Quenched limits for transient, ballistic, sub-gaussian one-dimensional random walk in random environment

www.imstat.org/aihp 2009, Vol. 45, No. 3, 685–709

DOI: 10.1214/08-AIHP149

© Association des Publications de l’Institut Henri Poincaré, 2009

Quenched limits for transient, ballistic, sub-Gaussian one-dimensional random walk in random environment ¹

Jonathon Peterson

Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 33705, USA Received 17 August 2007; accepted 5 May 2008

Abstract. We consider a nearest-neighbor, one-dimensional random walk{Xn}n≥0in a random i.i.d. environment, in the regime where the walk is transient with speedv_P >0 and there exists ans∈(1,2)such that the annealed law ofn^−1/s(X_n−nv_P) converges to a stable law of parameters. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences{tk}and{t_k}depending on the environment only, such that a quenched central limit theorem holds along the subsequencet_k, but the quenched limiting distribution along the subsequencet_k is a centered reverse exponential distribution. This complements the results of a recent paper of Peterson and Zeitouni (arXiv:math/0704.1778v1 [math.PR]) which handled the case when the parameters∈(0,1).

Résumé. On examine des marches aléatoires unidimensionnelles en milieu aléatoire avec un environnement i.i.d., dans le ré- gime où la marche est transiente avec vitesse v_P >0 et où il existes∈(1,2) tel que la loi “annealed” (i.e., moyennée) de n^−1/s(Xn−nvP)converge vers une loi stable de paramètres. Sous la loi “quenched” (i.e. conditionnelement à l’environnement) on montre qu’il n’existe pas de loi limite. En particulier on prouve qu’il existe des suites{tk}et{t_k}, dépendant de l’environnement, tel qu’un théorème de limite centrale quenched est valide le long de la suitet_k, mais où la distribution limite suivant la suite t_k est une distribution centrée exponentielle inverse. Ceci complète les résultats d’un article récent de Peterson et Zeitouni (arXiv:math/0704.1778v1 [math.PR]) qui traitait le case de paramètres∈(0,1).

MSC:Primary 60K37; secondary 60F05; 82C41; 82D30 Keywords:Random walk; Random environment

1. Introduction, notation and statement of main results

LetΩ = [0,1]^Z, and let F be the Borel σ-algebra onΩ. A random environment is anΩ-valued random variable ω= {ω_i}i∈Zwith distributionP. In this paper we will assume thatP is a product measure onΩ. Thequenchedlaw P_ω^xfor a random walkX_nin the environmentωis defined by

P_ω^x(X₀=x)=1 and P_ω^x(X_n+1=j|X_n=i)=

ω_i ifj=i+1, 1−ω_i ifj=i−1.

Z^N is the space for the paths of the random walk{X_n}n∈N, and letGdenote theσ-algebra generated by the cylinder sets. Note that for eachω∈Ω,P_ωis a probability measure on(Z^N,G), and for eachG∈G,P_ω^x(G):(Ω,F)→ [0,1]

1Supported in part by NSF Grant DMS-05-03775 and by a Doctoral Dissertation Fellowship from the University of Minnesota.

is a measurable function ofω. Expectations under the lawP_ω^xare denotedE_ω^x. Theannealedlaw for the random walk in random environmentX_nis defined by

P^x(F×G)=

F

P_ω^x(G)P (dω), F∈F, G∈G.

For ease of notation we will usePω andPin place of P_ω⁰andP⁰respectively. We will also useP^x to refer to the marginal on the space of paths, i.e.P^x(G)=P^x(Ω×G)=EP[P_ω^x(G)]forG∈G. Expectations under the lawPwill be writtenE.

A simple criterion for recurrence of a one-dimensional RWRE and a formula for the speed of transience was given by Solomon in [10]. For any integersi≤j define

ρi:=1−ω_i

ω_i and Πi,j:=

j

k=i

ρk. (1)

Then, Xn is transient to the right (resp. to the left) if EP(logρ0) <0, (resp. EPlogρ0>0) and recurrent if E_P(logρ₀)=0 (henceforth we will write ρ instead ofρ₀ in expectations involving onlyρ₀). In the case where E_Plogρ <0 (transience to the right), Solomon established the following law of large numbers

v_P := lim

n→∞

X_n n = lim

n→∞

n T_n = 1

ET1

, P-a.s., (2)

whereTn:=min{k≥0:Xk=n}. For any integersi < jdefine W_i,j :=

j

k=i

Π_k,j and W_j:=

k≤j

Π_k,j. (3)

WhenE_Plogρ <0, it was shown in [11] that

E_ω^jTj+1=1+2Wj<∞, P-a.s. (4)

and thus v_P =1/(1+2EPW₀). Since P is a product measure, E_PW₀=_∞

k=1(E_Pρ)^k. In particular, v_P >0 if E_Pρ <1.

