2

= e^{2iπd (ξ0+ξ1)} almost surely. Since the distribution of ξ₁ is symmetric, P(ξ₀ +ξ₁ = 0) > 0 and so (e^{2iπd ξ1})² = 1 almost surely. Hence either d = 1 (and e^{2iπd ξ1} = 1 a.s.) or d= 2 (and e^{2iπd ξ1} =−1 a.s.). The following lemma relates the conditional probability P[Zn= 0|S] to the self-intersection local timeVn of the random walk S up to time n. Let us recall that V_n is given by

V_n:=

n

X

k,ℓ=1

1_{Sk=Sℓ} =

n

X

k,ℓ=1

X

y∈Z

1_{Sk=Sℓ=y} =X

y

(N_n(y))². Under the assumptions ot Theorem 2, we set ˜Vn:=P

y( ˜Nn(y))².

Lemma 5. Let γ ∈ (0,1/48). There exists a sequence of S-measurable sets (Ω⁽⁰⁾_n (γ))_n, an integer n0 > 0 and a positive constant c such that P(Ω⁽⁰⁾n (γ)) = 1 +O

e⁻ⁿ

δ2

for any δ < ^γ₂ and such that, for every n≥n₀ such that n∈dN, the following inequalities hold on Ω⁽⁰⁾_n (γ):

P[Z_n= 0|S]≥ c

√V_n,

n^12−γ≤N_n^∗≤n^12+γ, R_n≤n^12+γ and n^32−γ≤V_n≤n^32+γ.

Under the assumptions of Theorem 2, there exists a sequence of (S,(ε_k)_k)-measurable sets ( ˜Ω⁽⁰⁾n (γ))_n, an integern₀ >0and a positive constantcsuch that P(Ω⁽⁰⁾n (γ)) = 1 +O

e^−nδ2

for any δ < ^γ₂ and such that, for everyn≥n0 such that the following inequalities hold on Ω˜⁽⁰⁾n (γ):

P

hZ˜n= 0

(S,(ε_k)_k)i

≥ c

pV˜_n1{^Pnk=1εk∈dN∗},

n^12−γ≤sup ˜N_n(Z)≤N_n^∗≤n^12+γ, R_n≤n^12+γ and n^32−γ ≤V˜_n≤n^32+γ.

Remark: If we assume that ϕ_ξ is non negative (in this case P(ξ₁ = 0) >0), then there exists c >0 such that for everyn≥1

P[Z_n= 0|S]≥ c

√V_n. Indeed, observe that

P[Zn= 0|S] = 1 2π

Z π

−π

E

e^itZn|S

dt= 1 2π

Z π

−π

Y

y

ϕ_ξ(tNn(y))dt. (15)

Remark that for every y ∈Z, N_n(y) ≤√

V_n. We know that ϕ_ξ(t)−1∼ −^σ

2 ξ

2 t². Let β >0 be such that, for every real numberu satisfying |u|< β, we have ϕ_ξ(u) ≥e^−σ2ξu2. Since ϕ_ξ is non

negative, we have

P[Z_n= 0|S] ≥ 1 2π

Z _β/^√_Vn

−β/√ Vn

Y

y

[ϕ_ξ(tN_n(y))] dt

≥ 1 2π

Z _β/^√_Vn

−β/√ Vn

Y

y

e^{−σ2ξt2(Nn(y))2}dt

= 1

2π Z β/√

Vn

−β/√ Vn

e^−σ2ξt2Vndt

≥ 1

2πσ_ξ√ V_n

Z

|u|<σξβ

e^−u2du.

The proof of Lemma 5 is based on the same idea. The fact that ϕ_ξ can take negative values complicates the proof.

Proof of Lemma 5. We have P[Zn= 0|S] = 1

2π Z π

−π

E

e^itZn|S

dt= 1 2π

Z π

−π

Y

y

ϕ_ξ(tNn(y))dt. (16) Observe that e^2iπξ1/d =E[e^2iπξ1/d] almost surely and soE[e^2iπξ1/d]^d=E[e^2iπξ1] = 1. Hence, for any integer m≥0 and anyu∈R, we have

ϕ_ξ 2mπ

d +u

=

ϕ_ξ 2π

d m

ϕ_ξ(u) and so

P[Zn= 0|S] = 1 2π

Z π

−π

E

e^itZn|S dt

= 1

2π

d−1

X

k=0

Z π/d

−π/d

Y

y

"

