Berry-Esseen estimates for regenerative processes under weak moment assumptions
aa r X i v : . [ m a t h . P R ] S e p BERRY-ESSEEN ESTIMATES FOR REGENERATIVE PROCESSES UNDERWEAK MOMENT ASSUMPTIONS
XIAOQIN GUO AND JONATHON PETERSON
Abstract.
We prove Berry-Esseen type rates of convergence for central limit theorems (CLTs)of regenerative processes which generalize previous results of Bolthausen under weaker momentassumptions. We then show how this general result can be applied to obtain rates of convergencefor (1) CLTs for additive functionals of positive recurrent Markov chains under certain conditions onthe strong mixing coefficients, and (2) annealed CLTs for certain ballistic random walks in randomenvironments. Introduction
A real-valued stochastic process { X n } n ≥ is called a (discrete time) regenerative process if thereexists an increasing sequence of random times (not necessarily stopping times) 0 = τ < τ < τ <τ < · · · such that if G m = σ ( τ , τ , . . . , τ m , X , X , . . . , X τ m ) for m ≥ P ( { X n + τ m − X τ m } n ≥ ∈ A, { τ m + k − τ m } k ≥ ∈ B | G m )= P ( { X n + τ − X τ } n ≥ ∈ A, { τ k − τ } k ≥ ∈ B ) , for any Borel measurable sets A ⊂ R Z + and B ⊂ N Z + . That is, the random times τ m , m ≥ τ . We call therandom variables { τ n } n ≥ regeneration times for the regenerative process { X n } n ≥ . Examples ofregenerative process include:i) Sums X n = P ni =1 ξ i of iid random variables ( ξ i ) i ∈ N , where we take τ k = k .ii) Additive functionals X n = P ni =1 f ( ζ i ) of a recurrent, irreducible Markov chain { ζ i } i ≥ on acountable state space S . In this case one defines τ n to be the n -th visit of the Markov chainto a fixed state o ∈ S in the state space of the Markov chain.iii) A ballistic random walk ( X n ) n ∈ N in a random environment under the annealed measure, where( τ n ) n ∈ N are defined to be the non-backtracking times in a fixed direction of transience (seeSection 3 for definitions of these terms).Since a regenerative process has the same law after any regeneration time τ m with m ≥
1, andsince this law may be different from the law of the process after time τ = 0, it is convenient todenote by P the law of the process after a regeneration time. That is,(1) P ( { X n } n ≥ ∈ A, { τ k } k ≥ ) = P ( { X τ + n − X τ } n ≥ ∈ A, { τ k − τ } k ≥ ∈ B ) . We will denote expectations with respect to the measures P and P by E and E , respectively. Date : October 9, 2018.2010
Mathematics Subject Classification.
Primary: 60F05; Secondary: 60K37, 60K15.
Key words and phrases. regeneration times, CLT rates of convergence, random walks in random environments.J. Peterson was partially supported by NSA grants H98230-15-1-0049 and H98230-16-1-0318.
For a regenerative process ( X n ) i ∈ N , we let X = 0 and denote the increments by ξ i := X i − X i − for i ∈ N . If E [ P τ i =1 | ξ i | ] < ∞ then it follows from standard arguments that(2) lim n →∞ X n n = E [ X τ ] E [ τ ] =: µ, P -a.s.Moreover, if E [ τ ] < ∞ and E h ( P τ i =1 | ξ i − µ | ) i < ∞ then a CLT holds for the sums of theregenerative sequence. That is, if Φ( t ) is the standard normal distribution function, then(3) lim n →∞ P (cid:18) X n − nµσ √ n ≤ t (cid:19) = Φ( t ) ∀ t ∈ R , where σ := E [( X τ − τ µ ) ] E [ τ ] > . The main result in this paper is the following theorem which gives polynomial rates of convergencefor the regenerative CLT in (3) under appropriate moment assumptions.
Theorem 1.1.
Assume for some δ ∈ (0 , that E [ τ δ ] < ∞ , E τ X i =1 | ξ i | ! δ < ∞ , E [ τ δ ] < ∞ , and E [ X δτ ] < ∞ , then there exists a constant C < ∞ such that (4) sup t ∈ R (cid:12)(cid:12)(cid:12)(cid:12) P (cid:18) X n − nµσ √ n ≤ t (cid:19) − Φ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cn δ/ , ∀ n ≥ , where µ and σ are defined as in (2) and (3) . Theorem 1.1 generalizes several known results. First of all, for i.i.d. sequences (i.e., when τ k ≡ k )the conclusion of Theorem 1.1 is the classical Berry-Esseen Theorem [Ber41, Ess42]. For regen-erative sequences, the results of Theorem 1.1 for the case δ = 1 were proved by Bolthausen in[Bol80]. Some of the techniques introduced by Bolthausen were then used in [Hip85, Mal93] to ob-tain asymptotic expansions of the CLT (i.e., identifying lower order terms in the CLT error beyondthe Berry-Esseen rates) under higher moment assumptions. The results of this paper extend theresults of [Bol80] in a different direction, obtaining weaker bounds on the rate of decay in the CLTerror but under less restrictive moment assumptions.For i.i.d. sequences, the Berry-Esseen Theorem states that the constant C in Theorem 1.1 canbe given by C δ E [ | ξ − µ | δ ] E [( ξ − µ ) ] δ/ for some absolute constant C δ < ∞ depending only on δ ∈ (0 , C explicitly. However, if one examines carefully the proofs in the paper, it can be seenthat these show that(5) lim sup n →∞ n δ/ sup t ∈ R (cid:12)(cid:12)(cid:12)(cid:12) P (cid:18) X n − nµσ √ n ≤ t (cid:19) − Φ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ′ < ∞ , and that the constant C ′ can be expressed explicitly in terms of certain moments of τ , X τ , P τ i =1 | ξ i | and ( X τ − µτ ) under the measures P and P . However, since for one of the main applications thatwe are interested in (random walks in random environments) the moments of τ and X τ cannotbe explicitly computed, we focus on the polynomial rate of decay rather than computing explicituniform upper bounds. We note also that (5) is sufficent to imply that the uniform upper bound(4) holds for some (non-explicit) C < ∞ , and thus our proof below will focus on proving (5) ratherthan (4). In [Bol80], the results were for additive functionals of positive recurrent Markov chains. However, the proofs in[Bol80] only use the regenerative structure of positive recurrent Markov chains and thus go through without changefor regenerative processes.
EGENERATIVE CLT RATES 3
Outline of the paper.
The remainder of the paper is structured as follows. In Section 2 weshow how Theorem 1.1 can be applied to additive functionals of Markov chains satisfying certainmixing conditions and moment bounds, and then in Section 3 we give applications of Theorem 1.1 toballistic RWRE on Z d for any d ≥
1. In both Sections 2 and 3 certain applications require Theorem1.1 with δ <
1, showing the necessity of generalizing the previous results in [Bol80]. The proof ofTheorem 1.1 is then given in Sections 4 and 5. The general approach of these two sections followsthat of [Bol80], but certain parts need to be adapted due to the weaker moment assumptions. Inparticular, the main result of Section 4 (Theorem 4.2) is a semi-local Berry-Esseen estimate forsums of two-dimensional i.i.d. random variables that is quite technical and required significant workto generalize the corresponding semi-local Berry-Esseen estimates in [Bol80]. Finally, in Section 6we again consider the rates of convergence of CLTs for RWRE, comparing the results of this paperwith other recent results and posing a few open questions regarding CLTs for RWRE which cannotbe handled using the regenerative methods in this paper.Throughout the paper we will use notation such as c, c ′ , C, C ′ to denote generic positive constantswhose specific values are not important and which can change from one line to the next. Specificconstants whose value remains the same throughout the paper are denoted by numbered subscriptslike c , c , C , C . When we wish to denote the dependence of a constant on a particular parameterwe will use subscript such as C ε or C f to denote this dependence.2. Application to additive functionals of Markov chains
As a first application of Theorem 1.1 we consider additive functionals of Markov chains. Let ζ = { ζ n } n ≥ be an irreducible, positive recurrent Markov chain on a countable state space S andlet X n = P ni =1 f ( ζ i ) for some function f : S → R . For a probability distribution ν on S we willdenote the law of the Markov chain with initial condition ζ ∼ ν by P ν . If we start at a fixed point ζ = x ∈ S then we will use P x in place of P δ x . Central limit theorems have been proved for additivefunctionals of Markov chains under a number of conditions (see for instance [Chu67, Jon04, MT09]).We will be interested here in conditions for a CLT which are given in terms of the strong mixingcoefficients of the Markov chain, α ( n ) = sup m sup A ∈ σ ( ζ i , i ≤ m ) sup B ∈ σ ( ζ i , i ≥ m + n ) | P π ( A ∩ B ) − P π ( A ) P π ( B ) | . For positive recurrent, aperiodic Markov chains it is known that lim n →∞ α ( n ) = 0 [Ros71, p. 195].The following Theorem, which is a direct application of [IL71, Theorem 18.5.3], shows that if thestrong mixing coefficients decay fast enough then there is a CLT for the additive functional X n . Theorem 2.1 (Theorem 18.5.3 in [IL71]) . Let X n = P ni =1 f ( ζ i ) , where { ζ i } i ≥ is an irreducible,positive recurrent Markov chain on a countable state space S with stationary distribution π . Assumethat for some p ∈ (2 , ∞ ] (i) f ∈ L p ( S , π ) ,(ii) and P n ≥ α ( n ) p − p < ∞ , where α ( n ) are the strong mixing coefficients.Then, (6) µ f := E π [ f ( ζ )] < ∞ and σ f := Var π ( f ( ζ )) + 2 ∞ X k =1 Cov π ( f ( ζ ) , f ( ζ k )) < ∞ , and if σ f > then (7) lim n →∞ P π (cid:18) X n − µ f nσ f √ n ≤ x (cid:19) = Φ( x ) , ∀ x ∈ R . XIAOQIN GUO AND JONATHON PETERSON
The main goal of this Section is to show how Theorem 1.1 allows us to obtain quantitative boundson the polynomial rate of convergence for the CLT in (7) under slightly stronger assumptions onthe strong mixing coefficients.
Theorem 2.2.
Let X n = P ni =1 f ( ζ i ) , where ζ is an irreducible, positive recurrent Markov chain ona countable state space S with stationary distribution π . Assume for some p ∈ (2 , ∞ ] and λ > p − that(i) f ∈ L p ( S , π ) (ii) and P n ≥ n λ α ( n ) < ∞ , where α ( n ) are the α -mixing coefficients.Then µ f and σ f defined in (6) are finite, and if σ f > and the initial distribution ν of the Markovchain is bounded by some multiple of the stationary distribution π , then there exists a constant C > such that sup x (cid:12)(cid:12)(cid:12)(cid:12) P ν (cid:18) X n − µ f nσ f √ n ≤ x (cid:19) − Φ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( Cn − min { λ ( p − − λ +1+ p ) , } if < p < ∞ Cn − min { λ , } if p = ∞ . Remark . The assumptions on the mixing coefficients in Theorem 2.2 are only slightly strongerthan in Theorem 2.1. Indeed, if P n n λ α ( n ) for some λ > p − then X n α ( n ) p − p = X n (cid:16) n λ α ( n ) (cid:17) p − p n − λ ( p − p ≤ X n n λ α ( n ) ! p − p X n n − λ ( p − ! p < ∞ . Conversely, since α ( n ) is non-increasing it can be shown that if P n α ( n ) p − p < ∞ then P n n λ α ( n ) < ∞ for any λ < p − . Remark . Theorem 2.2 extends another result of Bolthausen from [Bol80]. In [Bol80] it wasshown that the optimal O (1 / √ n ) rates of convergence for the CLT of X n hold when p > λ ≥ p +3 p − (including the case when p = ∞ and λ ≥ O (1 / √ n ) rates of convergence. In contrast, Theorem 2.2 gives slower polynomial rates ofconvergence when either(8) (i) p ∈ (2 ,
3] and λ > p − , or (ii) p > λ ∈ (cid:18) p − , p + 3 p − (cid:19) , where in the second case we are including p = ∞ and λ ∈ (0 , Proof.