Kesten, Kozlov and Spitzer [5] determined the annealed limiting distribution of a RWRE withE_Plogρ <0, i.e.

transient to the right. They derived the limiting distributions for the walk by first establishing a stable limit law of indexsforTn, wheres is defined by the equationEPρ^s=1. In particular, they showed that whens∈(1,2)there exists ab >0 such that

nlim→∞P

T_n−ET_n n^1/s ≤x

=Ls,b(x) (5)

and

nlim→∞P

X_n−nv_P v_P^1+1/sn^1/s ≤x

=1−L_s,b(−x), (6)

whereLs,b is the distribution function for a stable random variable with characteristic function Lˆ_s,b(t )=exp

−b|t|^s

1−i t

|t|tan(πs/2) .

While the annealed limiting distributions for transient one-dimensional RWRE have been known for quite a while, the corresponding quenched limiting distributions have remained largely unstudied until recently. In the case when s >2, Goldsheid [3] and Peterson [8] independently proved that a quenched CLT holds with a random (depending

on the environment) centering. Previously, in [7] and [11] it had only been shown that the limiting statements for the quenched CLT with random centering held in probability (rather than almost surely). In the case whens <1 it was shown in [9] that no quenched limiting distribution exists for the RWRE. In particular, it was shown thatP-a.s. there exist two different random sequencest_k andt_k such that the behavior of the RWRE is either localized (concentrated in a interval of size log²t_k) or spread out (scaling of ordert_k^s).

In this paper, we analyze the quenched limiting distributions of a one-dimensional transient RWRE in the case s∈(1,2). We show that, as in the case whens <1, there is no quenched limiting distribution of the random walk.

However, as shown in Section 2, the existence of a positive speed for the random walk allows us to transfer limiting distributions fromT_ntoX_n. Throughout the paper, we will make the following assumptions:

Assumption 1. P is a product measure onΩsuch that

E_Plogρ <0 and E_Pρ^s=1 for somes >0. (7)

Assumption 2. The distribution oflogρis non-lattice underP andE_P(ρ^slogρ) <∞.

Remarks.

1. Assumption1contains the essential assumptions for our results.The technical conditions contained in Assump- tion2were also invoked in[5]and[9].

2. Since E_Pρ^γ is a convex function of γ,the two statements in(7) give thatE_Pρ^γ <1 for all0< γ < s and E_Pρ^γ >1for allγ > s.In particular this implies thatv_P >0 ⇐⇒ s >1.The main results of this paper are for s∈(1,2),but many statements hold for a wider range ofs.If no mention is made of bounds onsthen it is assumed that the statement holds for alls >0.

3. The casess∈ {1,2}are not covered by [9] or by this paper.It is not clear whether or not a quenched CLT holds in the cases=2,but we suspect that the results fors=1will be similar to those of the casess∈(0,1)and s∈(1,2)– i.e.no quenched limiting distribution for the random walk.However,sinces=1is the bordering case between the zero-speed and positive-speed regimes the analysis is likely to be more technical(as was also the case in[5]).

Let(x)andΨ (x)be the distribution functions for a Gaussian and exponential random variable respectively. That is,

(x):=

_x

−∞

√1

2πe^−t2/2dt and Ψ (x):=

0, x <0, 1−e^−x, x≥0.

Our main results are the following:

Theorem 1.1. Let Assumptions 1 and 2 hold,and let s∈(1,2). Then P-a.s. there exists a random subsequence n_km=n_km(ω)ofn_k=2^2k and non-deterministic random variablesv_km,ωsuch that

mlim→∞P_ω

T_nkm−E_ωT_nkm

√v_km,ω ≤x

=(x) ∀x∈R and

mlim→∞P_ω

X_tm−n_km v_P√v_km,ω ≤x

=(x) ∀x∈R,

wheret_m=t_m(ω):= E_ωT_nkm.