ϕ_ξ

2π d

kNn(y)

ϕ_ξ(tN_n(y))

# dt

= 1

2π

d−1

X

k=0

ϕ_ξ

2π d

knZ _π/d

−π/d

Y

y

[ϕ_ξ(tN_n(y))]dt

= d

2π Z π/d

−π/d

Y

y

[ϕ_ξ(tNn(y))]1_{n∈dN^∗}dt. (17) Under the assumptions of Theorem 2, proceeding analogously we obtain

P[ ˜Z_n= 0|(S,(ε_k)_k)] = d 2π

Z _π/d

−π/d

Y

y

h

ϕ_ξ(tN˜_n(y))i 1_{^Pn

k=1εk∈dN∗}dt, (18) sinceP

y∈ZN˜_n(y) =P_n

k=1ε_k. We know that ϕ_ξ(t)−1 ∼ −^σ₂^2ξ t². Let β > 0 be such that, for every real numberusatisfying |u|< β, we havee^−σξ2u2 ≤ϕ_ξ(u)≤e⁻

σ2 ξ

4 u² (observe that the fact that the distribution of ξ is symmetric implies that ϕ_ξ takes real values). Using the fact that

Nn(y)≤N_n^∗ ≤√

Vn, we have d

2π

Z _β/Nn^∗

−β/N_n^∗

Y

y

[ϕ_ξ(tN_n(y))] dt ≥ d 2π

Z _β/^√_Vn

−β/√ Vn

Y

y

e^{−σξ2t2(Nn(y))2}dt

= d

2π Z β/√

Vn

−β/√ Vn

e^−σ2ξt2Vndt

≥ d

2πσ_ξ√ V_n

Z

|u|<σ_ξβ

e^−u2du.

This gives

d 2π

Z β/N_n^∗

−β/N_n^∗

Y

y

ϕ_ξ(tN_n(y))dt≥ c

√V_n, (19)

for some positive constantc, and analogously d

2π

Z _β/N˜_n^∗

−β/N˜_n^∗

Y

y

ϕ_ξ(tN_n(y))dt ≥ c

pV˜_n, (20) under the assumptions of Theorem 2 if ˜Vn6= 0.

Let Ωn(γ) be the set defined by Ω_n(γ) =

(

R_n≤n^12+γ, N_n^∗ ≤n^12+γ, sup

y6=z;|y−z|≤n

|N_n(y)−N_n(z)|

|y−z| ≤n^14+γ )

.

Due to Lemmas 3 and 4 (applied with µ = 1 and ϑ= 1), P(Ωn(γ)) = 1 +O

e⁻ⁿ

δ2

for any δ < ^γ₂. On Ω_n(γ), due to the Cauchy-Schwartz inequality, we have n=P

yN_n(y)1_{Nn(y)>0} ≤

VnP

y1_{Nn(y)>0}¹₂

≤ √

RnVn and so Vn ≥ n^32−γ. Observe also that Vn ≤ N_n^∗P

yNn(y) = n N_n^∗ ≤n^32+γ. Moreover n=P

yN_n(y) ≤R_nN_n^∗. Hence N_n^∗ ≥n^12−γ. This gives the three last inequalities in the first case.

Under the assumptions of Theorem 2, we set analogously Ω˜_n(γ) =

(

R_n ≤n^21+3γ4 , N_n^∗ ≤n^21+3γ4 , sup

y6=z;|y−z|≤n

|N˜_n(y)−N˜_n(z)|

|y−z| ≤n^14+γ,

n

X

k=1

ε_k> n(1−γ)δ 4

) .

If nis large enough, on ˜Ω_n(γ), we also obtain that n^32−γ ≤V˜_n ≤n^32+γ and sup ˜N_n(Z)≥n^12−γ using the same arguments as above and the fact thatP

yN˜_n(y) =Pn k=1ε_k.

To end the proof of the lemma, it remains to prove the first inequality. Due to (17) and (19), it remains to prove that there exists n₁>0 such that, for everyn≥n₁, on Ω_n(γ), we have

Z

β N∗

n≤|t|≤^π_d

Y

y

|ϕ_ξ(tN_n(y))|dt≤ Z

βn^−12−γ≤|t|≤^π_d

Y

y

|ϕ_ξ(tN_n(y))|dt≤ c 2√

V_n, (21) and the analogous inequality obtained by replacing N_n(·) by ˜N_n(·) and V_n by ˜V_n, under the assumptions of Theorem 2. To this end, we will use elements of the proof of [9] and more precisely the proofs of Propositions 9 and 10 therein.