As noted in Remark 2.2, due to the results in [Bol80] we need only give the proof of Theorem2.2 when λ > p > δ = ( λ ( p − − λ +1+ p if p < ∞ λ if p = ∞ . Note that the conditions on λ and p in (8) imply that δ defined in this way satisfies δ ∈ (0 , X n = P ni =1 f ( ζ i ), fix an arbitrarystate o ∈ S and define the regeneration times to be the successive return times of the Markov chainto o . That is, τ = 0 and τ k = inf { n > τ k − : ζ n = o } for k ≥
1. In this case, the distribution P defined in (1) is simply P o and thus since we are assuming that the initial distribution ν is boundedby a multiple of the stationary distribution Theorem 1.1 will give rates of convergence for a CLT EGENERATIVE CLT RATES 5 of X n if(10) E o [ τ δ ] < ∞ , E o τ X i =1 | f ( ζ i ) | ! δ < ∞ , E π [ τ δ ] < ∞ , and E π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ X i =1 f ( ζ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ < ∞ , with δ ∈ (0 ,
1) defined as in (9).It was shown in [Bol80, Theorem 2] that the mixing condition P n n λ α ( n ) < ∞ implies that E o [ τ λ ] < ∞ and therefore also that E π [ τ λ ] < ∞ . Since it can easily be checked that λ ( p − − λ +1+ p ≤ λ , it follows that the first and third conditions in (10) hold.In the case when p = ∞ , the function f is then bounded and the second and fourth conditionsin (10) are finite whenever the first and third conditions are finite. Therefore, for the remainder ofthe proof we will assume that p ∈ (2 , ∞ ). To verify the second condition in (10) in this case, notethat E o τ X i =1 | f ( ζ i ) | ! δ = E o τ X i =1 | f ( ζ i ) | ! p δp ≤ E o τ p − τ X i =1 | f ( ζ i ) | p ! δp ≤ E o (cid:20) τ ( p − δ ) p − − δ (cid:21) p − − δp E o " τ X i =1 | f ( ζ i ) | p δp = E o h τ λ i p − λ +1+ p E o " τ X i =1 | f ( ζ i ) | p λλ +1+ p , where the second inequality follows from H¨older’s inequality since p δ = λ +1+ p λ > δ in (9). We have already shown that the first expectationin the last line is finite, and the second expectation is also finite since f ∈ L p ( S , π ) and E o " τ X i =1 | f ( ζ i ) | p = X x ∈S E o " τ X i =1 { ζ i = x } | f ( x ) | p = E o [ τ ] X x ∈S π ( x ) | f ( x ) | p . Finally, for the second condition in (10), in the proof of Lemma 1 on page 61 of [Bol80] it wasshown that E π " τ X i =1 | f ( ζ i ) | ≤ π ( o ) E o τ − X i =0 | f ( ζ i ) | ! + 2 π ( o ) E o [ τ ] + max {| f ( o ) | , } , and the terms on the right are all finite by the arguments above. Since δ < (cid:3) Remark . For Harris recurrent Markov chains on more general state spaces, under a certainregularity assumption Nummelin [Num78] developed a “splitting” technique which allows one toconstruct a related Markov chain which does have regeneration times. The proof of Theorem 2.2can be extended to such Harris recurrent Markov chains using this splitting technique in the samemanner as was done by Bolthausen in [Bol82] in the case when p > λ ≥ p +3 p − . XIAOQIN GUO AND JONATHON PETERSON
Remark . The proof of the CLT for X n = P ni =1 f ( ζ i ) using the regenerative structure as in theproof above shows that µ f and σ f as defined in (6) must also have the alternative expressions(11) µ f = E o [ P τ i =1 f ( ζ i )] E o [ τ ] and σ f = E o h ( P τ i =1 f ( ζ i ) − µ f τ ) i E o [ τ ] . The equality of the expressions in (6) and (11) can also be verified more directly using the repre-sentation of the stationary distribution π ( x ) = E o [ τ ] E o (cid:2)P τ i =1 { ζ i = x } (cid:3) .3. Application to RWRE: Annealed CLT rates
In this section we will show how the results of Theorem 1.1 can be applied to certain non-Markovian random walks. For simplicity we will restrict ourselves to nearest neighbor RWRE,though clearly the same arguments will apply to other non-Markovian random walks with a similarregeneration structure and known bounds on the moments of regeneration times (e.g. excitedrandom walks [BR07, KZ08]).We begin by recalling the model of random walks in random environments. For nearest-neighborRWRE on Z d , an environment ω is a collection of probability distributions on E d = { z ∈ Z d : | z | = 1 } indexed by the vertices of Z d . That is, ω = { ω x ( z ) } x ∈ Z d , z ∈E d such that ω x ( z ) ≥ P z ∈E d ω x ( z ) = 1 for every x ∈ Z d . Given an environment ω , a random walk in the environment ω is a Markov chain { X n } n ≥ on Z d with law P ω given by P ω ( X = ) = 1 and P ω ( X n +1 = x + z | X n = x ) = ω x ( z ) , ∀ x ∈ Z d , z ∈ E d , n ≥ . A random walk in a random environment is then obtained by first choosing an environment ω randomly according to some fixed probability distribution P on the space of environments and thenrunning a random walk in that fixed environment. In general it is assumed that the distribution onenvironments P is ergodic under spatial shifts of Z d , but for this paper we will adopt the commonassumption that the environment is i.i.d. – that is, the family { ω x ( · ) } x ∈ Z d of transition probabilitiesindexed by the vertices of Z d is i.i.d. under the distribution P on environments. The distribution P ω of the walk conditioned on the environment ω is called the quenched law, while the distribution(12) P ( · ) = E [ P ω ( · )] , where both the environment and the walk are random is called the annealed (or averaged) law of theRWRE. Note that in (12) and below E [ · ] will denote expectation with respect to the distribution P on environments. Expectations with respect to the quenched and annealed laws on the RWREwill be denoted by E ω [ · ] and E [ · ] respectively.While the (multidimensional) Central Limit Theorem implies that classical simple random walkson Z d always have a Gaussian limiting distributions under diffusive scaling, random walks in randomenvironments (RWRE) on Z d are much more difficult to study and can have limiting distributionswhich are non-Gaussian (see for instance [KKS75, Sin83, Bv11]). Nonetheless, there are sufficientconditions for the distribution on the environment which ensure that a CLT holds for the RWRE.Our main goal in this section is to consider some ballistic (non-zero limiting speed) RWREfor which a CLT is known to hold under the annealed measure and to prove polynomial ratesof convergence for this CLT. While the limiting distributions of RWRE have been studied quiteextensively, there has been up until recently very few results giving quantitative bounds on the ratesof convergence. In particular, we are only aware of two such prior results for RWRE [Mou12, AP17].Our results below differ from both of these in the following ways. The results in [Mou12] consideredthe random conductance model while our results are applied to certain RWRE in i.i.d. environments.Also, the results in [AP17] gave bounds on the polynomial rate of convergence for the quenched CLT of one-dimensional RWRE while we consider in this paper the rates of convergence for the
EGENERATIVE CLT RATES 7 annealed
CLT and apply to certain multidimensional RWRE as well. A more in depth discussionof the relation between the quenched and annealed rates of convergence for RWRE is given at theend of this paper in Section 6.To apply the results of Theorem 1.1 to RWRE, we need to first review the appropriate conceptsof regeneration times for RWRE. If { X n } n ≥ is a RWRE on Z d and u ∈ S d − = { z ∈ R d : | z | = 1 } is a fixed direction, then, setting (13) τ u , = 0 , τ u ,k = inf (cid:26) n > τ u ,k − : sup m 1. With this i.i.d. structure, thefollowing results are known. • LLN [SZ99] : If E [ τ u , − τ u , ] < ∞ , then lim n →∞ X n n = v = , almost surely, where(14) v = E [ X τ u , − X τ u , ] E [ τ u , − τ u , ] . • CLT [Szn00] : If E [( τ u , − τ u , ) ] < ∞ then X n − n v √ n converges in distribution under theannealed law P to a d -dimensional Normal distribution with zero mean and covariancematrix(15) Σ = 1 E [ τ u , − τ u , ] E h(cid:0) X τ u , − X τ u , − ( τ u , − τ u , ) v (cid:1) (cid:0) X τ u , − X τ u , − ( τ u , − τ u , ) v (cid:1) T i . Since Theorem 1.1 gives rates of convergence for a one-dimensional CLT, we can only apply thisto one-dimensional projections of a multidimensional RWRE. To this end, suppose that there is adirection u ∈ S d − such that P ( A u ) = 1. Then for any other direction w ∈ S d − we can applyTheorem 1.1 to the sequence X n · w . Theorem 3.1. Let X n be a d -dimensional RWRE, and let u ∈ S d − be such that (16) E h ( τ u , − τ u , ) δ i < ∞ and E h τ δ u , i < ∞ for some δ ∈ (0 , . Then, there exists a constant C < ∞ such that for any w ∈ S d − , sup x ∈ R (cid:12)(cid:12)(cid:12)(cid:12) P (cid:18) ( X n − n v ) · w σ w √ n ≤ x (cid:19) − Φ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cn δ/ , where v is as in (14) and σ w = w T Σ w where Σ is the covariance matrix in (15) .Remark . The following remarks are in order regarding the moment assumptions (16) in Theorem3.1. • Since the RWRE is a nearest neighbor walk, the random variables ξ n = ( X n − X n − ) · w have | ξ n | ≤ In this definition we are using the convention that inf ∅ = ∞ ; that is, if τ u ,k = ∞ for some k then τ u ,k +1 is takento be ∞ also. XIAOQIN GUO AND JONATHON PETERSON • For one-dimensional RWRE, it can be shown under mild ballisticity condition that therequirement (16) is equivalent to E [ τ δ ] < ∞ . See Proposition 3.5. • When the dimension d ≥ 2, for uniformly elliptic and ballistic environment, it is conjecturedthat all moments of the regeneration times are finite. However, this is not true when theballistic environment is only assumed to be elliptic . See the following for more detailedcomments.Theorem 3.1 reduces the problem of obtaining rates of convergence for the annealed CLT tocomputing certain moment bounds of the regeneration times. For multidimensional RWRE, agreat deal of effort has gone into obtaining improved conditions under which moment bounds onregeneration times can be obtained and we will review the best known conditions here, though thefull picture is not yet complete.