Theorem 1.2. Let Assumptions 1 and 2 hold,and let s∈(1,2). Then P-a.s. there exists a random subsequence nk_m=nk_m(ω)ofnk=2^2k and non-deterministic random variablesvk_m,ωsuch that

mlim→∞P_ω

Tn_km−EωTn_km

√v_km,ω ≤x

=Ψ (x+1) ∀x∈R

and

mlim→∞P_ω

X_tm−n_km v_P√v_km,ω ≤x

=1−Ψ (−x+1) ∀x∈R,

wheret_m=t_m(ω):= E_ωT_nkm. Remarks.

1. Note that Theorems1.1and1.2preclude the possibility of quenched analogues of the annealed statements(5) and(6).

2. The choice of Gaussian and exponential distributions in Theorems 1.1and 1.2are the two extremes of what quenched limiting distributions can be found along random subsequences.In fact,it will be shown in Corollary4.5 that T_n is approximately the sum of a finite number of exponential random variables with random (depending on the environment)parameters.Thus,we expect in fact that any distribution which is the sum of(or limit of sums of) exponential random variables can be achieved as a quenched limiting distribution ofT_nalong a random subsequence.

3. The sequencen_k=2^2k in Theorems1.1and1.2is chosen only for convenience.In fact,for any sequencen_k growing sufficiently fast,P-a.s.there will be a random subsequencen_km(ω)such that the conclusions of Theorems1.1 and1.2hold.

4. The definition of vk_m,ω is given below in (11), and similar to Theorem 1.3, it can be shown that limn→∞P (n⁻_k^2/svk,ω≤x)=Ls/2,b(x)for some b >0. Also,from(2) we have thattm∼ET1nk_m.Thus,the scal- ing in Theorems1.1and1.2is of the same order as the annealed scaling but cannot be replaced by a deterministic scaling.

As in [9], define the “ladder locations”ν_i of the environment by ν₀=0 and ν_i=

inf{n > ν_i−1: Π_νi−1,n−1<1}, i≥1,

sup{j < ν_i+1: Π_k,j−1<1,∀k < j}, i≤ −1. (8) Throughout the remainder of the paper we will letν=ν₁. We will sometimes refer to sections of the environment betweenν_i−1andν_i −1 as “blocks” of the environment. Note that the block betweenν₋₁andν₀−1 is different from all the other blocks between consecutive ladder locations (in particular it can be thatΠ_{ν−1,ν0−1}≥1), and that all the other blocks have the same distribution as the block from 0 toν−1. As in [9] we define the measureQon environments byQ(·):=P (·|R), where

R:= {ω∈Ω: Π₋k,−1<1,∀k≥1} =

ω∈Ω:

−1

i=−k

logρi<0,∀k≥1

.

Note thatP (R) >0 sinceE_Plogρ <0.Qis defined so that the blocks of the environment between ladder locations are i.i.d. underQ, all with distribution the same as that of the block from 0 toν−1 underP. In particularP andQ agree onσ (ω_i: i≥0).

For any random variable Z, define the quenched variance VarωZ:=E_ω(Z−E_ωZ)². In [9], Theorem 1.1, it was proved that whens∈(0,1),n^−1/sE_ωT_νnconverges in distribution (underQ) to a stable distribution of indexs.

Correspondingly, whens <2 we will prove the following theorem:

Theorem 1.3. Let Assumptions1and2hold,and lets <2.Then there exists ab >0such that

nlim→∞Q

VarωTν_n

n^2/s ≤x

= lim

n→∞Q

1 n^2/s

n

i=1

E_ω^νi−1T_νi2

≤x

=Ls/2,b(x). (9)

Remarks.

1. The constantbin the above theorem may not be the same as in(5)and(6).

2. Theorem1.3can be used to show thatlimn→∞P (^VarωTn

n^2/s ≤x)=L_s/2,b(x)for someb>0,but we will not prove this since we do not use it for the other results in this paper.