We fix ε > 3γ such that 3γ + 3ε < ¹₄ (this is possible since γ < ₄₈¹ ). We first follow the proof of Proposition 9 in [9] (hereε₀=β) and more precisely of Lemma 14 therein. Lety₁ ∈Z be such that Nn(y1) = N_n^∗ and set y0 := min{y ≥ y1 : Nn(y) ≤ ^β₂n^12−ε}. On Ωn(γ), for n

large enough, y₀ > y₁ (since ε > γ) and soN_n(y₀ −1) > ^β₂n^12−ε ≥ N_n(y₀). Moreover, still on Ω_n(γ), N_n(y₀ −1)−N_n(y₀) ≤ n^41+γ which is smaller than ^β₄n^12−ε for n large enough so that

β

4n^12−ε≤N_n(y₀)≤ ^β₂n^12−ε. Now, on Ω_n(γ), for everyz∈Zsuch that|y₀−z| ≤e_n:= ₁₀^βn^14−ε−γ, then

|N_n(z)−N_n(y₀)| ≤ |y₀−z|n^14+γ ≤ β 10n^12−ε and so

β

10n^12−ε< N_n(z)< βn^12−ε, hence|tN_n(z)| ≤β if n^−12−γ<|t|< n^−12+ε and so, on Ω_n(γ),

Y

y∈Z

|ϕ_ξ(tN_n(y))| ≤ exp −σ_ξ² 4 t²

y0+en

X

z=y0−en

(N_n(z))²

!

≤ exp −σ_ξ²

4 n^−1−2γ2e_n β² 100n^1−2ε

!

≤ exp −σ_ξ²

2 n^{14−3γ−3ε} β³ 10³

! .

Hence we have proved that, forn large enough, on Ω_n(γ), Z

βn^−12−γ≤|t|≤n^{−12 +ε}

Y

y

|ϕ_ξ(tN_n(y))|dt ≤ c 4√

V_n (22)

since 3γ+ 3ε < ¹₄.

Under the assumptions of Theorem 2, the same argument gives Z

βn^−12−γ≤|t|≤n^{−12 +ε}

Y

y

|ϕ_ξ(tN˜n(y))|dt≤ c 4p

V˜_n (23)

Now, Lemma 15 of [9] still holds with our set Ω_n(γ) since the proof only uses the fact that N_n^∗ ≤ n^12+γ and that R_n ≤ n^12+γ. Due to this remark and using the notations and results contained in Section 2.8 of [9], we take for Ω⁽⁰⁾n (γ) the subset of Ω_n(γ)∩ Dn on which #{z : Nn(z)∈ I} ≥n^21−2γ/4 (with Dn and I being defined in Section 2.8 of [9] applied with α= 2).

Since γ < ¹₈ and 3γ < ε < ¹₂, we obtain that P(Ωn(γ)\Ω⁽⁰⁾n (γ)) = o(e^−cn) for some c > 0³. Moreover, there exists an integer n₂ such that if n≥n₂, on Ω⁽⁰⁾_n (γ) we have

∀t∈[n^−12+ε,π

d], Y

y

|ϕ_ξ(tN_n(y))| ≤exp(−n^γ)≤ c 4√

V_n (24)

(see the lines before the proof of Lemma 17 of [9]). The same argument (with the flat peaks instead of the peaks as explained in Section 5.4 of [9] gives also (for everyn large enough)

∀t∈[n^−12+ε,π

d], Y

y

|ϕ_ξ(tN˜_n(y))| ≤exp(−n^γ)≤ c 4p

V˜_n (25)

on some set ˜Ω⁽⁰⁾n (γ) such thatP( ˜Ω_n(γ)\Ω˜⁽⁰⁾n (γ)) =o(e^−cn).

3Indeed, using the notationsDn,EnandI of Section 2.8 of [9],P(Dn) = 1−o(e^−cn); moreover following the proof of Lemma 15 of [9] we obtain that Ωn(γ)∩Dn⊂ En, and finally, due to the remark following Lemma 17 of [9], p2(n) :=P(En,#{z : Nn(z)∈ I}< n¹2−2γ/4) =o(e^−cn). ThereforeP(Ωn(γ)\Ω⁽⁰⁾n (γ))≤P(Ωn(γ)\Dn)+p2(n) = o(e^−cn).