Weak quenched limit theorems for a random walk in a sparse random environment

We study the quenched behaviour of a perturbed version of the simple symmetric random walk on the set of integers. The random walker moves symmetrically with an exception of some randomly chosen sites where we impose a random drift. We show that if the gaps between the marked sites are i.i.d. and regularly varying with a sufficiently small index, then there is no strong quenched limit laws for the position of the random walker. As a consequence we study the quenched limit laws in the context of weak convergence of random measures.


Introduction
One of the most classical and well-understood random processes is the simple symmetric random walk (SRW) on the set of integers, where the particle starting at zero every unit time moves with probability 1/2 to one of its neighbours.This process is a time and space homogeneous Markov chain, that is its increments are independent of the past and the transitions do not depend on time and the current position of the process.In many cases, the homogeneity of the environment reduces the applicability of the process.In numerous applied models some kind of obstacles can appear like impurities, fluctuations, etc.Thus, it is natural to express such irregularities as a random environment and it is well known that even small perturbations of the environment affect properties of the random process.In 1981 Harrison and Shepp [15] described the behaviour of the SRW in a slightly disturbed environment, replacing only the probability of passing from 0 to 1 by some fixed p ∈ (0, 1).They observed that the scaling limit is not the Brownian motion, but the skew Brownian motion.
We intend to study random walks in a randomly perturbed environment.Our main results concern the so-called random walk in a sparse random environment (RWSRE) introduced in [17], in which homogeneity of an environment is perturbed only on a sparse subset of Z.More precisely, first we choose randomly a subset of integers marked by the positions of a standard random walk with positive integer jumps and next we impose a random drift at the chosen sites.The present paper can be viewed as a continuation of the recent publications [17,6,5], where annealed limit theorems were described.These annelad-type results do not settle however the question if the environment alone is sufficient to determine the distributional behaviour of the process with high certainty.Here we froze the environment and we are interested in limit behaviour of the random process in the quenched settings.As we show in the present article, even in a very diluted random environment the fluctuations of the random perturbation of the medium affect the conditional distribution of the random walker.
The model RWSRE we consider here can be viewed as an interpolation between SRW and the one suggested in the seventies by Solomon [25] called a one dimensional random walk in random environment (RWRE), where all the sites were associated with random i.i.d.weights {ω i } describing the probability of passing to the right neighbour.It quickly became clear that the additional environmental noise in the system has a significant impact on the behaviour of the model.In fact, the answers to a variety of questions about the model like limit theorem [16] and large deviations [7,4] are given only in terms of the environment marginalizing the impact of the random motion of the process.
1.1.General setting.To define our model let Ω = (0, 1) Z be the set of all possible configurations of the environment equipped with the corresponding cylindrical σ-algebra F and a probability measure P. A random element ω = (ω n ) n∈Z of (Ω, F) distributed according to P is called a random environment.Each element ω of Ω and integer x gives a rise to a probability measure P x ω on the set X = Z N 0 with the cylindrical σ-algebra G such that P x ω [X 0 = x] = 1 and where X = (X n ) n∈N 0 ∈ X .One sees that under P x ω , X forms a nearest neighbour random walk which is a time-homogeneous Markov chain on Z and it is called a random walk in random environment.The randomness of the environment ω influences significantly various properties of X.In view of this, it is natural to investigate the behaviour of X under the annealed measure P x = P x ω P(dω) which is defined as the unique probability measure on (Ω × X , F ⊗ G) satisfying In the sequel we will write P ω = P 0 ω and P = P 0 .It turns out that, in general, under the annealed probability X is no longer a Markov chain, because it usually exhibits a long range dependence.
We are interested in limit theorems for X n as → ∞, however in this paper we discuss the asymptotic behaviour of the corresponding sequence of first passage times T = (T n ) n∈N , that is (1.1) T n = inf{k ∈ N : X k = n}.
We will study the distribution of T n in the quenched setting which means that we will investigate the behaviour of for suitable choices of sequences (a n ) n∈N and (b n ) n∈N possibly depending on ω.In the present setting µ n , defined by µ n (ω) = µ n,ω , becomes a random element of M 1 , the space of probability measures on (R, Bor(R)), where Bor(R) stands for the Borel σ-algebra.M 1 equipped in the Prokhorov distance is a complete, separable metric space.One can distinguish two types of limiting behaviour of (µ n ) n∈N .We will say that a strong quenched limit theorem for T holds if µ n → µ almost surely in M 1 , that is for P a.s.ω the sequence of measures {µ n,ω } converges weakly to µ, and say that a weak quenched limit law for T holds if µ n ⇒ µ in M 1 .Here and in the sequel ⇒ denotes weak convergence.