A major difficulty in analyzingT_νn is that the crossing time fromν_i−1toν_i depends on the entire environment to the left ofν_i. Thus Varω(T_νi−T_νi−1)and Varω(T_νj−T_νj−1)are not independent even if|i−j|is large. In order to make the crossing times of blocks that are far apart essentially independent, we introduce some reflections to the RWRE. Forn=1,2, . . . ,define

bn:=

log²(n)

. (10)

LetX¯⁽ⁿ⁾_t be the random walk that is the same asX_t with the added condition that after reachingν_k the environment is modified by settingω_νk−bn =1 (i.e. never allow the walk to backtrack more than log²(n)blocks). We coupleX¯⁽ⁿ⁾_t with the random walkXt in such a way thatX¯_t⁽ⁿ⁾≥Xt with equality holding until the first timet when the walkX¯⁽ⁿ⁾_t reaches a modified environment location. Denote byT¯_x⁽ⁿ⁾the corresponding hitting times for the walkX¯_t⁽ⁿ⁾. It was shown in [9], Lemma 4.5, that limn→∞Pω(Tν_n= ¯Tν⁽ⁿ⁾_n )=0, P-a.s. so that in fact with high probability the added reflections do not affect the walk at all beforeTν_n. For ease of notation we define

μi,n,ω:=Eω^νi−1T¯_ν⁽ⁿ⁾_i and σ_i,n,ω² :=Varω

T¯_ν⁽ⁿ⁾_i − ¯T_ν⁽ⁿ⁾_i−1 .

The structure of the paper is as follows: In Section 2 we prove the following general proposition that allows us to easily transfer quenched limit laws from subsequences ofTntoXn.

Proposition 1.4. Let Assumptions1and2hold,and lets∈(1,2).Also,letn_k be a sequence of integers growing fast enough so thatlimk→∞ nk

n¹_k⁺₋^δ₁ = ∞for someδ >0,and define d_k:=n_k−n_k−1 and v_k,ω:=

nk

i=nk−1+1

σ_i,d²

k,ω=Varω

T¯_ν^(dk)

nk − ¯T_ν^(dk)

nk−1

. (11)

Assume thatF is a continuous distribution function for whichP-a.s.there exists a subsequencenk_m=nk_m(ω)such that forαm:=nk_m−1,

mlim→∞Pω^ναm

T¯_x^(d_m^km)−E_ω^ναmT¯_x^(d_m^km)

√v_km,ω ≤y

=F (y) ∀y∈R,

for any sequencexm∼nk_m.Then,P-a.s.for ally∈Rwe also have

mlim→∞P_ω

T_xm−E_ωT_xm

√v_km,ω ≤y

=F (y), (12)

for anyx_m∼n_km,and

mlim→∞Pω

X_tm−n_km v_P√v_km,ω ≤y

=1−F (−y), (13)

wheretm:= EωTn_km.

Then in Sections 3 and 4 we use Theorem 1.3 to find subsequencesn_km(ω)that allow us to apply Proposition 1.4.

To find a subsequence that gives Gaussian behavior ofT_nkm we find a subsequence where none of the crossing times of the firstn_kmblocks is too much larger than all the others and then use the Linberg–Feller condition for triangular arrays. In contrast, to find a subsequence that gives exponential behavior ofT_nkm we first prove that the crossing times of “large” blocks is approximately exponential in distribution. Then we find a subsequence where the crossing time

of one of the firstn_km blocks dominates the total crossing time of the firstn_km blocks. Finally, Section 5 contains the proof of Theorem 1.3 which is similar to that of [9], Theorem 1.1.

Before continuing with the proofs of the main theorems we recall some notation and results from [9] that will be used throughout the paper. First, recall that from [9], Lemma 2.1, there exist constantsC1, C2>0 such that

P (ν > x)≤C1e^−C2x ∀x≥0. (14)

Then, sinceν_n=_n

i=1ν_i−ν_i−1and theν_i−ν_i−1are i.i.d., the law of large numbers gives that

nlim→∞

ν_n

n =E_Pν=: ¯ν <∞, P-a.s. (15)

In [9] the following formulas for the quenched expectation and variance ofT_νwere given:

E_ωT_ν=ν+2

ν−1

j=0

W_j and VarωT_ν=4

ν−1

j=0

W_j+W_j² +8

ν−1

j=0

i<j

Π_i+1,j

W_i+W_i²

. (16)

Note that since the added reflections only decrease crossing times we obviously haveT_ν≥ ¯T_ν⁽ⁿ⁾andE_ωT_ν≥E_ωT¯_ν⁽ⁿ⁾ for anyn. Also, since (16) holds for any environmentω, the formula for VarωT¯_ν⁽ⁿ⁾is the same as in (16) but with ρ_ν−bn replaced by 0. In particular, this shows that VarωT_ν≥VarωT¯_ν⁽ⁿ⁾for anyn. As in [9] define for any integeri

Mi:=max

Πν_i−1,j: j∈ [νi−1, νi)

. (17)