(1) Uniformly elliptic environments. A nearest neighbor RWRE is called uniformly elliptic if there exists a constant c > P ( ω ( z ) ≥ c ) = 1 for all | z | = 1; that is, thetransition probabilities in all directions are uniformly bounded away from zero. For uni-formly elliptic RWRE, a number of conditions have been shown to imply that E [ τ p v , ] < ∞ for all p < ∞ where v = is the limiting speed; these conditions include Kalikow’s condi-tion [Szn00], Sznitman’s conditions ( T ), ( T ′ ) and ( T ) γ [Szn01, Szn02], and the Polynomialcondition ( P ) introduced by Berger, Drewitz, and Ram´ırez [BDR14].We refer the interested reader to the above references for the exact statement of theseconditions and simply note that the weakest condition is the polynomial condition ( P ) andthat this condition is “effective” in the sense that it can be verified by computing certainexit probabilities of the RWRE from a large but finite multidimensional box. We alsonote that all of the known conditions implying ballisticity (non-zero limiting speed) foruniformly elliptic RWRE imply moments of all orders for the regeneration times. In fact itis conjectured that for uniformly elliptic RWRE in dimension d ≥ P ( A u ) = 1 (i.e.,transience in direction u ) implies that E [ τ p u , ] < ∞ for all p < ∞ . This is in contrast to whatis known for one-dimensional RWRE (see Proposition 3.5 below) and for multidimensionalRWRE which are not uniformly elliptic.(2) Elliptic environments. A nearest neighbor RWRE is called elliptic if P ( ω ( z ) > 0) = 1for all | z | = 1; that is, the transition probabilities in all directions are non-zero but notnecessarily uniformly bounded away from zero. In [BRS16] and [FK16], checkable ellipticityconditions are given which together with the polynomial condition ( P ) imply the finitenessof certain moments of the regeneration times. Moreover, these papers also give explicitexamples of elliptic RWRE which satisfy condition ( P ) or even the stronger Kalikow’scondition but which do not have all moments of regeneration times finite. In particular, fori.i.d. Dirichlet random environments there are certain choices of the parameters for whichthe results in [BRS16] show that the regeneration times have infinite third moment butfinite (2 + δ ) moments for some δ ∈ (0 , One-dimensional RWRE. The purpose of this subsection is to consider more in depth theannealed CLT rates of convergence for one-dimensional RWRE. In one dimension we are able toobtain more explicit results as a result of the fact that it is possible to give an explicit criterion forwhat moments of the regeneration times of the RWRE are finite (see Proposition 3.5 below). Ourmain result in this subsection (Corollary 3.3) gives explicit polynomial rates of convergence for theannealed CLTs of both the position and the hitting times of the walk.For a RWRE on Z there is no need to take a projection to apply Theorem 1.1 and so we willwrite X n for the position of the walk rather than X n . Also, if the walk is transient, without loss EGENERATIVE CLT RATES 9 of generality we can assume it is transient to the right and so we need only consider regenerationtimes to the right and will therefore write τ k rather than τ ,k .For one-dimensional RWRE in i.i.d. environments, much of the behavior of the walk can beexplicitly characterized in terms of the distribution of the random variable ρ = ω ( − ω (1) . In particular,it was shown in [Sol75, KKS75] that • the random walk is transient to the right if and only if E [log ρ ] < • the limiting speed v = lim n →∞ X n n is positive if and only if E [ ρ ] < − E [ ρ ]1+ E [ ρ ] for the speed, • and if E [ ρ ] < X n and thehitting times T n = inf { k ≥ X k = n } . That is,(17) lim n →∞ P (cid:18) X n − n vv / σ √ n ≤ x (cid:19) = Φ( x ) and lim n →∞ P (cid:18) T n − n/ v σ √ n ≤ x (cid:19) = Φ( x ) , ∀ x ∈ R , where σ = E [Var ω ( T )] + Var( E ω [ T ]).In fact the CLTs in (17) are a specific case of a more general result on limiting distributions byKesten, Kozlov, and Spitzer [KKS75]. If the RWRE is transient to the right (i.e., E [log ρ ] < 0) then,under mild technical assumptions on the distribution on the environment, the limiting distributiondepends on a parameter κ > E [ ρ κ ] = 1. The assumption E [ ρ ] < κ > 2, and this is the only case where annealed CLTs like (17) hold; if κ ∈ (0 , 2) then the limiting distribution is not Gaussian and the scaling is not diffusive, while if κ = 2 then the limting distributions of X n and T n are Gaussian but with logarithmic correctionsto the diffusive scaling √ n . When κ > 2, the following Corollary of Theorems 1.1 and 3.1 givespolynomial rates of convergence for both of the annealed CLTs in (17). Corollary 3.3. Assume that E [log ρ ] < and E [ ρ κ ] = 1 for some κ > .(1) If κ > , then there exists a constant C < ∞ such that (1a) sup x ∈ R (cid:12)(cid:12)(cid:12)(cid:12) P (cid:18) X n − n v σ v / √ n ≤ x (cid:19) − Φ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C √ n and (1b) sup x ∈ R (cid:12)(cid:12)(cid:12)(cid:12) P (cid:18) T n − n/ v σ √ n ≤ x (cid:19) − Φ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C √ n . (2) If κ ∈ (2 , , then for any ε > , (2a) lim n →∞ n κ − − ε sup x ∈ R (cid:12)(cid:12)(cid:12)(cid:12) P (cid:18) X n − n v σ v / √ n ≤ x (cid:19) − Φ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 and (2b) lim n →∞ n κ − − ε sup x ∈ R (cid:12)(cid:12)(cid:12)(cid:12) P (cid:18) T n − n/ v σ √ n ≤ x (cid:19) − Φ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 . The key to the proof of Corollary 3.3 will be establishing moment bounds for the regenerationtimes of the RWRE in terms of the parameter κ . As a first step in this direction, the followinglemma shows that κ determines what moments of hitting times are finite. Lemma 3.4. Assume that E [log ρ ] < and that E [ ρ κ ] = 1 for some κ ≥ . Then E [ T γ ] < ∞ ifand only if γ < κ . Proof. It was shown in [DPZ96] that γ < κ implies that E [ T γ ] < ∞ . For the reverse implication,we will use the fact that the quenched expectation of T has the explicit formula (see [Sol75] or[Zei04]), E ω [ T ] = 1 + 2 ∞ X k =1 0 Y x = − k +1 ω x ( − ω x (1) . Therefore, if γ ≥ E [ T γ ] ≥ E [( E ω [ T ]) γ ] ≥ γ E " ∞ X k =1 Y x = − k +1 ω x ( − ω x (1) ! γ = 2 γ ∞ X k =1 E [ ρ γ ] k , where we used that the environment is i.i.d. in the last equality. If γ ≥ κ , then it follows fromJensen’s inequality that E [ ρ γ ] ≥ E [ ρ κ ] γ/κ = 1, and thus the sum on the right above is infinite. (cid:3) The following Proposition shows that the parameter κ also determines what moments of theregeneration times are finite. Proposition 3.5. Assume that E [log ρ ] < and that E [ ρ κ ] = 1 for some κ ≥ . Then E [ τ γ ] < ∞ and E [( τ − τ ) γ ] < ∞ if and only if γ < κ .Proof of Proposition 3.5. In the context of one-dimensional RWRE, the measure P as defined in(1) for the regenerative sequence X n is the same as P ( · | T − = ∞ ). Therefore, E [( τ − τ ) γ ] = E [ τ γ ] = E [ τ γ { T − = ∞} ] P ( T − = ∞ ) , which implies that P ( T − = ∞ ) E [ τ γ ] ≤ E [ τ γ ]. Since P ( T − = ∞ ) > E [ τ γ ] < ∞ if γ < κ and E [ τ γ ] = ∞ if γ ≥ κ .To prove that E [ τ γ ] < ∞ when γ < κ , by decomposing according the the location of the walk atthe first regeneration time, we obtain that for any ε > E [ τ γ ] = ∞ X n =1 E h ( T n ) γ { X τ = n } i ≤ ∞ X n =1 E h ( T n ) γ (1+ ε ) i ε P ( X τ = n ) ε ε ≤ E h T γ (1+ ε )1 i ε ∞ X n =1 n γ ∨ ( ε ) P ( X τ = n ) ε ε , where the last inequality follows either from Minkowski’s inequality when γ (1 + ε ) ≥ x x γ (1+ ε ) when γ (1 + ε ) < 1. Since [Szn01, Prop. 2.6] implies that P ( X τ = n ) ≤ e − cn for some c > 0, it follows from Lemma 3.4 that if γ < κ the right side is finitefor ε > In general, the results in [Szn01] assume that the RWRE is “uniformly elliptic,” i.e., that all transition probabilitiesare uniformly bounded away from zero. However, an examination of the proof of Proposition 2.6 in that paper showsthat the uniform ellipticity assumption is not needed there. EGENERATIVE CLT RATES 11 To prove that E [ τ γ ] = ∞ when γ ≥ κ , note that if γ ≥ P -almost surely, E [ τ γ ] = lim n →∞ n n X k =1 ( τ k − τ k − ) γ = lim n →∞ n n X k =1 X τk X x = X τk − +1 ( T x − T x − ) γ ≥ lim n →∞ n n X k =1 X τk X x = X τk − +1 ( T x − T x − ) γ = lim n →∞ n X τn X x =1 ( T x − T x − ) γ = E [ X τ ] E [ T γ ] . where in the last equality we used that the sequence { T x − T x − } x ≥ is ergodic under the annealedmeasure [Sol75]. Therefore, if γ ≥ κ ≥ E [ τ γ ] = ∞ . (cid:3) Proof of Corollary 3.3. Applying Proposition 3.5 to Theorem 3.1 for any δ < ( κ − ∧ 1, weimmediately obtain (1a) and (2a).The proofs of (2a) and (2b) also follow from Theorem 1.1, but applied to a different regenerativeprocess. Represent T n = P nk =1 ζ i where ζ i = T i − T i − . Then under the annealed measure P thesequence ( T n ) n ≥ is a regenerative process with “regeneration times” 0 = σ < σ < σ < · · · where σ k = X τ k is the position of the walk at the time of the k -th regeneration time of the walk. Sincethe crossing times ζ i ≥ 1, to apply Theorem 1.1 we need only to check that E [( P σ i =1 ζ i ) δ ] < ∞ and E [ (cid:0)P σ i = σ +1 ζ i (cid:1) δ ] < ∞ for some δ ∈ (0 , σ k X i = σ k − +1 ζ i = T σ k − T σ k − = T X τk − T X τk − = τ k − τ k − , this is equivalent to checking that E [ τ δ ] and E [( τ − τ ) δ ] < ∞ , and by Proposition 3.5 this holdsfor δ = 1 if κ > δ ∈ (0 , − κ ) if κ ∈ (2 , (cid:3) A non-uniform semi-local Berry-Esseen bound Consider a random variable Z = ( V, W ) ∈ R with zero-mean E [ Z ] = and a positive-definitecovariance matrix Σ = (cid:18) Var( V ) Cov( V, W )Cov( V, W ) Var( W ) (cid:19) = (cid:18) σ σ σ σ (cid:19) > . (That is, both eigenvalues of Σ are strictly positive.) Let Z i = ( V i , W i ), i ∈ N , denote iid copies of Z and S n = ( X n , Y n ) := n X i =1 V i , n X i =1 W i ! . Throughout this section, we assume that almost surely, W ∈ ρ + Z for some ρ ∈ R and that W hasa lattice distribution with span 1.By the central limit theorem, if E [ | Z | ] < ∞ , then S n / √ n converges weakly to a two-dimensionalnormal random variable N = ( N , N ) with covariance matrix Σ. Here | Z | := √ V + W . More-over, when E [ | W | ] < ∞ , the classical local limit theorem (LLT) states that the probability massfunction of Y n / √ n converges to the density of N . See [Pet75, VII]. Under weaker moment condi-tion E [ | W | δ ] < ∞ for some δ ∈ (0 , Proposition 4.1. Assume that E [ | W | δ ] < ∞ for δ ∈ (0 , . Writing y n := ( y + nρ ) / √ n for y ∈ Z . Then sup y ∈ Z (1 + y n ) (cid:12)(cid:12)(cid:12) P (cid:16) Y n √ n = y n (cid:17) − σ √ nπ e − y n / σ (cid:12)(cid:12)(cid:12) ≤ Cn − δ , where the constant C depends only on δ and E [ | W | δ ] . For any positive definite 2 × γ A ( x ) = C A exp {− x T A − x / } , x ∈ R be the densityfunction of a centered Gaussian with covariance matrix A and let(18) ψ A ( x, y ) = Z x −∞ γ A ( t, y ) dt. The purpose of this section is to generalize Proposition 4.1 to a non-uniform estimate of a semi-local limit theorem, which is of interest in its own right. Theorem 4.2. Assume that E [ | Z | δ ] < ∞ for δ ∈ (0 , , then sup x ∈ R ,y ∈ Z (1 + y n ) (cid:12)(cid:12)(cid:12)(cid:12) P (cid:16) X n √ n ≤ x, Y n √ n = y n (cid:17) − √ n ψ Σ ( x, y n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cn − (1+ δ ) / . For the case δ = 1, Theorem 4.2 was previously obtained by Bolthausen[Bol80, Theorem 4]. Ourproof follows the main idea of [Bol80], where characteristic functions (ch.f.) are used to expressthe probabilities. In fact, the term y n comes from second-order derivatives of ch.f.’s. However,unlike [Bol80], estimates about the third order derivative of ch.f.’s (which were used to bound thedifference of the second-order derivatives) are not available because of the lack of moments when δ ∈ (0 , Estimates of characteristic functions. Let t = ( t , t ) ∈ R . We denote the characteristicfunctions of Z , S n / √ n and N by ϕ ( t ), λ n ( t ) = ϕ ( t / √ n ) n and λ ( t ) = exp( − t T Σ t / Proposition 4.3. Assume E [ | Z | δ ] < ∞ for δ ∈ (0 , . Then there exist positive constants ε, c, C depending on δ, Σ and E [ | Z | δ ] such that for any t ∈ R with | t | ≤ ε √ n , | t | ≤ π √ n , (a) (cid:12)(cid:12)(cid:12) ϕ ( t √ n ) n − j − λ ( t ) (cid:12)(cid:12)(cid:12) ≤ Cn − δ/ e − c | t | , ∀ j = 0 , , (cid:12)(cid:12)(cid:12) ∂ ∂t ( λ n ( t ) − λ ( t )) (cid:12)(cid:12)(cid:12) ≤ Cn − δ/ e − c | t | ; (c) Set Λ( t ) = Λ n ( t ) := ∂ ∂t ( λ n ( t ) − λ ( t )) . Then there exists a constant c > such that (cid:12)(cid:12)(cid:12) Λ( t , t ) − Λ(0 , t ) + ϕ (0 , t √ n ) n − E [ W ( e i t · Z / √ n − e it W/ √ n )] (cid:12)(cid:12)(cid:12) ≤ Cn − δ/ | t | (1 + | t | ) e − c t . Before giving the proof, let’s recall some basic inequalities. For any x ∈ R and any δ ∈ [0 , | e ix − | = 2 | sin x | ≤ | x/ | δ , and so(20) (cid:12)(cid:12)(cid:12) e ix − ix − (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ix Z ( e isx − s (cid:12)(cid:12)(cid:12) ≤ C δ | x | δ , (21) (cid:12)(cid:12)(cid:12) e ix − (1 + ix − x / (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) x Z ( s − e isx − s (cid:12)(cid:12)(cid:12) ≤ C δ | x | δ . EGENERATIVE CLT RATES 13 Proof. (a) We first consider the case | t | ≤ ε √ n for small enough ε > (cid:12)(cid:12)(cid:12) ϕ ( t √ n ) − (cid:12)(cid:12)(cid:12) ≤ C δ | t / √ n | δ E [ | Z | δ ] , (23) (cid:12)(cid:12)(cid:12) ϕ ( t √ n ) − (1 − t T Σ t / n ) (cid:12)(cid:12)(cid:12) ≤ C δ | t / √ n | δ E [ | Z | δ ] . We take ε > | ϕ ( t / √ n ) − | < . | t | ≤ ε √ n . In this caselog ϕ ( t / √ n ) is well-defined for | t | / √ n ≤ ε , and (cid:12)(cid:12)(cid:12) log ϕ ( t √ n ) + t T Σ t / n (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) log (cid:16) − [1 − ϕ ( t √ n )] (cid:17) + t T Σ t / n (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ∞ X k =2 ( ϕ ( t √ n ) − k k + ϕ ( t √ n ) − t T Σ t / n (cid:12)(cid:12)(cid:12) (23) ≤ (cid:12)(cid:12)(cid:12) ϕ ( t √ n ) − (cid:12)(cid:12)(cid:12) ∞ X k =2 − k k + C ( | t |√ n ) δ (22) ≤ C ( | t |√ n ) δ . Further, for | t | / √ n ≤ ε , using the inequality | e x − | ≤ | x | e | x | , for 0 ≤ j ≤ (cid:12)(cid:12)(cid:12) ϕ ( t √ n ) n − j − λ ( t ) (cid:12)(cid:12)(cid:12) = e − ( n − j ) t T Σ t / n (cid:12)(cid:12)(cid:12) e ( n − j ) (cid:18) log ϕ ( t √ n )+ t T Σ t / n (cid:19) − (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) e − ( n − j ) t T Σ t / n − e − t T Σ t / (cid:12)(cid:12)(cid:12) ≤ Ce − c | t | n ( | t |√ n ) δ e Cε δ | t | + C | t | n e − c | t | ≤ Cn − δ/ ( | t | + 1) δ e − c | t | , where the last inequality holds if the constant ε > | t | ≤ ε √ n .It remains to consider the case ε √ n ≤ | t | ≤ π √ n . Since the random variable W has a latticedistribution with span 1, by [Bol80, Lemma 1, § ε ′ ∈ (0 , ε ) is small enough, then thereexists γ = γ ( ε, ε ′ ) ∈ (0 , 1) such that(24) | ϕ ( t √ n ) | ≤ − γ, ∀ | t | ≤ ε ′ √ n, ε √ n ≤ | t | ≤ π √ n. Hence, when | t | ≤ ε ′ √ n and ε √ n ≤ | t | ≤ π √ n , for j = 0 , , | ϕ ( t √ n ) n − j | + λ ( t ) ≤ Ce − cn ≤ Cn − δ/ e − ct ≤ Cn − δ/ e − c | t | . Therefore, we have proved that (a) holds whenever | t | ≤ ε ′ √ n, | t | ≤ π √ n . (b) Note that(25) ∂ ∂t λ n ( t ) = − ( n − ϕ ( t √ n ) n − E [ W e i t · Z / √ n ] − ϕ ( t √ n ) n − E [ W e i t · Z / √ n ] . First, for any t = ( t , t ) ∈ R , (cid:12)(cid:12)(cid:12) E [ W e i t · Z − i ( t σ + σ t )] (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) E [ W ( e i t · Z − i t · Z − (cid:12)(cid:12)(cid:12) (20) ≤ C δ | t | δ E h W | Z | δ i ≤ C | t | δ . (26) Thus for any t ∈ R , (using | z − w | ≤ | z − w | + 2 | z − w || w | )(27) (cid:12)(cid:12)(cid:12) E [ W e i t · Z ] + ( t σ + σ t ) (cid:12)(cid:12)(cid:12) ≤ C | t | δ (1 + | t | δ ) . Next, for any t ∈ R ,(28) | E [ W e i t · Z − σ ] | = | E [ W ( e i t · Z − | (19) ≤ CE [ W | t · Z | δ ] ≤ C | t | δ . Combining (27) and (28), we obtain for | t | ≤ π √ n (cid:12)(cid:12)(cid:12) ∂ ∂t λ n ( t ) − n − n ϕ ( t √ n ) n − ( t σ + σ t ) + ϕ ( t √ n ) n − σ (cid:12)(cid:12)(cid:12) ≤ C | t | δ + | t | δ n δ/ (cid:12)(cid:12)(cid:12) ϕ ( t √ n ) (cid:12)(cid:12)(cid:12) n − Furthermore, since(29) ∂ ∂t λ ( t ) = ( σ t + σ t ) λ ( t ) − σ λ ( t ) , we have for | t | ≤ π √ n , (cid:12)(cid:12)(cid:12) ∂ ∂t ( λ n ( t ) − λ ( t )) (cid:12)(cid:12)(cid:12) (30) ≤ C | t | (cid:12)(cid:12)(cid:12) n − n ϕ ( t √ n ) n − − λ ( t ) (cid:12)(cid:12)(cid:12) + C (cid:12)(cid:12)(cid:12) ϕ ( t √ n ) n − − λ ( t ) (cid:12)(cid:12)(cid:12) + C | t | δ + | t | δ n δ/ (cid:12)(cid:12)(cid:12) ϕ ( t √ n ) (cid:12)(cid:12)(cid:12) n − . Note that (a) implies that(31) | ϕ ( t √ n ) n − | ≤ Ce − c | t | when | t | ≤ ε √ n and | t | ≤ π √ n. Statement (b) now follows from (a) and (30). (c) In what follows, for t = ( t , t ), we let ¯ t = (¯ t , ¯ t ) := t / √ n denote the rescaled vector. Set H j ( t ) = ϕ (¯ t ) n − j − λ ( t ) , j = 0 , , 2, and let K ( t ) := E [ W e i ¯ t · Z ] , K ( t ) := E [ W e i ¯ t · Z ] , K ( t ) := − ( σ t + σ t ) . We define the functions ˜Λ( t ) = Λ(0 , t ) , ˜Λ ( t ) = λ (0 , t ) , ˜ H j ( t ) = H j (0 , t ) and˜ K i ( t ) = K i (0 , t ), 0 ≤ j ≤ , ≤ i ≤ 3. Our goal is to obtain a bound forΛ( t ) − ˜Λ( t ) + ϕ (0 , ¯ t ) n − ( K − ˜ K ) . By (25) and (29),Λ( t ) = − ( n − H K − H K − λ [( n − K − K + ( K − σ )]Setting ∆ j := H j − ˜ H j , ≤ j ≤ 2, we have (Note that ϕ (0 , ¯ t ) n − = ˜ H + ˜Λ .)Λ( t ) − ˜Λ( t ) + ϕ (0 , ¯ t ) n − ( K − ˜ K )= − [( n − K + ∆ K ] − ( λ − ˜Λ )[( n − K − K + ( K − σ )] − ( n − 1) ˜ H ( K − ˜ K ) − ˜Λ [( n − K − ˜ K ) − ( K − ˜ K )]:= I + I + I + I . (32)We will estimate the four terms in the following steps.Step 1. To estimate I , we will first show that for | t | ≤ ε √ n, | t | ≤ π √ n ,(33) | ∆ j | ≤ Cn − δ/ | t | e − ct , ≤ j ≤ . EGENERATIVE CLT RATES 15 For simplicity we only provide proof for the case j = 0. By (a) and (31), | ∂∂t H | = | ∂∂t ( ϕ (¯ t ) n − λ ( t )) | = |√ nϕ (¯ t ) n − E [ iV ( e i ¯ t · Z − i ¯ t · Z − λ − ϕ (¯ t ) n − )( t σ + t σ ) | (20) ≤ Cn − δ/ e − c | t | . Thus | ∆ ( t ) | = (cid:12)(cid:12)(cid:12) R t ∂∂t H ( s, t )d s (cid:12)(cid:12)(cid:12) ≤ Cn − δ/ | t | e − ct . Display (33) is proved for j = 0. Theproofs for j = 1 , | K | ≤ Cn − / | t | and | K | ≤ C when | ¯ t | ≤ ε, | ¯ t | ≤ π . Hence, the term I defined in (32) has bound | I | ≤ Cn − δ/ | t | (1 + | t | ) e − ct . Step 2. To estimate I , noting that | λ − ˜Λ | ≤ C | t || t | e − ct , it suffices to show that(34) | ( n − K − K + ( K − σ ) | ≤ Cn − δ/ (1 + | t | ) . By (27) and (28), when | t | ≤ π √ n , we have | nK − K | ≤ Cn − δ/ | t | δ and | K − σ | ≤ Cn − δ/ | t | δ . Thus (34) is obtained and we can conclude that for | t | ≤ π √ n , | I | ≤ Cn − δ/ | t | (1 + | t | ) e − ct . Step 3. To estimate I , it suffices to prove that for | t | ≤ π √ n ,(35) | ( n − K − ˜ K ) − ( K − ˜ K ) | ≤ Cn − δ/ | t || t | δ . Indeed, by (20), | K − ( ˜ K + iσ ¯ t ) | = | E [ W e i ¯ t W ( e i ¯ t V − i ¯ t V − | ≤ C | ¯ t | δ . Further,by (26), we have | K | + | ˜ K | ≤ C | ¯ t | when | ¯ t | ≤ π . Hence | K − ( ˜ K + iσ ¯ t ) | ≤ Cn − (2+ δ ) / | t || t | δ . On the other hand, | n (( ˜ K + iσ ¯ t ) − ˜ K ) − ( K − ˜ K ) | = (cid:12)(cid:12)(cid:12) i √ nσ t E [ W ( e i ¯ t W − i ¯ t W − (cid:12)(cid:12)(cid:12) ≤ Cn − δ/ | t || t | δ . Thus we conclude that when | t | ≤ π √ n , | n ( K − ˜ K ) − ( K − ˜ K ) | ≤ Cn − δ/ | t || t | δ . Noticing that | K − ˜ K | ≤ C | t || t | , we get(36) n | K − ˜ K | ≤ C | t || t | when | t | ≤ π √ n. Display (35) then follows, and we obtain for | t | ≤ π √ n , | I | ≤ Cn − δ/ | t || t | δ e − ct . Step 4. Finally, by (a), we have | ˜ H | ≤ Cn − δ/ e − ct . This inequality, together with (36), yields | I | ≤ Cn − δ/ | t || t | e − ct when | t | ≤ ε √ n