We will now discuss different choices of the probability P which is the distribution of the environment.To keep the introduction brief we will limit the discussion to the i.i.d.random environment, which is the most classical choice for P, and the sparse random environment which we will study in depth in the sequel.1.2.Independent identically distributed environment.One of the simplest and most studied choices of the environmental distribution P is random walk in i.i.d.random environment, corresponding to a product measure, under which ω = (ω n ) n∈Z forms a collection of independent, identically distributed (i.i.d.) random variables.In their seminal work Kesten et al. [16] used the following link between walks and random trees [14]: the one-dimensional distributions of T are connected to a branching process in random environment with immigration and a reproduction law with the mean distributed as (1 − ω 0 )/ω 0 .This observation later leads to a conclusion that T lies in the domain of attraction of an α-stable distribution, where E[ω −α 0 (1 − ω 0 ) α ] = 1, provided that such α ∈ (0, 2) exists (see Figure 1.3).After a close examination of the main results of Kesten et al. [16] it transpires that the centering and scaling are determined by the distribution of (1−ω 0 )/ω 0 , which means that the behaviour of the walker does not affect the limiting behaviour in a significant way.In turn, to understand the random motion, one is led to investigate the behaviour of T under P ω .If α > 2, then a strong quenched limit theorem [24,13] of the form lim As seen from the results in [18,20] there is no strong quenched limit theorem for T in the case α < 2. Indeed it turns out that for α < 2 one can find different strong quenched limits for T along different sequences.This in turn leads to the analysis of T in the weak quenched setting, that is weak limits of µ n .Consider first the mapping H : M p → M 1 given as follows: for a point process ζ = i≥1 δ x i , where {x i } i∈N is an arbitrary enumeration of the points, define where {τ i } i∈N is a sequence of i.i.d.mean one exponential random variables.Then the main result of [9,12,19] states that for α < 2, where N is a Poisson point process on (0, ∞) with intensity c N x −α−1 dx for some constant c N > 0.
1.3.Sparse random environment.We now specify the object of interest in the present paper.We will work under a choice of environmental probability P for which the random walk X will move symmetrically except some randomly marked points where we impose a random drift.The marked sites will be distributed according to a two-sided random walk.Denote by ((ξ k , λ k )) k∈Z a sequence of independent copies of a random vector (ξ, λ), where λ ∈ (0, 1) and ξ ∈ N, P-a.s.Considering the aforementioned two-sided random walk S = (S n ) n∈Z given via The sequence S determines the marked sites in which the random drifts 2λ k −1 are placed.Since for the unmarked sites n (that is, for most of sites) the probabilities of jumping to the right are deterministic and equal to ω n = 1/2, it is natural to call ω a sparse random environment.Following [17] we use the term random walk in sparse random environment (RWSRE) for X as defined above with ω being a sparse random environment.
Example 1.1.In the case when P[ξ = 1] = 1 random walk in sparse random environment is equivalent to a random walk in i.i.d.environment.
Example 1.2.Suppose that ξ is independent of λ and has a geometric distribution Random walk in a sparse random environment was studied in detail in the annealed setting in [17,6,5].In [17] the authors address the question of transience and recurrence of RWSRE and prove a strong law of large numbers and some distributional limit theorems for X.As in the case of i.i.d.random environment, the fraction ρ = 1 − λ λ appears naturally in the description of the asymptotic behaviour of the random walk.According to [17,Theorem 3 Note that the first condition in (1.3) excludes the degenerate case ρ = 1 a.s. in which X is a simple random walk.Under (1.3), the RWSRE also satisfies a strong law of large numbers, that is, [17] and Proposition 2.1 in [6].We note right away that conditions present in (1.3) are satisfied under the conditions of our main results.Thus, the random walks in a sparse random environment that we treat here are transient to the right.
The asymptotic behaviour of T is controlled by two ingredients.The first one, similarly as in the case of i.i.d.environment, is α > 0 such that The parameter α > 0, if it exists, is used to quantify the effect that the random transition probabilities λ k 's have on the asymptotic behaviour of the random walker.The second for some function : R → R slowly varying at infinity.Here and in the rest of the article we write f (t) ∼ g(t) for two functions f, g ∈ R → R whenever f (t)/g(t) → 1 as t → ∞.
Recall that a function is slowly varying at infinity if (ct) ∼ (t) as t → ∞ for any constant c > 0. It transpires that if the tail of ξ is regularly varying with β ∈ (0, 4) with E[ξ] < ∞, then with respect to the annealed probability T lies in the domain of attraction of γ-stable distribution with γ = min{α, β/2}, see [6].
For small values of β one sees an interplay between the contribution of the sparse random environment and the random movement of the process in the unmarked sites.To state this result take ϑ to be a non-negative random variable with the Laplace transform Note that 2ϑ is equal in distribution to the exit time of the one-dimensional Brownian motion from the interval [−1, 1], see [23,Proposition II.3.7].Next consider a measure η for x 1 , x 2 > 0. Now let N = k δ (t k ,j k ) be a Poisson point process on [0, ∞) × K with intensity LEB ⊗ η, where LEB stands for the one-dimensional Lebesgue measure.Under mild integrability assumptions, see [5,Lemma 6.4], the integral converges and defines a two-dimensional non-stable Lévy process with Lévy measure η.