Then [4], Theorem 1, gives that there exists a constantC3<∞such that

Q(Mi> x)=P (M1> x)∼C3x^−s. (18)

Note that M1≤max0≤j <νWj. Therefore, from the formulas for EωTν and VarωTν in (16) it is easy to see that E_ωT_ν≥M₁and VarωT_ν≥M₁²(the same also being true withT¯_ν⁽ⁿ⁾). Finally, recall the following results from [9]:

Theorem 1.5 (Lemma 3.3 and Theorem 5.1 in [9]). There exists a constantK_∞∈(0,∞)such that Q(VarωT_ν> x)∼Q

(E_ωT_ν)²> x

∼K_∞x^−s/2 asx→ ∞. Moreover,for anyε >0andx >0

Q

VarωT¯_ν⁽ⁿ⁾> xn^2/s, M₁> n^(1−ε)/s

∼Q

E_ωT¯_ν⁽ⁿ⁾2

> xn^2/s, M₁> n^(1−ε)/s

∼K_∞x^−s/21 n asn→ ∞.

2. Converting time limits to space limits

In this section we develop a general method for transferring a quenched limit law for a subsequence of T_n to a quenched limit law for a subsequence ofX_n. We begin with some lemmas analyzing the a.s. asymptotic behavior of the quenched variance and mean of the hitting times.

Lemma 2.1. Assumes≤2.Then for anyδ >0, Q

VarωT¯_ν⁽ⁿ⁾_n ∈/

n^2/s−δ, n^2/s+δ

≤ 1 P (R)P

VarωT¯_ν⁽ⁿ⁾_n ∈/

n^2/s−δ, n^2/s+δ

=o n^−δs/4

.

Proof. The first inequality in the lemma is trivial since for anyA∈Fwe have from the definition ofQthatQ(A)=

P (A∩R)

P (R) ≤_{P (}^{P (A)}_R). Next, note that whens≤2 [9], Lemma 5.11, gives P

VarωT¯_ν⁽ⁿ⁾

n ≥n^2/s+δ

≤P

VarωT_νn≥n^2/s+δ

=o n^−δs/4

. (19)

Also, since Varω(T¯_ν⁽ⁿ⁾_i − ¯T_ν⁽ⁿ⁾_i−1)≥M_i²we have P

VarωT¯_ν⁽ⁿ⁾_n ≤n^2/s−δ

≤P

M₁²≤n^2/s−δn

= 1−P

M1> n^1/s−δ/2n

=o e^−nδs/4

,

where the last equality is from (18).

Corollary 2.2. Assumes≤2.Then for anyδ >0 P

vk,ω∈/

d_k^2/s−δ, d_k^2/s+δ

=o d_k^−δs/4

. Consequently,ifs <2we have√v_k,ω=o(dk),P-a.s.

Proof. Recall from (11) that by definitionvk,ω=Varω(T¯ν^(d_nk^k)− ¯Tν^(d_nk−^k)₁). Also, note that the conditions onnk ensure that nk grows faster than exponentially and that dk∼nk. Thus, for allk large enough vk,ω only depends on the environment to the right of zero. Therefore for allklarge enough

P v_k,ω∈/

d_k^2/s−δ, d_k^2/s+δ

=Q Varω

T¯_ν^(dk)

nk − ¯T_ν^(dk)

nk−1

∈/

d_k^2/s−δ, d_k^2/s+δ

=Q

VarωT¯_ν^(dk)

dk ∈/

d_k^2/s−δ, d_k^2/s+δ

=o d_k^−δs/4

,

where the last equality is from Lemma 2.1. Now, for the second claim in the corollary, first note that 2>²_s +^s−_s¹ sinces >1. Therefore, for anyε >0 and for allklarge enough we have

P

vk,ω> εd_k²

≤P

vk,ω> d_k^{2/s+(s−1)/s}

=o

d_k^−(s−1)/4 .

This last term is summable sincedkgrows faster than exponentially. Thus the Borel–Cantelli lemma gives thatvk,ω=

o(d_k²),P-a.s.

Corollary 2.3. Assumes≤2.Then

klim→∞

VarωT_ν

nk−1

v_k,ω =0, P-a.s.

Proof. By the Borel–Cantelli lemma it is enough to prove that for anyε >0 ∞

k=1

P (VarωTν_nk−1 ≥εvk,ω) <∞.