and | t | ≤ π √ n. Our proof is complete. (cid:3) Proof of Proposition 4.1. When B is a continuous random variable, the proof of Proposi-tion 4.1 can be found in [She17] or [BCG11]. For our case where B is a discrete random variable,we include the proof as follows for the purpose of completeness, since it is rather elementary. Proof of Proposition 4.1. First, we will express the right-hand side of the equality in terms of thecharacteristic function. We let ˜ λ ( t ) = exp( − σ t / 2) and let ˜ λ n ( t ), t ∈ R , denotes the characteristicfunctions of Y n / √ n . Then for any y ∈ Z ,(37) 1 Y n = y + nρ = 12 π Z π − π e it ( Y n − nρ ) e − ity d t = 12 π √ n Z π √ n − π √ n e itY n / √ n e − ity n d t and so P ( Y n / √ n = y n ) = 12 π √ n Z π √ n − π √ n ˜ λ n ( t ) e − ity n d t. Using integration by parts, we get y n P ( Y n / √ n = y n ) = − π √ n Z π √ n − π √ n ˜ λ ′′ n ( t ) e − ity n d t and y n √ πσ e − y n / σ = − π Z ∞−∞ ˜ λ ′′ ( t ) e − ity n d t. Thus (1 + y n ) (cid:12)(cid:12)(cid:12) √ nP ( Y n / √ n = y n ) − √ πσ e − y n / σ (cid:12)(cid:12)(cid:12) = 12 π (cid:12)(cid:12)(cid:12) Z π √ n − π √ n (˜ λ n − ˜ λ ′′ n ) e − ity n d t − Z ∞−∞ (˜ λ − ˜ λ ′′ ) e − ity n d t (cid:12)(cid:12)(cid:12) ≤ π Z π √ n − π √ n (cid:12)(cid:12)(cid:12) ˜ λ n − ˜ λ ′′ n − ˜ λ + ˜ λ ′′ (cid:12)(cid:12)(cid:12) d t + Z | t | >π √ n | ˜ λ − ˜ λ ′′ | d t. Note that R | t | >π √ n | ˜ λ − ˜ λ ′′ | d t ≤ Ce − cn . On the other hand, by Proposition 4.3(a)(b), Z π √ n − π √ n (cid:12)(cid:12)(cid:12) ˜ λ n − ˜ λ ′′ n − ˜ λ + ˜ λ ′′ (cid:12)(cid:12)(cid:12) d t ≤ Z π √ n − π √ n Cn − δ/ e − ct d t ≤ Cn − δ/ . The proposition follows. (cid:3) Proof of Theorem 4.2. The proof relies on the expression (cf. (38) and (40)) of the Kol-mogorov distance in terms of characteristic functions, where a probability measure v J is introducedto make the distribution functions smooth and to truncate their characteristic functions. To bespecific, define the measure v J (d x ) := − cos( Jx ) πJx d x on R , where J > v J ( x ) = (1 − | x | J ) + is supported on [ − J, J ]. Proof. In what follows, for any measure (or distribution function) µ , we denote its characteristicfunction by ˆ µ . Recall that the characteristic functions of S n √ n , N are denoted by λ n ( t ) and λ ( t ), t ∈ R . Also, for simplicity we will suppress the subscript Σ and write ψ Σ simply as ψ . Step 1. First, we will express the left-side of Theorem 4.2 in terms of measures with compactly sup-ported characteristic functions, i.e. (39). For any fixed y ∈ Z , let F n ( x, y n ) := P ( X n / √ n ≤ x, Y n / √ n = y n ) and denote the corresponding conditional distribution functions by ¯ F n ( x ) := F n ( x,y n ) F n ( ∞ ,y n ) , ¯ ψ n ( x ) := ψ ( x,y n ) ψ ( ∞ ,y n ) . Of course, since the case F n ( ∞ , y n ) = 0 follows immediately EGENERATIVE CLT RATES 17 from Proposition 4.1, we only consider the non-trivial case when F n ( ∞ , y n ) > 0, so that¯ F n is well-defined. Then(1 + y n ) (cid:12)(cid:12)(cid:12) √ nF n ( x, y n ) − ψ ( x, y n ) (cid:12)(cid:12)(cid:12) = (1 + y n ) (cid:12)(cid:12)(cid:12) ( ¯ F n ( x ) − ¯ ψ n ( x )) ψ ( ∞ , y n ) + ( √ nF n ( ∞ , y n ) − ψ ( ∞ , y n )) ¯ F n ( x ) (cid:12)(cid:12)(cid:12) ≤ (1 + y n ) ψ ( ∞ , y n ) (cid:12)(cid:12)(cid:12) ¯ F n ( x ) − ¯ ψ n ( x ) (cid:12)(cid:12)(cid:12) + Cn − δ/ , where in the last inequality we used Proposition 4.1. Further, let ¯ F Jn (and ¯ ψ Jn ) be theconvolution of ¯ F n (and ¯ ψ n , resp.) and the measure v J . Then, by [Fel71, Lemma 1, XVI.3],(38) sup x (cid:12)(cid:12)(cid:12) ¯ F n ( x ) − ¯ ψ n ( x ) (cid:12)(cid:12)(cid:12) ≤ x (cid:12)(cid:12)(cid:12) ¯ F Jn ( x ) − ¯ ψ Jn ( x ) (cid:12)(cid:12)(cid:12) + 24 πJ sup x (cid:12)(cid:12)(cid:12) ∂∂x ¯ ψ n ( x ) (cid:12)(cid:12)(cid:12) . From now on we take J = ε √ n , where ε is the constant in Proposition 4.3. Collecting theabove inequalities we get(39) sup x ∈ R ,y ∈ Z (1+ y n ) (cid:12)(cid:12)(cid:12) √ nF n ( x, y n ) − ψ ( x, y n ) (cid:12)(cid:12)(cid:12) ≤ C sup x ∈ R ,y ∈ Z (1+ y n ) ψ ( ∞ , y n ) (cid:12)(cid:12)(cid:12) ¯ F Jn ( x ) − ¯ ψ Jn ( x ) (cid:12)(cid:12)(cid:12) + Cn − δ/ . Step 2. Let ∆ Jn ( x ) := ¯ F Jn ( x ) − ¯ ψ Jn ( x ) . Our second step is to write ∆ Jn in terms of characteristic functions, cf (45). By Fourier’sinversion formula for distribution functions [Fel71, (3.11), XV.4], for any x > a ,¯ F Jn ( x ) − ¯ F Jn ( a ) = 12 π Z J − J e − it x − e − it a it ˆ¯ F Jn ( t )d t , (40) ¯ ψ Jn ( x ) − ¯ ψ Jn ( a ) = 12 π Z J − J e − it x − e − it a it ˆ¯ ψ Jn ( t )d t . Note that (let t := ( t , t ))ˆ¯ F Jn ( t ) = ˆ¯ F n ( t )ˆ v J ( t ) = ˆ v J ( t ) F n ( ∞ , y n ) E [ e iX n t / √ n Y n / √ n = y n ] (37) = ˆ v J ( t )2 π √ nF n ( ∞ , y n ) Z π √ n − π √ n λ n ( t ) e − it y n d t . (41) On the other hand,ˆ¯ ψ Jn ( t ) = ˆ¯ ψ n ( t )ˆ v J ( t ) = ˆ v J ( t )2 πψ ( ∞ , y n ) Z ∞−∞ λ ( t ) e − it y n d t . (42) These equalities, together with those in (40), yield √ nF n ( ∞ , y n )( ¯ F Jn ( x ) − ¯ F Jn ( a )) − ψ ( ∞ , y n )( ¯ ψ Jn ( x ) − ¯ ψ Jn ( a ))(43) = Z | t |≤ J,t ∈ R ˆ v J ( t )(2 π ) · e − it a − e − it x it e − it y n (cid:16) λ n ( t )1 | t |≤ π √ n − λ ( t ) (cid:17) d t . Further, integration by parts in (41) and (42) gives y n ˆ¯ F Jn ( t ) = − ˆ v J ( t )2 π √ nF n ( ∞ , y n ) Z π √ n − π √ n e − it y n ∂ ∂t λ n ( t )d t , y n ˆ¯ ψ Jn ( t ) = − ˆ v J ( t )2 πψ ( ∞ , y n ) Z ∞−∞ e − it y n ∂ ∂t λ ( t )d t . Similar to (43), we then have y n (cid:2) √ nF n ( ∞ , y n )( ¯ F Jn ( x ) − ¯ F Jn ( a )) − ψ ( ∞ , y n )( ¯ ψ Jn ( x ) − ¯ ψ Jn ( a )) (cid:3) (44) = Z | t |≤ J,t ∈ R ˆ v J ( t )(2 π ) · e − it a − e − it x it e − it y n (cid:16) ∂ ∂t λ n ( t )1 | t |≤ π √ n − ∂ ∂t λ ( t ) (cid:17) d t . Combining (43) and (44), we get for any x > a ,(1 + y n ) ψ ( ∞ , y n )(∆ Jn ( x ) − ∆ Jn ( a ))(45) = (1 + y n )[ ψ ( ∞ , y n ) − √ nF n ( ∞ , y n )]( ¯ F Jn ( x ) − ¯ F Jn ( a ))+ Z | t |≤ J,t ∈ R G n,J ( t ) e − it y n h ( λ n ( t ) − ∂ ∂t λ n ( t ))1 | t |≤ π √ n − λ ( t ) + ∂ ∂t λ ( t ) i d t , where(46) G n,J ( t ) = G n,J ( t , x, a ) := ˆ v J ( t )(2 π ) · e − it a − e − it x it . Step 3. Our next goal is to bound (45) by Cn − δ/ . Set U ( t ) := ( λ n − λ ) − ∂ ∂t ( λ n − λ ) . Note that by (45) and Proposition 4.1, we have for x > a ,(1 + y n ) ψ ( ∞ , y n ) (cid:12)(cid:12)(cid:12) ∆ Jn ( x ) − ∆ Jn ( a ) (cid:12)(cid:12)(cid:12) ≤ Cn − δ/ + (cid:12)(cid:12)(cid:12) Z | t |≤ J, | t | >π √ n G n,J ( t ) e − it y n ( λ − ∂ ∂t λ )d t (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Z | t |≤ J, | t |≤ π √ n G n,J ( t ) e − it y n U ( t )d t (cid:12)(cid:12)(cid:12) =: Cn − δ/ + I + I . (47) We start with I . Recall J = ε √ n and for any K > G n ( K ) denote the set of “good”functions f : R → C such thatsup x,y,a (cid:12)(cid:12)(cid:12) Z | t |≤ J, | t |≤ π √ n G n,J ( t , x, a ) e − it y n f ( t )d t (cid:12)(cid:12)(cid:12) ≤ Kn − δ/ . We will show that(48) U ( t ) ∈ G n ( C ) . Notice that every f : R → C that satisfies | f ( t ) | ≤ Cn − δ/ | t | e − c | t | for | t | ≤ ε √ n, | t | ≤ π √ n is in G n ( C ). Set(49) R ( t ) := ϕ (0 , t √ n ) n − E [ W ( e i t · Z / √ n − e it W/ √ n )] . Then, letting c be the same as in Proposition 4.3(c), U ( t ) = e − c t ( U − U (0 , t ) + R ) + (1 − e − c t ) U + e − c t U (0 , t ) − e − c t R. We will show that U ∈ G n ( C ) by showing that all the four terms on the right aboveare in G n ( C ). Note that the constant C may differ for each of these four terms. When | t | ≤ ε √ n and | t | ≤ π √ n , by (33), | λ n ( t ) − λ ( t ) − [ λ n (0 , t ) − λ (0 , t )] | ≤ Cn − δ/ | t | e − ct . EGENERATIVE CLT RATES 19 This inequality and Proposition 4.3(c) yield e − c t | U − U (0 , t ) + R | ≤ Cn − δ/ | t | e − c | t | .Hence there exists a constant C such that e − c t ( U − U (0 , t ) + R ) ∈ G n ( C ). Also, using1 − e − c t ≤ Ct and Proposition 4.3(a)(b), we have (1 − e − c t ) U ( t ) ∈ G n ( C ) for someconstant C . Further, (cid:12)(cid:12)(cid:12) Z | t |≤ J, | t |≤ π √ n G n,J ( t ) e − it y n e − c t U (0 , t )d t (cid:12)(cid:12)(cid:12) (50) ≤ (cid:12)(cid:12)(cid:12) Z | t |≤ J G n,J ( t ) e − c t d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z | t |≤ π √ n U (0 , t ) e − it y n d t (cid:12)(cid:12)(cid:12) . By the inversion formula, for x > a , the first integral R | t |≤ J G n,J ( t ) e − c t d t = µ J ( a, x ) / π < / π , where µ J denotes the probability measure of v J ∗ Z c and Z c denotes the normaldistribution with mean 0 and variance 2 c . On the other hand, by Proposition 4.3(a)(b), wehave | U (0 , t ) | ≤ Cn − δ/ e − ct for | t | ≤ π √ n , which implies (cid:12)(cid:12)(cid:12) R | t |≤ π √ n U (0 , t ) e − it y n d t (cid:12)(cid:12)(cid:12) ≤ Cn − δ/ . Hence the integral in (50) is bounded by Cn − δ/ and so e − c t U (0 , t ) ∈ G n ( C )for some constant C .To prove U ( t ) ∈ G n ( C ) it remains to show that e − c t R ∈ G n ( C ) for some constant C . Indeed, by the fact that ˆ v J is supported on [ − J, J ] and Fubini’s theorem, (Recall thedefinition of G n,J at (46).) Z | t |≤ J, | t |≤ π √ n G n,J ( t ) e − it y n e − c t R ( t )d t (51) = E " W Z ∞−∞ G n,J ( t ) e − c t ( e it V/ √ n − t Z | t |≤ π √ n ϕ (0 , t √ n ) n − e − it y n e it W/ √ n d t . By the inversion formula for distribution functions, Z ∞−∞ G n,J ( t ) e − c t ( e it V/ √ n − t = C [ µ J ( x, x + V √ n ) − µ J ( a, a + V √ n )]1 V ≥ + C [ µ J ( a + V √ n , a ) − µ J ( x + V √ n , x )]1 V < . Since µ J has (by the inversion formula) bounded density, for any x ∈ R , µ J ( x, x + V √ n )1 V ≥ + µ J ( x + V √ n , x )1 V < ≤ C | V √ n | ∧ ≤ C | V √ n | δ . Also, by (31), the second integral on the right side of (51) is bounded in absolute value by R t ∈ R | ϕ (0 , t √ n ) | n − d t < C . Then, by (51) we have (cid:12)(cid:12)(cid:12) Z | t |≤ J, | t |≤ π √ n G n,J ( t ) e − it y n e − c t R ( t )d t (cid:12)(cid:12)(cid:12) ≤ CE [ W | V √ n | δ ] ≤ Cn − δ/ . So e − c t R ∈ G n ( C ) for some constant C > I ≤ Cn − δ/ . Step 4. To estimate I in (47), recall that by (29), | ∂ ∂t λ ( t ) − ( σ t − σ ) λ ( t ) | ≤ C | t || t | λ ( t ) ≤ C | t | e − c | t | . Thus (cid:12)(cid:12)(cid:12) Z | t |≤ J, | t | >π √ n G n,J ( t ) e − it y n [ ∂ ∂t λ ( t ) − ( σ t − σ ) λ ( t )]d t (cid:12)(cid:12)(cid:12) ≤ C Z | t |≤ J, | t | >π √ n | t | | t | e − c | t | d t ≤ Ce − cn . On the other hand, recalling that N = ( N , N ) is the limiting normal distribution, wehave λ ( t ) = E [ e it N + it N ]. By Fubini’s theorem, Z | t |≤ J, | t | >π √ n G n,J ( t ) e − it y n ( σ t − σ − λ d t = E "Z | t | >π √ n e it ( N − y n ) ( σ t − σ − t Z | t |≤ J e − it ( a −N ) − e − it ( x −N ) (2 π ) it ˆ v J ( t )d t . Note that by Fourier’s inversion formula (and the fact that ˆ v J is supported on [ − J, J ]), f ( N ) := 12 π Z | t |≤ J e − it ( a −N ) − e − it ( x −N ) it ˆ v J ( t )d t = v J ( a − N , x − N ) . Thus | f | ≤ 1. Also note that conditioning on N , the variable N has a normal distributionwith mean σ N /σ and variance σ − σ σ . Hence (cid:12)(cid:12)(cid:12) Z | t |≤ J, | t | >π √ n G n,J ( t ) e − it y n ( σ t − σ − λ d t (cid:12)(cid:12)(cid:12) = 12 π (cid:12)(cid:12)(cid:12) E "Z | t | >π √ n e it ( N − y n ) ( σ t − σ − f ( N )d t = 12 π (cid:12)(cid:12)(cid:12) Z | t | >π √ n ( σ t − σ − E h exp (cid:16) i ( σ σ N − y n ) t − ( σ − σ σ ) t (cid:17) f ( N ) i d t (cid:12)(cid:12)(cid:12) ≤ Z | t | >π √ n Ce − ct d t ≤ Ce − cn . Therefore, I ≤ Ce − cn . Step 5. Finally, plugging the bounds I ≤ Ce − cn and I ≤ Cn − δ/ into (47) we obtainsup x ∈ R ,y ∈ Z (1 + y n ) ψ ( ∞ , y n ) (cid:12)(cid:12)(cid:12) ∆ Jn ( x ) − ∆ Jn ( a ) (cid:12)(cid:12)(cid:12) ≤ Cn − δ/ . Since the right hand side is uniform for all a , we simply havesup x ∈ R ,y ∈ Z (1 + y n ) ψ ( ∞ , y n ) | ∆ Jn ( x ) | ≤ Cn − δ/ . This, together with (39), yieldssup x ∈ R ,y ∈ Z (1 + y n ) (cid:12)(cid:12)(cid:12) √ nF n ( x, y n ) − ψ ( x, y n ) (cid:12)(cid:12)(cid:12) ≤ Cn − δ/ . Our proof of Theorem 4.2 is complete. (cid:3) Proof of the Regenerative CLT rates In this section we will use the semi-local Berry Esseen estimates from Theorem 4.2 in the previoussection to give the proof of our main result (Theorem 1.1). To more easily adapt to the i.i.d.setting of Theorem 4.2, we first prove the statement of Theorem 1.1 under the measure P (that is,conditioned on a regeneration at time zero). Then, at the end of the section we show how to obtainthe same results taking into account that the process is different prior to the first regeneration time. EGENERATIVE CLT RATES 21 Proof of Theorem 1.1 under the measure P . In this subsection, our aim is to prove thefollowing Proposition which is the analog of Theorem 1.1 under the measure P . Proposition 5.1. Let X n = P ni =1 ξ i be a regenerative process with regeneration times { τ k } k ≥ .Assume for some δ ∈ (0 , that E [ τ δ ] < ∞ and E τ X i =1 | ξ i | ! δ < ∞ . Then, lim sup n →∞ n δ/ sup x ∈ R (cid:12)(cid:12)(cid:12)(cid:12) P (cid:18) X n − µnσ √ n ≤ x (cid:19) − Φ( x ) (cid:12)(cid:12)(cid:12)(cid:12) < ∞ , where µ and σ are defined as in (2) and (3) , respectively.Proof. For notational convenience, in the proof below we will let ¯ X n = X n − nµ . The strategy ofthe proof of Proposition 5.1 will be to condition on the time and value of the regenerative process atthe last regeneration time prior to time n . To this end, let k ( n ) ≥ n ; that is, τ k ( n ) ≤ n < τ k ( n )+1 . By decomposing according to thevalues of k ( n ), n − τ k ( n ) and X n − X τ k ( n ) , we can write P (cid:18) ¯ X n σ √ n ≤ x (cid:19) = n X k =0 n X m =0 Z P (cid:18) ¯ X n σ √ n ≤ x, k ( n ) = k, τ k = n − m, X n − X τ k ∈ du (cid:19) . Using the structure provided by the regeneration times, for any fixed k, m , and u we can re-writethe probability inside the sums and integral on the right as P (cid:18) ¯ X n σ √ n ≤ x, k ( n ) = k, τ k = n − m, X n − X τ k ∈ du (cid:19) = P (cid:0) X τ k − τ k µ ≤ xσ √ n − u + ( n − τ k ) µ, τ k = n − m, τ k +1 > n, X n − X τ k ∈ du (cid:1) = P (cid:18) ¯ X τ k √ k ≤ xσ √ n − u + mµ √ k , τ k = n − m (cid:19) P ( τ > m, X m ∈ du ) , and therefore, P (cid:18) ¯ X n σ √ n ≤ x (cid:19) = n X k =1 ⌊√ n ⌋ X m =0 Z √ n −√ n P (cid:18) ¯ X τ k √ k ≤ xσ √ n − u + mµ √ k , τ k = n − m (cid:19) P ( τ > m, X m ∈ du )+ P (cid:16) { ¯ X n ≤ xσ √ n } ∩ n n − τ k ( n ) > √ n, or | X n − X τ k ( n ) | > √ n o(cid:17) . Note that in the above we could have included the terms m > √ n and | u | > √ n in the first term onthe right and omitted the second term. However, the main contribution will come from m, | u | ≤ √ n and thus to simplify later parts of the proof we choose to handle the cases where n − τ k ( n ) > √ n or | X n − X τ k ( n ) | > √ n separately. Note also that we have ommited k = 0 from the first sum sincethis is included in the last term since τ = 0. To use this decomposition to compare with Φ( x ), note first of all that letting ¯ τ = E [ τ ] we canwrite Φ( x ) = Φ( x )¯ τ ∞ X m =0 P ( τ > m ) = Φ( x )¯ τ ∞ X m =0 Z P ( τ > m, X m ∈ dy )= Φ( x )¯ τ ⌊√ n ⌋ X m =0 Z √ n −√ n P ( τ > m, X m ∈ dy ) + Φ( x )¯ τ ⌊√ n ⌋ X m =0 P ( τ > m, | X m | > √ n )+ Φ( x )¯ τ X m> √ n P ( τ > m ) . Therefore, we can conclude that P (cid:18) ¯ X n σ √ n ≤ x (cid:19) − Φ( x )= ⌊√ n ⌋ X m =0 Z √ n −√ n ( n X k =1 P (cid:16) ¯ X τk √ k ≤ xσ √ n − y + mµ √ k , τ k = n − m (cid:17) − Φ( x )¯ τ ) P ( τ > m, X m ∈ dy )(52) + P (cid:16) { ¯ X n ≤ xσ √ nz } ∩ n n − τ k ( n ) > √ n, or | X n − X τ k ( n ) | > √ n o(cid:17) (53) − Φ( x )¯ τ ⌊√ n ⌋ X m =0 P ( τ > m, | X m | > √ n ) − Φ( x )¯ τ X m> √ n P ( τ > m ) . (54)To control the terms in (53), note that the moment assumptions in the statement of the theoremimply that (53) ≤ n P ( τ > √ n ) + n P τ X i =1 | ξ i | > √ n ! = O ( n − δ/ ) . Similarly, the terms in (54) can be bounded by(54) ≤ √ n ¯ τ P τ X i =1 | ξ i | > √ n ! + E [ τ δ ]¯ τ X m> √ n ( m + 1) − − δ = O ( n − (1+ δ ) / ) . Therefore, it remains only to show that the term in (52) is also O ( n − δ/ ), uniformly in x . To thisend, let ψ A ( x, y ) be defined as in (18), where A = (cid:18) E [( X τ − τ µ ) ] E [( X τ − τ µ )( τ − ¯ τ )] E [( X τ − τ µ )( τ − ¯ τ )] E [( τ − ¯ τ ) ] (cid:19) is the covariance matrix of ( X τ − τ µ, τ ) under the measure P . For convenience of notation, let(55) α = E (cid:2) ( X τ − τ µ ) (cid:3) be the top left entry of the covariance matrix A . If N = ( N , N ) is a centered Gaussian withcovariance matrix A , then it follows that N α is a standard Normal random variable and thus Z R ψ A ( αx, y ) dy = P ( N ≤ αx ) = Φ( x ) . Using this notation, the necessary bounds on (52) which complete the proof of Proposition 5.1 areobtained by a series of approximations given by the following three lemmas. Note that in these EGENERATIVE CLT RATES 23 lemmas and below we will use the following notation.(56) y k,n,m = n − m − k ¯ τ √ k . Lemma 5.2. There exists a constant C < ∞ such that for n large enough, ⌊√ n ⌋ X m =0 Z √ n −√ n n X k =1 (cid:12)(cid:12)(cid:12)(cid:12) P (cid:16) ¯ X τk √ k ≤ xσ √ n − u + mµ √ k , τ k = n − m (cid:17) − √ k ψ A (cid:16) xσ √ n − u + mµ √ k , y k,n,m (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) P ( τ > m, X m ∈ du ) ≤ Cn δ/ , for all x ∈ R . Lemma 5.3. There exists a constant C < ∞ such that for n large enough, ⌊√ n ⌋ X m =0 n X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z √ n −√ n √ k ψ A (cid:16) xσ √ n − u + mµ √ k , y k,n,m (cid:17) P ( τ > m, X m ∈ du ) − √ k ψ A ( αx, y k,n,m ) P (cid:0) τ > m, | X m | ≤ √ n (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C √ n , for all x ∈ R . Lemma 5.4. There exists a constant C < ∞ such that for n large enough, ⌊√ n ⌋ X m =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 √ k ψ A ( αx, y k,n,m ) − τ Z R ψ A ( αx, y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P (cid:0) τ > m, | X m | ≤ √ n (cid:1) ≤ C √ n , for all x ∈ R .Proof of Lemma 5.2. It follows from Theorem 4.2 that the sum in the statement of the Lemma isbounded by(57) ⌊√ n ⌋ X m =0 Z √ n −√ n n X k =1 Ck (1+ δ ) / (cid:18) n − m − k ¯ τ ) k (cid:19) − P ( τ > m, X m ∈ du ) . (A direct application of Theorem 4.2 requires that the random variable τ has span 1 under thelaw P , but clearly Theorem 4.2 can be generalized to any lattice random variable W .) Note thatfor m ≤ √ n we can bound1 k (1+ δ ) / (cid:18) n − m − k ¯ τ ) k (cid:19) − ≤ k (1 − δ ) / ( n −√ n − k ¯ τ ) if 1 ≤ k < n − √ n ¯ τ k (1+ δ ) / if | n − k ¯ τ | ≤ √ n k (1 − δ ) / ( n − k ¯ τ ) if n +2 √ n ¯ τ < k ≤ n, and from this it follows easily (using integrals to bound the appropriate sums) that n X k =1 k (1+ δ ) / (cid:18) n − m − k ¯ τ ) k (cid:19) − ≤ Cn δ/ , for some C < ∞ . Therefore, we obtain that(57) ≤ ⌊√ n ⌋ X m =0 Z √ n −√ n Cn δ/ P ( τ > m, X m ∈ dy ) ≤ Cn δ/ ⌊√ n ⌋ X m =0 P ( τ > m ) ≤ C E [ τ ] n δ/ . (cid:3) Before giving the proofs of Lemmas 5.3 and 5.4, we first state the following facts which wereused in the proofs of the corresponding statements in [Bol80]. Lemma 5.5. Let y k,n,m = n − m − k ¯ τ √ k . For any constant c > , there exists a constant C < ∞ depending only on c and ¯ τ such that (58) sup m ≤√ n n X k =1 k e − cy k,n,m ≤ C √ n and (59) sup m ≤√ n n X k =1 √ k (cid:12)(cid:12)(cid:12)(cid:12)r nk ¯ τ − (cid:12)(cid:12)(cid:12)(cid:12) e − cy k,n,m ≤ C √ n . Remark . We refer the reader to pages 69-70 in [Bol80] for the proofs of (58) and (59). Proof of Lemma 5.3. Let I ( k, m, x, y ) denote the interval between αx and xσ √ n − y + mµ √ k , and recallthat γ A ( x, y ) is the p.d.f. of a centered two dimensional Gaussian with covariance matrix A . Then, (cid:12)(cid:12)(cid:12) ψ A (cid:16) xσ √ n − y + mµ √ k , y k,n,m (cid:17) − ψ A ( αx, y k,n,m ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z I ( k,m,x,y ) γ A ( z, y k,n,m ) dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) | x | (cid:12)(cid:12)(cid:12)(cid:12) σ √ n √ k − α (cid:12)(cid:12)(cid:12)(cid:12) + | y − mµ |√ k (cid:19) sup z ∈ I ( k,m,x,y ) γ A ( z, y k,n,m )Next, note that there exist constants c , c > A such that(60) γ A ( x, y ) ≤ c e − c ( x + y ) . Therefore,(Left side of Lemma 5.3) ≤ √ n X m =0 n X k =1 c k e − c y k,n,m Z √ n −√ n | y − mµ | P ( τ > m, X m ∈ dy )(61) + √ n X m =0 n X k =1 c | x |√ k (cid:12)(cid:12)(cid:12)(cid:12) σ √ n √ k − α (cid:12)(cid:12)(cid:12)(cid:12) e − c y k,n,m Z √ n −√ n sup z ∈ I ( k,m,x,y ) e − c z P ( τ > m, X m ∈ dy )(62)To control (61), note that the integral inside the sums is zero if m = 0 whereas for m ≥ Z √ n −√ n | y − mµ | P ( τ > m, X m ∈ dy ) = E (cid:2) | X m − mµ | { τ >m } (cid:3) ≤ E " τ X i =1 | ξ i − µ | { τ >m } ≤ C P ( τ > m ) δ δ ≤ C ′ m δ , using the moment assumptions in the statement of Proposition 5.1 together with H¨older’s inequalityand Chebychev’s inequality in the last two inequalities, respectively. From this and (58), we obtain EGENERATIVE CLT RATES 25 that (61) ≤ √ n X m =1 Cm δ n X k =1 k e − c y k,n,m ≤ C ′ √ n √ n X m =1 m δ = C ′′ √ n . To control (62), we claim that(63) sup z ∈ I ( k,m,x,y ) | x | e − c z ≤ C. To see this, first note that since m, | y | ≤ √ n and k ≤ √ n it follows that (cid:12)(cid:12)(cid:12)(cid:12) xσ √ n − y + mµ √ k (cid:12)(cid:12)(cid:12)(cid:12) ≥ | x | σ √ n − (1 + µ ) √ n √ k ≥ | x | σ − (1 + µ ) . If | x | > µ ) σ then the right side can be bounded below by | x | σ/ | z | > min { α, σ/ }| x | for all z ∈ I ( k, m, x, y ). Therefore,sup z ∈ I ( k,m,x,y ) e − c z ≤ ( | x | ≤ µ ) σ e − c min { α,σ/ }| x | if | x | > µ ) σ , and from this the claim in (63) follows. Using (63) and then (59) we then have that(62) ≤ C √ n X m =0 n X k =1 √ k (cid:12)(cid:12)(cid:12)(cid:12) σ √ n √ k − α (cid:12)(cid:12)(cid:12)(cid:12) e − c y k,n,m ! P ( τ > m ) ≤ C ′ √ n √ n X m =0 P ( τ > m ) ≤ C ′ ¯ τ √ n . (Note that in the application of (59) we are using that σ ¯ τ = α which follows from the definitionsof σ and α in (3) and (55), respectively.) (cid:3) Proof of Lemma 5.4. In the proof of this Lemma, to make the notation less burdensome, in a slightabuse of notation we will write y k for y k,n,m as defined in (56). To begin, note for any fixed n ≥ m that y > y > · · · > y n . Since for n large enough and m ≤ √ n we have y = n − m − ¯ τ ≥ n/ 2, if N = ( N , N ) is a centered Gaussian random variable with Covariance matrix A , then ⌊√ n ⌋ X m =0 τ (cid:18)Z ∞ y ψ A ( αx, y ) dy (cid:19) P ( τ > m, | X m | ≤ √ n ) ≤ ⌊√ n ⌋ X m =0 τ P ( N ≥ n/ P ( τ > m ) ≤ P ( N ≥ n/ 2) = o ( n − / ) . Similarly, since y n ≤ − (¯ τ − √ n and ¯ τ = E [ τ ] > ⌊√ n ⌋ X m =0 τ (cid:18)Z y n −∞ ψ A ( αx, y ) dy (cid:19) P ( τ > m, | X m | ≤ √ n ) ≤ P ( N ≤ − (¯ τ − √ n ) = o ( n − / ) . Therefore, to finish the proof of Lemma 5.4 it is enough to show that(64) ⌊√ n ⌋ X m =0 n − X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ k ψ A ( αx, y k ) − τ Z y k y k +1 ψ A ( αx, y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ( τ > m, | X m | ≤ √ n ) = O ( n − / ) . To prove (64), first note that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ k ψ A ( αx, y k ) − τ Z y k y k +1 ψ A ( αx, y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) √ k − y k − y k +1 ¯ τ (cid:12)(cid:12)(cid:12)(cid:12) ψ A ( αx, y k ) + 1¯ τ Z y k y k +1 | ψ A ( αx, y ) − ψ A ( αx, y k ) | dy. (65)To control the first term in (65), the definition of y k implies that y k = ¯ τ √ k + y k +1 q k +1 k , or equiva-lently,(66) y k − y k +1 = ¯ τ √ k + y k +1 r k − ! . Since q k − ≤ k we can conclude from this that(67) (cid:12)(cid:12)(cid:12)(cid:12) √ k − y k − y k +1 ¯ τ (cid:12)(cid:12)(cid:12)(cid:12) ψ A ( αx, y k ) = y k +1 ¯ τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)r k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ A ( αx, y k ) ≤ Ck y k +1 e − cy k , where in the last inequality we used that the bounds on γ A in (60) imply that ψ A ( z, y ) ≤ Ce − cy .To control the second term in (65), note that for y ∈ [ y k +1 , y k ],(68) | ψ A ( αx, y ) − ψ A ( αx, y k ) | ≤ C | y k − y k +1 | sup y ∈ [ y k +1 ,y k ] e − cy To further simplify the supremum on the right, note that for any y ∈ [ y k +1 , y k ] y k +1 ≤ y + 2( y − y k +1 ) ≤ y + 2( y k − y k +1 ) ≤ y + 4¯ τ k + y k +1 k , where we used (66) in the last inequality. For k ≥ y ∈ [ y k +1 ,y k ] y ≥ y k +1 − ¯ τ ,and this is also trivially true for k = 1 since 0 < y < y so that we can conclude(69) sup y ∈ [ y k +1 ,y k ] e − cy ≤ Ce − c y k +1 . Using (68), (69) and then (66) we can bound the second term in (65) by1¯ τ Z y k y k +1 | ψ A ( αx, y ) − ψ A ( αx, y k ) | dy ≤ C | y k − y k +1 | e − cy k +1 ≤ C ′ k + y k +1 k ! e − cy k +1 ≤ C ′′ k e − c ′ y k +1 . Combining this with (67) and (65) we obtain that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ k ψ A ( αx, y k ) − τ Z y k y k +1 ψ A ( αx, y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Ck y k +1 e − cy k + Ck e − cy k +1 ≤ C ′ k e − c ′ y k +1 EGENERATIVE CLT RATES 27 and thus, ⌊√ n ⌋ X m =0 n − X k =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ k ψ A ( αx, y k ) − τ Z y k y k +1 ψ A ( αx, y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ( τ > m, | X m | ≤ √ n ) ≤ ⌊√ n ⌋ X m =0 n X k =1 C ′ k e − c ′ y k +1 P ( τ > m, | X m | ≤ √ n ) ≤ C ′′ √ n ⌊√ n ⌋ X m =0 P ( τ > m ) ≤ C ′′ ¯ τ √ n , where we used (58) in the second to last inequality. (cid:3)(cid:3) Accounting for the first regeneration interval. In this subsection, we will show how toaccount for the difference of the first regeneration interval to improve Proposition 5.1 to a proof ofTheorem 1.1. Proof of Theorem 1.1. By conditioning on the values of τ and X τ we obtain that P (cid:18) X n − nµσ √ n ≤ t (cid:19) = ⌊√ n ⌋ X m =1 Z √ n −√ n P ( X τ ∈ dz, τ = m ) P ( X n − m − ( n − m ) µ ≤ σt √ n − z + mµ )+ P (cid:18) X n − nµσ √ n ≤ t, and max {| X τ | , τ } > √ n (cid:19) Since Φ( t ) = ⌊√ n ⌋ X m =1 Z P ( X τ ∈ dz, τ = m )Φ( t ) + P (cid:0) max {| X τ | , τ } > √ n (cid:1) Φ( t ) , by comparing like terms we obtain (cid:12)(cid:12)(cid:12)(cid:12) P (cid:18) X n − nµσ √ n ≤ t (cid:19) − Φ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ⌊√ n ⌋ X m =1 Z √ n −√ n P ( X τ ∈ dz, τ = m ) (cid:12)(cid:12) P ( X n − m − ( n − m ) µ ≤ σt √ n − z + mµ ) − Φ( t ) (cid:12)(cid:12) + 2 P ( τ > √ n ) + 2 P ( | X τ | > √ n ) ≤ ⌊√ n ⌋ X m =1 Z √ n −√ n P ( X τ ∈ dz, τ = m ) (cid:26) sup s (cid:12)(cid:12)(cid:12)(cid:12) P (cid:18) ¯ X n − m σ √ n − m ≤ s (cid:19) − Φ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) + ⌊√ n ⌋ X m =1 Z √ n −√ n P ( X τ ∈ dz, τ = m ) (cid:12)(cid:12)(cid:12) Φ (cid:16) t q nn − m − z − mµσ √ n − m (cid:17) − Φ( t ) (cid:12)(cid:12)(cid:12) + 2( E [ τ δ ] + E [ | X τ | δ ]) n δ/ . For n large enough and m ≤ √ n we have from Proposition 5.1 that the supremum in braces on theright is bounded by C/ √ n for n large enough. Therefore, we need only to show that(70) lim sup n →∞ n δ/ sup t ∈ R ⌊√ n ⌋ X m =1 Z √ n −√ n P ( X τ ∈ dz, τ = m ) (cid:12)(cid:12)(cid:12) Φ (cid:16) t q nn − m − z − mµσ √ n − m (cid:17) − Φ( t ) (cid:12)(cid:12)(cid:12) < ∞ . To prove (70), it is easy to show (see [Pet75, Section V.3, equations (3.3),(3.4)]) that for any a > b, t ∈ R that | Φ( at + b ) − Φ( t ) | ≤ | Φ( at + b ) − Φ( at ) | + | Φ( at ) − Φ( t ) | ≤ √ π | b | + 1 √ πe ( a − . Therefore, sup t ∈ R ⌊√ n ⌋ X m =1 Z √ n −√ n P ( X τ ∈ dz, τ = m ) (cid:12)(cid:12)(cid:12) Φ (cid:16) t q nn − m − z − mµσ √ n − m (cid:17) − Φ( t ) (cid:12)(cid:12)(cid:12) ≤ C ⌊√ n ⌋ X m =1 Z √ n −√ n P ( X τ ∈ dz, τ = m ) n | z − mµ |√ n − m + q nn − m − o ≤ C p n − √ n E h | X τ − µτ | {| X τ |≤√ n,τ ≤√ n } i + C (cid:16)q nn −√ n − (cid:17) . (71)Finally, using the moment assumptions regarding the first regeneration time we have that(72) E h | X τ − µτ | {| X τ |≤√ n,τ ≤√ n } i ≤ (1 + µ ) − δ n (1 − δ ) / E h | X τ − µτ | δ i . Applying (72) to (71), we see that (70) follows easily. (cid:3) Discussions: rates of convergence of quenched and annealed CLT of RWRE The results in Section 3 give rates of convergence for annealed CLTs of RWRE. However, undercertain assumptions it is known that CLTs hold under the quenched measures as well. Below we willreview some recent results on the corresponding quenched rates of convergence for one-dimensionalRWRE. We will then close the paper with a few related open questions.6.1. One-dimensional quenched CLTs. Recall from (17) that one-dimensional RWREs withparameter κ > P ω (for P -a.e. environment ω ), but that the centeringand scaling needs to be somewhat different than in the annealed CLTs [Ali99, Gol07, Pet08]. Inparticular,(73) lim n →∞ sup x (cid:12)(cid:12)(cid:12)(cid:12) P ω (cid:18) T n − E ω [ T n ] σ √ n ≤ x (cid:19) − Φ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 , P -a.s. , where σ = E [Var ω ( T )] , and lim n →∞ sup x (cid:12)(cid:12)(cid:12)(cid:12) P ω (cid:18) X n − n v + Z n ( ω )v / σ √ n ≤ x (cid:19) − Φ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 , P -a.s. , where Z n ( ω ) = v (cid:0) E ω [ T ⌊ n v ⌋ ] − E [ T ⌊ n v ⌋ ] (cid:1) . Recent results of Ahn and Peterson [AP17] gave upper bounds for the rates of convergence ofthese quenched CLTs. While the quenched CLT for the hitting times stated in (73) had a quenchedcentering and a deterministic scaling, the results in [AP17] show that improved rates of convergencecan be obtained for the hitting times by using a quenched scaling as well. EGENERATIVE CLT RATES 29 Theorem 6.1 (Ahn and Peterson [AP17]) . Let F n,ω ( x ) = P ω (cid:18) T n − E ω [ T n ] σ √ n ≤ x (cid:19) and F n,ω ( x ) = P ω T n − E ω [ T n ] p Var ω ( T n ) ≤ x ! be the centered quenched distribution functions of T n with deterministic and quenched scalings,respectively.(1) Rates of convergence with deterministic scaling:(a) If κ > , then for any ε > , lim n →∞ n − ε k F n,ω − Φ k ∞ = 0 , P -a.s.