Next consider the β-inverse subordinator Finally, if β < 2α and β ∈ (0, 1), then under some additional mild integrability assumptions [5, Theorem 21], with respect to the annealed probability The aim of the present article is to present a quenched version of this result.As we will see in our main theorem, the terms L 2 (L ← 1 (1)−) and L 1 (L ← 1 (1−)) present on the right hand side can be viewed as the contribution of the environment, whereas ϑ represents the contribution of the movement of the random walker in the unmarked sites that are close to n.For the full treatment of the annealed limit results, in particular the complementary case β ≥ 2α, we refer the reader to [5].
The article is organised as follows: in Section 2 we give a precise description of our setup and main results.In Section 3 we provide a preliminary analysis of the environment.The essential parts of the proof of our main results are in Sections 4 and 5 where we prove an absence of the strong quenched limits and prove weak quenched limits respectively.

Weak quenched limit laws
In this section we will present our main results.From this point we will consider only a sparse random environment given via (1.2).We assume that (2.1) for some β ∈ (0, 4) and slowly varying .We will focus on the case in which the asymptotic of the system is not determined solely by the environment and thus we will assume also that for some parameter γ ∈ (β/4, 1 ∧ β).The first condition in (2.2) guarantees that a part of the fluctuations of T n will come from the time that the process spends in the unmarked sites.The second condition is purely technical.Note that we do not assume that there exists α > 0 for which (1.5) holds.
Our first result states that there is no quenched limit for T n 's in the strong sense.Take (a n ) n∈N to be any sequence of positive real numbers such that Then, since the tail of ξ is assumed to be regularly varying, the sequence (a n ) n∈N is also regularly varying with index 1/β.That is for some slowly varying function 1 , The sequence (a n ) n∈N will play the role of the scaling factor in our results.The first one shows an absence of strong quenched limit laws for T .Theorem 2.1.Assume (1.3), (2.1) and (2.2).Then for P almost every ω there are no sequences {A n (ω)} n∈N and {C n (ω)} n∈N such that the sequence of normalized random variables (T n − C n (ω))/A n (ω) converges in distribution (with respect to P ω ) to a nontrivial random variable.
Therefore, as in the case of i.i.d.environment, the asymptotic quenched behaviour of T n 's ought to be expressed in terms of weak quenched convergence.As it is the case for annealed limit theorem, one needs to distinguish between a moderately (Eξ < ∞) and strongly (Eξ = ∞) sparse random environment.
To describe the former take {ϑ j } j∈N to be a sequence of i.i.d.copies of ϑ distributed according to (1.6) and let G : M p → M 1 be given via where {x i } is an arbitrary enumeration of the point measure.
Before we introduce the notation necessary to state our results in the strongly sparse random environment, we will first treat the critical case which is relatively simple to state.Denote m n = nE ξ1 {ξ≤an} .Note that by Karamata's theorem [2, Theorem 1.5.11] the sequence {m n } n∈N is regularly varying with index 1/β.Furthermore Next let {c n } n∈N be the asymptotic inverse of {m n } n∈N , i.e. any increasing sequence of natural numbers such that By the properties of an asymptotic inversion of regularly varying sequences [2, Theorem 1.5.12],c n is well defined up to asymptotic equivalence and is regularly varying with index β.Finally, by the properties of the composition of regularly varying sequences {a cn } n∈N is regularly varying with index 1 and The limiting random measures in Theorems 2.2 and 2.3 share some of the properties of their counterpart in the case of i.i.d.environment [19,Remark 1.5].Namely, using the superposition and scaling properties of Poisson point processes, one can directly show that for each n ∈ N and G, G 1 , . . ., G n being i.i.d.copies of the limit random measure G(N ) in Theorem 2.2 or Theorem 2.3, The statement of our results in the strongly sparse case needs some additional notation.As it is the case for the annealed results, it is most convenient to work in the framework of non-decreasing càdlàg functions rather than point processes.Denote by D ↑ the class of non-decreasing càdlàg functions are all points on the non-negative half line such that h has a (left) discontinuity with jump of size where L is a β-stable Lévy subordinator with Lévy measure ν(x, +∞) = x −β .
Interestingly the limit measure F (L) does not enjoy a self-similarity property in the sense of (2.4).Namely, for any a, b ∈ R, b > 0 the laws of are different, where F, F 1 and F 2 are independent copies of the limiting random measure F (L) in Theorem 2.4.

Auxiliary results
We will now present a few lemmas that we will use in our proofs.We will discuss properties of some random series as well as the asymptotic behaviour of the hitting times (1.1).

3.1.
Estimates for the related stochastic processes {R i } i∈Z and {W i } i∈Z .We will frequently make use of the following notation: for integers i ≤ j, We will also make use of the the limits Note that if E log ρ < 0 and E log ξ < ∞, both series are convergent as one can see by a straightforward application of the law of large numbers and the Borel-Cantelli lemma (see [3,Theorem 2.1.3]).The random variables R i 's and W j 's have the same distribution and obey the recursive formulae We can therefore invoke the proof of [3,Lemma 2.3.1] to infer the following result on the existence of moments of R i 's and W j 's.In what follows we write R (respectively W ) for a generic element of {R i } i∈Z (respectively {W j } j∈Z ).