However, for anyδ >0 we have P (VarωT_νnk

−1 ≥εv_k,ω)≤P

VarωT_νnk

−1 ≥εd_k^2/s−δ +P

v_k,ω≤d_k^2/s−δ

. (20)

By Corollary 2.2 the last term in (20) is summable for anyδ >0. To show that the second to last term in (20) is also summable first note that the conditions on the sequencenk give that there exists aδ >0 such thatεd_k^2/s−δ≥n^2/s_k−1^+δ for allklarge enough. Thus, for someδ >0 and allklarge enough we have

P

VarωTν_nk−1 > εd_k^2/s+δ

≤P

VarωTν_nk−1> n^2/s_k−1^−δ

=o n⁻_k−^δs/4₁

,

where the last equality is from (19).

Lemma 2.4. Assumes∈(1,2).ThenET1<∞,andP-a.s.

klim→∞

E_ωT_nk+x^√_vk,ω−E_ωT_nk

√vk,ω =xET1 ∀x∈R. (21)

Proof. Now, since ^EωTnk+x√^√vk,ωvk,ω^−EωTnk is monotone inx it is enough to prove that for arbitraryx∈Qthe limiting statement in (21) holds. Obviously this is true whenx=0 since both sides are zero. For the remainder of the proof we’ll assumex >0. The proof forx <0 is essentially the same (recall that by Corollary 2.2v_k,ω=o(dk)=o(nk) whens <2). Note that forx≥0 then we can re-writeEωTn_k+x√v_k,ω−EωTn_k =Eω^nkTn_k+x√v_k,ω. By the Borel–

Cantelli lemma it is enough to show that for anyε >0, ∞

k=1

PE_ω^nkT_nk+x^√_vk,ω− x√

vk,ω

ET1≥ε√ vk,ω

<∞. (22)

However, for anyδ >0 we have PE_ω^nkT_nk+x^√_vk,ω−

x√ v_k,ω

ET1≥ε√ v_k,ω

≤P

∃m∈

xd_k^1/s−δ ,

xd_k^1/s+δ

: E_ω^nkT_nk+m−mET₁≥εm x

+P

v_k,ω∈/

d_k^2/s−2δ, d_k^2/s+2δ

≤P

max

m≤xd_k^1/s+δ|E_ωT_m−mET₁| ≥εd_k^1/s−δ +o

d_k^−δs/2

, (23)

where the last inequality is due to Corollary 2.2 and the fact that {E_ω^nkT_nk+m}m∈Z has the same distribution as {EωTm}m∈ZsinceP is a product measure. Thus, we only need to show that the first term in (23) is summable inkfor someδ >0. For this, we need the following lemma whose proof we defer.

Lemma 2.5. Assumes∈(1,2].Then for any0< δ<^s−_2s¹ we have that P

maxm≤n|E_ωT_m−mET₁| ≥n^1−δ =o

n^−(s−1)/2 .

Assuming Lemma 2.5, fix 0< δ<^s−_2s¹ and then choose 0< δ < _s(2^δ_−δ). We chooseδandδthis way to ensure that(1/s+δ)(1−δ) <1/s−δ. Therefore, for allk large enough,εd_k^1/s−δ>xd_k^1/s+δ1−δ. Thus for allk large enough we have

P

max

m≤xd_k^1/s+δ|E_ωT_m−mET₁| ≥εd_k^1/s−δ ≤P

max

m≤xd_k^1/s+δ|E_ωT_m−mET₁| ≥

xd_k^1/s+δ1−δ

=o

d_k^{−(1/s+δ)(s−1)/2}

ask→ ∞.

Sinces >1 this last term is summable ink.

Proof of Lemma 2.5. Before proceeding with the proof we need to introduce some notation for a slightly different type of reflection. DefineX˜_t⁽ⁿ⁾to be the RWRE modified so that it cannot backtrack a distance ofb_n(the definition of X¯t

(n)is similar except the walk was not allowed to backtrackbnblocks instead). That is, after the walk first reaches locationi, we modify the environment by settingωi−b_n=1. LetT˜x

(n)be the corresponding hitting times of the walk

X˜t

(n). Then P

maxm≤n|E_ωT_m−mET₁| ≥n^1−δ

≤P

E_ωT_n−E_ωT˜_n⁽ⁿ⁾≥ n^1−δ 3

+P

ET1−ET˜₁⁽ⁿ⁾≥n^−δ 3

+P

maxm≤nE_ωT˜_m⁽ⁿ⁾−mET˜₁⁽ⁿ⁾≥n^1−δ 3

≤3n^−1+δ

ET_n−ET˜_n⁽ⁿ⁾ +1_ET

1−ET˜₁⁽ⁿ⁾≥n^−δ/3+P

maxm≤nE_ωT˜_m⁽ⁿ⁾−mET˜₁⁽ⁿ⁾≥n^1−δ 3

. (24)