(b) If κ ∈ (2 , , then for any ε > , lim n →∞ n − κ − ε k F n,ω − Φ k ∞ = 0 , P -a.s.(2) Rates of convergence with quenched scaling.(a) If κ > , then there exists a constant C < ∞ such that lim sup n →∞ √ n k F n,ω − Φ k ∞ ≤ C, P -a.s.(b) If κ ∈ (2 , then for any ε > , lim n →∞ n − κ − ε k F n,ω − Φ k ∞ = 0 , P -a.s. The corresponding results for the quenched CLT of the position of the walk are somewhat weakerbut don’t require a quenched scaling. Theorem 6.2 (Ahn and Peterson [AP17]) . Let G n,ω ( x ) = P ω (cid:16) X n − n v+ Z n ( ω )v / σ √ n ≤ x (cid:17) be the rescaledquenched distribution function of X n . If κ > , then for any ε > n →∞ n − κ − ε k G n,ω − Φ k ∞ = 0 , P -a.s.Moreover, by relaxing the convergence to that of in probability one obtains the following faster ratesof convergence.(1) If κ ∈ (2 , ) , then for any ε > , (74) lim sup n →∞ n − κ − ε k G n,ω − Φ k ∞ = 0 , in P -probability.(2) If κ > then for any ε > , lim sup n →∞ n − ε k G n,ω − Φ k ∞ = 0 , in P -probability. Remaining questions for quenched and annealed rates of convergence. (1) The rates of convergence of the annealed CLTs in Corollary 3.3 are clearly optimal when κ > 3. However, since − κ > κ − κ ∈ (2 , x ∈ R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P T n − E ω [ T n ] p V ar ω ( T n ) ≤ x ! − Φ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o ( n − κ +1 ) , when κ ∈ (2 , (2) For multidimensional RWRE, under strong enough moment conditions on the regenerationtimes it is known that a quenched CLT holds [BZ08, RAS09]. Moreover, in contrast to theone-dimensional case, the quenched CLT holds with the same (deterministic) centering andscaling as the annealed CLT. Can one prove rates of convergence for the quenched CLT inthese cases? Also, can the rate of convergence be improved by instead using a quenchedcentering and/or scaling instead of the deterministic one? Answering these questions willlikely require techniques very different from this paper since the intervals of the walk betweenregeneration times are no longer i.i.d. under the quenched measure.(3) There are certain multidimensional RWRE which are not directionally transient but forwhich a CLT holds; for instance RWRE in balanced random environments [Law83, GZ12,BD14] or environments in which certain projections of the walk are a simple symmetricrandom walk [BSZ03]. Since these walks are not directionally transient, the regenerationtimes do not even exist. Can one use other techniques to obtain rates of convergence forthe quenched or annealed CLTs of these RWRE? References [Ali99] S. Alili. Asymptotic behaviour for random walks in random environments. J. Appl. Probab. , 36(2):334–349,1999.[AP17] Sung Won Ahn and Jonathon Peterson. Quenched central limit theorem rates of convergence for one-dimensional random walks in random environments, April 2017.[BCG11] S. G. Bobkov, G. P. Chistyakov, and F. G¨otze. Non-uniform bounds in local limit theorems in case offractional moments. I. Math. Methods Statist. , 20(3):171–191, 2011.[BD14] Noam Berger and Jean-Dominique Deuschel. A quenched invariance principle for non-elliptic random walkin i.i.d. balanced random environment. Probab. Theory Related Fields , 158(1-2):91–126, 2014.[BDR14] Noam Berger, Alexander Drewitz, and Alejandro F. Ram´ırez. Effective Polynomial Ballisticity Condi-tions for Random Walk in Random Environment. Communications on Pure and Applied Mathematics ,67(12):1947–1973, 2014.[Ber41] Andrew C. Berry. The accuracy of the Gaussian approximation to the sum of independent variates. Trans.Amer. Math. Soc. , 49:122–136, 1941.[Bol80] E. Bolthausen. The Berry-Esseen theorem for functionals of discrete Markov chains. Z. Wahrsch. Verw.Gebiete , 54(1):59–73, 1980.[Bol82] E. Bolthausen. The Berry-Esse´en theorem for strongly mixing Harris recurrent Markov chains. Z. Wahrsch.Verw. Gebiete , 60(3):283–289, 1982.[BR07] Jean Berard and Alejandro Ramirez. Central limit theorem for the excited random walk in dimension d ≥ Electron. Commun. Probab. , 12:no. 30, 303–314, 2007.[BRS16] ´Elodie Bouchet, Alejandro F. Ram´ı rez, and Christophe Sabot. Sharp ellipticity conditions for ballisticbehavior of random walks in random environment. Bernoulli , 22(2):969–994, 2016.[BSZ03] Erwin Bolthausen, Alain-Sol Sznitman, and Ofer Zeitouni. Cut points and diffusive random walks in randomenvironment. Annales de l’institut Henri Poincar´e (B) Probabilit´es et Statistiques , 39(3):527–555, 2003.[Bv11] Martin T. Barlow and Jiˇr´ı ˇCern´y. Convergence to fractional kinetics for random walks associated withunbounded conductances. Probab. Theory Related Fields , 149(3-4):639–673, 2011.[BZ08] Noam Berger and Ofer Zeitouni. A quenched invariance principle for certain ballistic random walks in i.i.d.environments. In In and out of equilibrium. 2 , volume 60 of Progr. Probab. , pages 137–160. Birkh¨auser,Basel, 2008.[Chu67] Kai Lai Chung. Markov chains with stationary transition probabilities . Second edition. Die Grundlehren dermathematischen Wissenschaften, Band 104. Springer-Verlag New York, Inc., New York, 1967.[DPZ96] Amir Dembo, Yuval Peres, and Ofer Zeitouni. Tail estimates for one-dimensional random walk in randomenvironment. Comm. Math. Phys. , 181(3):667–683, 1996.[Ess42] Carl-Gustav Esseen. On the Liapounoff limit of error in the theory of probability. Ark. Mat. Astr. Fys. ,28A(9):19, 1942.[Fel71] William Feller. An introduction to probability theory and its applications. Vol. II . Second edition. JohnWiley & Sons, Inc., New York-London-Sydney, 1971.[FK16] Alexander Fribergh and Daniel Kious. Local trapping for elliptic random walks in random environments in Z d . Probab. Theory Related Fields , 165(3-4):795–834, 2016. EGENERATIVE CLT RATES 31 [Gol07] Ilya Ya. Goldsheid. Simple transient random walks in one-dimensional random environment: the centrallimit theorem. Probab. Theory Related Fields , 139(1-2):41–64, 2007.[GZ12] Xiaoqin Guo and Ofer Zeitouni. Quenched invariance principle for random walks in balanced randomenvironment. Probab. Theory Related Fields , 152(1-2):207–230, 2012.[Hip85] Christian Hipp. Asymptotic expansions in the central limit theorem for compound and Markov processes. Z. Wahrsch. Verw. Gebiete , 69(3):361–385, 1985.[IL71] I. A. Ibragimov and Yu. V. Linnik. Independent and stationary sequences of random variables . Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V.Petrov, Translation from the Russian edited by J. F. C. Kingman.[Jon04] Galin L. Jones. On the Markov chain central limit theorem. Probab. Surv. , 1:299–320, 2004.[KKS75] H. Kesten, M. V. Kozlov, and F. Spitzer. A limit law for random walk in a random environment. CompositioMath. , 30:145–168, 1975.[KZ08] Elena Kosygina and Martin P. W. Zerner. Positively and negatively excited random walks on integers, withbranching processes. Electron. J. Probab. , 13:no. 64, 1952–1979, 2008.[Law83] Gregory F. Lawler. Weak convergence of a random walk in a random environment. Comm. Math. Phys. ,87(1):81–87, 1982/83.[Mal93] V. K. Malinovski˘ı. Limit theorems for stopped random sequences. I. Estimates for the rate of convergenceand asymptotic expansions. Teor. Veroyatnost. i Primenen. , 38(4):800–826, 1993.[Mou12] Jean-Christophe Mourrat. A quantitative central limit theorem for the random walk among random con-ductances. Electron. J. Probab. , 17:no. 97, 17, 2012.[MT09] Sean Meyn and Richard L. Tweedie. Markov chains and stochastic stability . Cambridge University Press,Cambridge, second edition, 2009. With a prologue by Peter W. Glynn.[Num78] E. Nummelin. A splitting technique for Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete ,43(4):309–318, 1978.[Pet75] V. V. Petrov. Sums of independent random variables . Springer-Verlag, New York-Heidelberg, 1975. Trans-lated from the Russian by A. A. Brown, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82.[Pet08] Jonathon Peterson. Limiting distributions and large deviations for random walks in random environments .PhD thesis, University of Minnesota, 2008. Available at http://arxiv.org/abs/0810.0257 .[RAS09] Firas Rassoul-Agha and Timo Sepp¨al¨ainen. Almost sure functional central limit theorem for ballistic randomwalk in random environment. Ann. Inst. Henri Poincar´e Probab. Stat. , 45(2):373–420, 2009.[Ros71] Murray Rosenblatt. Markov processes. Structure and asymptotic behavior . Springer-Verlag, New York-Heidelberg, 1971. Die Grundlehren der mathematischen Wissenschaften, Band 184.[She17] I. G. Shevtsova. On convergence rate in the local limit theorem for densities under various moment condi-tions. J. Math. Sci. (N.Y.) , 221(4):588–608, 2017.[Sin83] Ya. G. Sinai. The limit behavior of a one-dimensional random walk in a random environment. TheoryProbab. Appl. , 27(2):256–268, 1983.[Sol75] Fred Solomon. Random walks in a random environment. Ann. Probability , 3:1–31, 1975.[SZ99] Alain-Sol Sznitman and Martin Zerner. A law of large numbers for random walks in random environment. Ann. Probab. , 27(4):1851–1869, 1999.[Szn00] Alain-Sol Sznitman. Slowdown estimates and central limit theorem for random walks in random environ-ment. J. Eur. Math. Soc. (JEMS) , 2(2):93–143, 2000.[Szn01] Alain-Sol Sznitman. On a class of transient random walks in random environment. Ann. Probab. , 29(2):724–765, 2001.[Szn02] Alain-Sol Sznitman. An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Related Fields , 122(4):509–544, 2002.[Zei04] Ofer Zeitouni. Random walks in random environment. In Lectures on probability theory and statistics ,volume 1837 of Lecture Notes in Math. , pages 189–312. Springer, Berlin, 2004. Xiaoqin Guo, University of Wisconsin, Madison, Department of Mathematics, 425 Van Vleck Hall,Madison, WI 53706, USA E-mail address : [email protected] URL : https://sites.google.com/site/guoxx097/ Jonathon Peterson, Purdue University, Department of Mathematics, 150 N University St, WestLafayette, IN 47907, USA E-mail address : [email protected] URL ::