Hitting times.
We describe now some properties of the sequence of stopping times T = {T n } n∈N that allow us to better understand the process X and indicate its ingredients which play an essential role in the proof of our main results.We will first analyse the hitting times T along the marked sites S, that is As it turns out, one can use R i,j 's given in (3.1) to represent the exit probabilities from interval (S i , S j ).That is, for i < k < j we have be the time that the particle needs to hit k'th marked point S k after reaching S k−1 .One uses W j 's to describe the expected value of T k : Observe that the random variable T k can be decomposed into a sum of two parts: the time the trajectory, after reaching S k−1 but before it hits S k , spends to the left of S k−1 and the time it spends to the right of S k−1 .For technical reasons that will become clear below, we divide the visits exactly at point S k−1 between these two sets depending on the direction from which the particle enters S k−1 .To be precise we define k is the sum of the time the particle spends in (−∞, S k−1 − 1] and the number of steps from S k−1 − 1 to S k−1 .Similarly we define ) .Thus we can write Observe that given ω, the random variables {T k } k∈N are P ω independent, however for fixed k, T l k and T r k mutually depend on each other.Summarizing, we obtain the following decomposition that will be used repeatedly: To proceed further we need to analyse T r j , T l j in details and describe their quenched expected value and quenched variance.Below we prove that after hitting any of the chosen sites (S k ) k the consecutive excursions to the left are negligible.This entails that behaviour of T S k is determined mainly by T r S k .3.3.The sequence {T r Sn }.Note that, under P ω , T r k equals in distribution to the time it takes a simple random walk on [0, ξ k ] with a reflecting barrier placed in 0 to reach ξ k for the first time when starting from 0. This is the reason we include into T r k the visits at S k−1 , but only those from S k−1 + 1.Indeed, let (Y n ) n be a simple random walk on Z independent of the environment ω.Define In what follows we investigate how the asymptotic properties of ξ k affect those of T r k .To do that, we will utilize the aforementioned equality in distribution and hence we first need to describe the asymptotic properties of U n as n tends to infinity.The proof of the next lemma is omitted, since it follows from a standard application of Doob's optimal stopping theorem to martingales ) and exp{±tY n }cosh(t) −n .Lemma 3.2.Let U n , for n ∈ N be given in (3.5).We have We can now use Potter bounds [2, Theorem 1.5.6] to control (α This in turn yields a large deviation result asymptotic on the logarithmic scale.We summarize this discussion in the following lemma. Corollary 3.3.The sequence {Var ω T r Sn /a 4 n } n∈N converges in distribution (with respect to P) to some stable random variable Z. Moreover for any sequence {α n } n∈N that tends to infinity, 3.4.The sequence {T l Sn }.The structure of T l k is more involved.We may express it as a sum of independent copies of F k , which denotes the length of a single excursion to the left from S k , and thus obtain formulae for its quenched expectation and quenched variance.
Lemma 3.4.The following formulae hold and (3.8) Proof.Since the environment {(ρ k , ξ k )} k∈Z is stationary, it is sufficient to calculate all the above formulae for k = 0 or k = 1.We first consider T l 1 .Notice that the particle starting at 0 can return repeatedly to 0 from the right or from the left.By the classical ruin problem, P 1 ω (T 0 > T ξ 1 ) = 1/ξ 1 .Thus the particle starting at 1 hits the point 0 M 1 times before it reaches ξ 1 , where M 1 is geometrically distributed with parameter 1/ξ 1 and mean ξ 1 − 1.This is exactly the number of visits to 0 counted by T r 1 .Between consecutive steps from 0 to 1, let's say between mth and (m + 1)'th step, the particle spends some time in (−∞, 0].In particular its visits at 0 from the left are exactly those included in T l 1 .Let us denote such an excursion by G 0 (m) and denote by G 0 its generic copy.That is, G 0 (G k , resp.) is the time the particle spends in (−∞, 0] ((−∞, S k ], resp.)before visiting 1 where N m is the number of jumps from 0 to −1 before the next step to 1. Since the particle can jump to −1 with probability (1 − λ 0 ), N m has geometric distribution with mean ρ 0 .Summarizing, T l 1 can be decomposed as where F 0 (j, m) measures the length of a single left excursion from 0. Observe that both N m 's and F 0 (j, m)'s are i.i.d.under P ω .Moreover, the first sum includes m = 0, because the process starts at 0. Recall that if S N = N k=1 X i for some random variable N and an i.i.d.sequence {X n } independent of N , then The above formula together with (3.9) easily entails (3.11) Since F 0 is the time of a single excursion from 0 that begins with a step left, using the solution to the classical ruin problem in combination with formula (3.4) we get A formula for quenched variance of crossing times for arbitrary neighbourhood was given in [13, Lemma 3] and yields (3.7).Inserting these formulae to (3.11), using the fact that ρ 0 W −1 = W 0 − ξ 0 ρ 0 , and finally simplifying the expression leads to (3.8).
Lemma 3.5.For every ε > 0 and θ ≥ 0, where γ is a parameter satisfying (2.2).In particular, Proof.To prove the lemma one needs to deal with the formula for the variance (3.8).To avoid long and tedious arguments we will explain how to estimate two of the terms, i.e. we will prove (3.12) All the remaining terms can be treated using exactly the same arguments.Recall γ ∈ (β/4, 1 ∧ β) and Eρ 2γ < 1.The Markov inequality and independence of ξ k , Π j+2,k−1 , ρ j+1 ξ j+1 and W j yield where the last inequality follows from our hypotheses (2.2) and Lemma 3.1.This proves (3.12).We proceed similarly with the second formula (3.13): Invoking the first part of the lemma with θ = 0 we conclude convergence of Var ω T l Sn /a 4 n to 0 in probability.

Absence of a strong limit
Our aim now is to prove Theorem 2.1 saying that the 'strong limit in distribution' does not exist.For most of the proof we will consider the standard normalization that is Var ω T n and study the normalized sequence (4.1) We will prove that for P-a.e. ω there exists its subsequence { T n k (ω) } convergent to 2ϑ − 1, however on the other hand as we will see this cannot be the limit of the whole sequence.Finally we will show that there is no other normalization leading to a nontrivial limit.Absence of the strong quenched limit follows essentially from the fact that for P-a.e. ω one can find an infinite subsequence {ξ n l } l∈N such that the values of ξ n l +1 are exceptionally large, hence T S n l +1 − T Sn l , the time the walk needs to move from S n l to S n l +1 , is either much bigger or comparable with T Sn l .
First we need to construct a favourable environment of probability one.For this purpose we consider two increasing sequences {p n }, {q n } diverging to +∞ such that (4.2) 2p n < q n < p n+1 /2, p n /q n → 0 and a qn a 2pn ≥ n θ for θ > max{1/(4γ), 1/β}.Notice that one may take e.g.p n = 2 2 n , q n = p n+1 /4.We will need to prove that behaviour of the process in the interval [S 2pn , S 2qn ] is determined by its position after time T 2pn and that its previous values up to time 2p n are negligible when looking at time q n and scale a qn .The trajectory of the random walk X cannot be divided into independent pieces with respect to P , because the process can have large excursions to the left and the environment is not homogeneous.To remedy that we will censor the left excursions of X that become too large.We consider a new process, say X = {X k } k∈N .This process essentially behaves as the previous one and evolves in the same environment, with a small difference.Namely after X reaches S qn and before it reaches S 2qn we put a barrier at point S pn , i.e. the process cannot come back below S pn .However this barrier is removed when X hits S 2qn .Of course we can couple both processes on the same probability space removing all left excursions from S pn after hitting S qn and before reaching S 2qn .
For any k, we define the random variables T k , T k , T r k , T l k in an obvious way, e.g.T k = inf{j : k for all k s and T l k = T l k for k / ∈ n (q n , 2q n ].Notice that T k − T k is the time that the process X spends below S p k after hitting S k−1 and before reaching S k .The next lemma ensures that asymptotic properties of the processes X and X are comparable.Lemma 4.1.For any ε ∈ (0, 1) and P-a.e. ω there is N = N (ω) such that Moreover T n = T n a.s.for large (random) n.
Proof.Fix k ∈ (q n , 2q n ].To describe the quenched mean and the quenched variance of we need to calculate the time the trajectory X, after it hits S k−1 , but before reaching S k , spends below S pn .For this purpose we proceed as in the proof of Lemma 3.4, that is we decompose (4.5) where M k denotes the number of times the walk visits S pn from the right in the time interval (T S k−1 , T S k ), N m is the number of consecutive left excursions from S pn after hitting it from the right, and F pn (j, m) is the length of the corresponding excursion.Note that N m is geometrically distributed with mean ρ pn and variance ρ pn (1 + ρ pn ).Thus, by formulae (3.10) and (3.7), Next, observe that for any m > 0 and, invoking once again the gambler's ruin problem, We may easily calculate the mean and variance of M k and use the formulae (3.3) to express them in terms of the environment.We get, after simplifying, Therefore, by (3.1), (4.6) and (4.7), Now, we are ready to prove (4.3).We have where γ ∈ (0, 1) is a small constant such that Eρ γ < 1, Eξ γ < ∞ and EW γ < ∞ (see (2.2) and Lemma 3.1).Then, by the Borel-Cantelli lemma which gives (4.3).Formula (4.4) can be proved applying essentially the same argument.One needs to compute, combining (3.7) with (4.8), the precise expression for the variance and next, arguing as above and considering each of the summands separately (as in the proof of Lemma 3.4), one can deduce (4.4).We skip the details.Finally we write pn to infer our final claim by yet another appeal to the Borel-Cantelli lemma.