Now, from (4) we get thatEωT1−EωT˜₁⁽ⁿ⁾=(1+2W0)−(1+2W₋b_n+1,0)=2Π₋b_n+1,0W₋b_n, and thus sinceP is a product measure

ET_n−ET˜_n⁽ⁿ⁾=nE_P

E_ωT₁−E_ωT˜₁⁽ⁿ⁾

= 2n

1−E_Pρ(E_Pρ)^bn+1. (25)

SinceE_Pρ <1 andb_n∼log²nthe above decreases faster than any power ofn. Thus by (24) we need only to show that P (maxm≤n|E_ωT˜_m⁽ⁿ⁾−mET˜₁⁽ⁿ⁾| ≥ ⁿ¹⁻₃^δ)=o(n^−(s−1)/2). For ease of notation we defineκ_i⁽ⁿ⁾:=E_ωⁱ⁻¹T˜_i⁽ⁿ⁾−ET˜₁⁽ⁿ⁾. Thus, sinceEωT˜_m⁽ⁿ⁾−mET˜₁⁽ⁿ⁾=_m

i=1κ_i⁽ⁿ⁾=_bn

i=1

(m−i)/b_n j=0 κ_{j b}⁽ⁿ⁾

n+i we have P

maxm≤n

E_ωT˜_m⁽ⁿ⁾−mET˜₁⁽ⁿ⁾≥ n^1−δ 3

≤P

maxm≤n b_n

i=1

(m−i)/bn j=0

κ_{j b}⁽ⁿ⁾

n+i

≥n^1−δ

3

≤

b_n

i=1

P

maxm≤n

(m−i)/bn j=0

κ_{j b}⁽ⁿ⁾

n+i

≥n^1−δ

3bn

=

bn

i=1

P

l≤(nmax−i)/bn

l

j=0

κ_{j b}⁽ⁿ⁾

n+i

≥n^1−δ

3bn

. (26)

Due to the reflections of the random walk,κ_i⁽ⁿ⁾depends only on the environment betweeni−bnandi−1. Thus, for eachi{κ_{j b}⁽ⁿ⁾

n+i}^∞_j=0is a sequence of i.i.d. random variables with zero mean, and so{_l

j=0κ_{j b}⁽ⁿ⁾

n+i}l≥0is a martingale.

Now, letγ∈(1, s). Then, by the Doob–Kolmogorov inequality, for any integerN we have P

maxl≤N

l

j=0

κ_{j b}⁽ⁿ⁾

n+i

≥n^1−δ

3bn

≤3^γb_n^γn^{−γ+γ δ}E_P

N

j=0

κ_{j b}⁽ⁿ⁾

n+i

γ

.

Now, since {κ_{j b}⁽ⁿ⁾

n+i}^∞_j=0 is a sequence of independent, zero-mean random variables, the Marcinkiewicz–Zygmund inequality [1], Theorem 2 on p. 356, gives that there exists a constantB_γ <∞depending only onγ >1 such that

E_P

N

j=0

κ_{j b}⁽ⁿ⁾

n+i

γ

≤B_γE_P

N

j=0

κ_{j b}⁽ⁿ⁾

n+i

2

γ /2

≤B_γE_{P N}

j=0

κ_{j b}⁽ⁿ⁾

n+i^γ

=B_γ(N+1)EPκ₁^(n)γ,

where the second inequality is becauseγ < s≤2 impliesγ /2<1. Now, recall from [5] thatP (EωT1> x)∼Kx^−s for some K >0. Therefore, since γ < s we have that EP|EωT1|^γ <∞. Thus, it’s easy to see that EP|κ₁⁽ⁿ⁾|^γ = E_P|E_ωT˜₁⁽ⁿ⁾−ET˜₁⁽ⁿ⁾|^γ is uniformly bounded inn. So, there exists a constantB_γ depending onγ∈(1, s)such that

P

max

l≤N

l

j=0

κ_{j b}⁽ⁿ⁾

n+i

≥n^1−δ

3bn

≤B_γb^γ_nn^{−γ+γ δ}(N+1)