The advantage of introducing a new process X is that it behaves similarly to X and from the point of view of limit theorems this change is indistinguishable.However, here one can indicate independent pieces: {X k } k∈(T qn ,T 2qn ] are P-independent.Now we are ready to describe the required properties of the environment.The sets below depend on several parameters.Given d < D and b < B, and ε > 0 let where G k is the length of the left excursion of X from S k before hitting We want to consider environments which belong to infinitely many sets U n .However, given ω, we want to have some freedom of choosing all the parameters.The lemma below justifies that the measure of these environments is one.Observe that the events {U n } n∈N are independent, because U n depends only on {ω j } j∈[pn,2qn] and thanks to (4.2) the sets {[p n , 2q n ]} n∈N are pairwise disjoint.Thus, invoking the Borel-Cantelli Lemma, it is sufficient to prove that there is δ 0 > 0 such that for large indices n, (4.9) We need to estimate probabilities of all the events which appear in the definition of U n .Denote To estimate the probability of V 1 k notice that thanks to (4.2) we have a k−2pn /a k+1 → 1 for any k ∈ (q n , 2q n ], therefore Finally, observe that thanks to our choice of parameters the events defining U n are disjoint for different values of k's.Indeed, if k, j ∈ (q n , 2q n ], k < j and ξ 4 k+1 ≥ ba 4 k+1 , then Summarizing, for some δ > 0 and large n In conclusion, the probabilities of U n are bounded from below, which entails (4.9) and completes the proof.
Proof of Theorem 2.1.In view of Lemma 3.5 and our hypothesis (4.2), the Borel-Cantelli lemma yields P ∀ε > 0 Var ω T S 2pn ≥ a 4 qn ε i.o.= 0. Therefore, invoking Lemma 4.2 the set has probability 1.From now we fix ω from the event above which also satisfies the claim of Lemma 4.1.Assume that, given ω, for some random variable Y ω .We fix parameters d < D, b < B such that b > 3 • 2 5/β−1 D and ε > 0. Take two sequences {n m } m∈N and k m ∈ (q nm , q 2nm ] such that , where all the sets were defined in (4.10).We can additionally assume (removing a finite number of elements of the sequence if needed), that for all indices m (4.12) Var ω T S 2pn m < a 4 km ε.
Random variables V m and (W m , Z m ) are P ω -independent.By (4.13), V m converges in distribution to Y ω , whereas W m , by Lemma 3.2, converges to 2ϑ − 1. Therefore we need to understand the behaviour of both deterministic (given ω) sequences {v m } m∈N , {w m } m∈N and of the sequence of random variables {Z m }.For this purpose we need to understand behaviour of the variances which appear in the above formulae.Note first that on the considered event, recalling (3.10), we have Applying the Schwartz inequality and the well-known inequality 2ab ≤ a 2 /C + Cb 2 , for any n and arbitrary large constant C, we can easily prove Combining the above inequalities with (4.12) and the definition of U n , we have The above inequality ensures that Var ω T S km+1 is of the order a 4 km+1 .For arbitrary small δ > 0, choosing appropriate parameters ε, C and large indices k m (4.16) We can pass with δ to 0 and assume that the parameters satisfy Observe that for any η > 0, using the Chebyshev inequality and (4.15), we have: So, if the limits (4.11), (4.13) exist, both must be equal to Y ω = 2ϑ − 1.
Next, fixing all the parameters b, B, d, D observe that both sequences {v m }, {w m } are bounded, therefore we can assume, possibly choosing their subsequences, that they are convergent to some v and w, respectively.Since the sequences of random variables {V m } and {W m } are independent, we conclude where ϑ v , ϑ w are independent.However this equation cannot be satisfied e.g. by (1.6).That leads to a contradiction and proves that the limit (4.11) cannot exist.
Summarizing, we have proved up to now that the sequence { T n } defined in (4.1) cannot converge in distribution.Nevertheless, it still can happen that different normalization leads to a convergent sequence and we need to exclude this possibility.Let us consider another normalization T n = T n /A n + C n for some sequences {A n } and {C n }.Let us also recall that we already know that if the limit exists, it must be equal to 2ϑ − 1.
Observe first that A n must be bounded.Indeed, assume that there exists its subsequence {A n k } converging to +∞.Then, since { T n } is tight, we have T n k /A n k P → 0 and thus the sequence {C n k } must converge to some limit C and finally T n k ⇒ C, which is a trivial limit.
Next we fix parameters b, B, d, D and consider the subsequence {k m } satisfying all the above requirements.If the sequence {A n } contains a subsequence {A km } convergent to a positive constant A, then again the sequence {C km } must converge to some C. Repeating the above calculations we are led once more to equation (4.21), which leads us to a contradiction.
Thus, the sequence {A km } must converge to 0. However in this case, since w n W m + Z m ⇒ w(2ϑ − 1), the sequence {(w m W m + Z m )/A km } cannot be tight and finally, since V n is independent, by (4.14) the sequence T km cannot be tight and converge in distribution.This completes the proof of the theorem.

Proofs of the weak quenched limit laws
In this final section we present a complete proof of our main results.We will begin by presenting a suitable coupling.Then we will treat the moderately sparse and strongly sparse case separately.5.1.Coupling.In the first step we will prove our result along the marked sites.That is we analyse The main part of the argument concentrates on the limit law of T r Sn = n k=1 T r k .Recall U n defined in (3.5), which is the first time the reflected random walk hits n.For every k > 0 and for fixed environment ω it holds T r k d = U ξ k .By the merit of Lemma 3.2 and Skorokhod's representation theorem we may assume that our space holds random variables U (k) n and ϑ k such that: • {U ξ k and T r k have the same distribution under P ω .Observe that the convergence in L 2 is secured by the convergence in distribution and uniform integrability provided in Lemma 3.2.
To simplify the notation we will write and, moreover, for N ∈ N large enough We can hence estimate, for n sufficiently large, Since the sequence n k=1 ξ 4 k /a 4 n converges weakly (under P) to some β/4-stable variable L β/4 , the probability on the right hand side above converges to P[L β/4 > δ/(2ε)].To estimate the second term in (5.2), note that By the Fubini theorem, we have and the Karamata theorem [2, Theorem 1.5.11]entails that the expression on the right is asymptotically equivalent to 4ε 4 a 4 n P[ξ 1 > εa n ] ∼ 4ε 4−β n −1 a 4 n .Finally, we can conclude that for any ε, δ > 0, lim sup and passing with ε to 0 we conclude the desired result.
We are now ready to determine the weak limit of the sequence φ n (ω) = φ n,ω given by (5.1).Recall the map G defined in (2.3).
Lemma 5.2.The map G is measurable.
Remark 5.3.The proof of Lemma 5.2 is identical to that of Lemma 1.2 in [19] and therefore will be omitted.Part of the proof is showing that the map Theorem 5.4.Assume (2.1) and (2.2).Then In the proof of this result we will use the following lemma.where µ(dx) = βx −β/2−1 dx/2.Since G is not continuous, we cannot simply apply the continuous mapping theorem and, similarly as in [19], we are forced to follow a more tedious argument.Define By [1,Theorem 3.2], to prove that for all δ > 0, (5.5)where ρ is the Prokhorov metric on M 1 (R).
First, for any ε > 0, N ∞ ∈ M ε p almost surely.Thus (5.3) is satisfied by the continuous mapping theorem since G ε is continuous.
For any sequence x = (x k ) k∈N ∈ 2 and ε > 0 define x ε ∈ 2 by x ε k = x k 1 {x k >ε} .By the dominated convergence theorem, x ε → x in 2 as ε → 0. Hence, since the map G 2 defined in Remark 5.3 is continuous, also G 2 (x ε ) ⇒ G 2 (x).This means that for any point which gives (5.4).
Recall that if L X , L Y are laws of random variables X, Y defined on the same probability space, then ρ(L X , L Y ) 3 < E|X − Y | 2 (c.f.[10,Theorem 11.3.5]).Thus The sequence a −1 n n k=1 ξ k converges weakly to some β-stable variable L β , therefore which proves (5.5).Therefore G(N n ) ⇒ G(N ∞ ).Now the claim of the theorem follows from Proposition 5.1 and Lemmas 5.5 and 3.5.
from which it follows, by Lemma 5.5, that µ n ⇒ G(N α ∞ ).Observe that on the event {n ≤ S αn }, for any k such that S k ≤ n, and similarly on {S αn ≤ n} for any k such that S k ≥ n, Therefore for any δ > 0 and ε > 0, The first term tends to 0 by the law of large numbers (recall 1/α = Eξ 1 ).
The last expression can be made arbitrary small by taking sufficiently small ε.As one may expect S n grows at a scale m n and thus ν n must grow at a scale c n (in the sense of limit theorem which we will soon make precise).For our purposes we need to justify that S n /m n and ν n /c n converge jointly with some other characteristics of the trajectory of S. For this reason we will need to use the setting of càdlàg functions.Recall that D stands for the space of right continuous functions that have a left limit at each point.For Consider D ↑ ⊆ D consisting of non-decreasing functions and take M : where for h ∈ D ↑ , {t k } are the discontinuity points of h and is the size of the jump at t k .
Lemma 5.6.The function M : D ↑ → M is continuous with respect to J 1 topology.
Proof.Let f n , f ∈ D ↑ be such that f n → f in J 1 topology.For any nonnegative, continuous function ϕ : (0, +∞] × [0, +∞) → R with compact support we can find ε > 0 and T > 0 and the function M is J 1 -continuous by Lemma 5.6.Moreover, and the map h → h ← is continuous in M 1 topology by [27].In what follows, we will use notation introduced in [26].For h ∈ D let h − be the lcrl (left-continuous, having righthands limits) version of h, that is, h − (t) = lim ε→0 + h(t − ε) and h − (0) = 0. Similarly, let h + denote rcll version of a lcrl path.Let Φ : D ↑ → D be given by Finally, observe that for any k ∈ N, Φ(L dn ) on the set [S k /a dn , S k+1 /a dn ) is constant and equal to S k /a dn , therefore S νn−1 a dn = Φ (L dn ) n a dn .
By [26], Φ is J 1 -continuous on D ↑↑ ⊂ D, the set of strictly increasing, unbounded functions.Since L ∈ D ↑↑ almost surely, by the continuous mapping theorem we have joint convergence in distribution . By Skorokhod's representation theorem we may assume the above convergence holds almost surely.
Since the limiting processes admit no fixed discontinuities, Proposition 2.4 in [26] gives Remark 5.8.Observe that all information on the sequence (ξ k ) k is carried by the process Λ n and therefore by L n or, equivalently, L n .We may thus assume that our space holds random variables U (k) n , ϑ k as described in Section 5.1 and at the same time the convergence given in Lemma 5.7 holds almost surely.Lemma 5.9.Assume that (2.1) holds true.If β < 1, then Since a Cdn ∼ C 1/β n, an appeal to Lemma 3.5 shows that the second term tends to 0 as n → ∞.The first term can be made arbitrary small by taking C > 0 sufficiently big.In the case β = 1 we can use an analogous argument with d n replaced with c n .
For the purpose of the next lemma.let ({U 0 n } n∈N , ϑ 0 ) be, as before, a copy of ({U n } n∈N , ϑ) given by the claim of Lemma 3.2 independent of the environment.→ 0.
Therefore the weak limit of the quenched law of (T n − E ω T n )/a 2 cn will coincide with the limit of P ω cn k=1 ξ 2 k (2ϑ k − 1)/a 2 cn ∈ • .
The weak limit of the latter is G(N ), which follows from the proof of Theorem 2.2.
Proof of Theorem 2.4.Let µ n,ω denote the quenched law of (T n − E ω T n )/n 2 .Then To treat the second term under the probability we can, similarly as previously, decouple the times that the random walker spends between consecutive S k 's for k ≤ n.The first part will be controlled with the help of Lemma 5.11.Let (U 0 n , ϑ 0 ) be, as before, a copy of (U n , ϑ) given by the claim of Lemma 3.2 independent of the environment.Then U n−S νn−1 has, under P ω , the same distribution as the time the walk spends in [S νn−1 , n) after reaching S νn−1 and before reaching n.By Lemma 5.10 and Lemma 5. Recall the random functions L n given in (5.7) and that for a càdlàg function h we denote by {x k (h), t k (h)} an arbitrary enumeration of its discontinuities, i.e. x k (h) = h(t k ) − h(t − k ) > 0, where t k (h) = t k .Note that, with Υ given in (2.5), one has by the merit of Lemmas 5.5, 5.9, 5.10 and 5.11 that the limit of μn,ω will coincide with the limit of For fixed ε > 0, F n ε → F ∞ ε , where since associated point processes converge and a dn /n → 1.Then we show that F ∞ ε ⇒ F (L) as ε → 0. We finally prove that (5.5) also holds in this context and conclude the result.
(3.5) U n = inf{m : |Y m | = n}, i.e.U n is the first time the reflected random walk hits n.Then for every k > 0, for fixed environment ω, T r k d = U ξ k .

Lemma 4 . 2 .
Assume that conditions (2.1) and (2.2) are satisfied.Then the event U = lim sup n U n (d, D, b, B, ε) : d, D, b, B ∈ Q + , d < D, b < B, b > 3 • 2 4/β D, ε > 0 has probability one.Proof.Since in the above formula the intersections are essentially over a countable set of parameters (one can obviously restrict to the rational parameter ε), it is sufficient to prove that for fixed parameters d < D, b < B such that b > 3 • 2 5/β−1 D and ε > 0, P lim sup n U n = 1, for U n = U n (d, D, b, B, ε).
m → ∞.One can easily see that for any fixed d and D one can construct sequences {b m }, {B m }, {k m } such that b m , B m → ∞, b m /B m → 1 and inequalities (4.18) and (4.19) hold.Then v m → 0 and w m → 1.Since the sequence {V m } is tight, we have

5. 3 .
Strong sparsity: preliminaries.From now we assume that Eξ = ∞.This case is technically more involved, however the underlying principle remains the same.Denote the first passage time of S viaν n = inf {k > 0 : S k > n} .Recall that we write m n = nE ξ1 {ξ≤an} and d n = 1/P[ξ > n]. and we denote by {c n } n∈N for the asymptotic inverse of {m n } n∈N , i.e. any increasing sequence of real numbers such that lim n→∞ c mn /n = lim n→∞ m cn /n = 1.
1) andS νn−1 a dn → Φ(L)(1) = Υ(L)almost surely.The case β = 1 is similar and follows from the fact that by[22, Corollary 7.1] and properties of J 1 topology, L n ⇒ id in (D, J 1 ).One can combine this with ν n and the arguments presented in the case β < 1 to get the desired claim.
2ϑ, for ϑ defined in (1.6).Furthermore the family of random variables {n −4 U 2 n } n∈N is uniformly integrable.Theorem 3.8.2].Moreover, we can use precise large deviation results for sums of i.i.d.regularly varying random variables [8, Theorem 9.1] to describe the deviations of Var ω T r Sn .That is for any sequence {α n } that tends to infinity, a.s.bounded and Var ω T n /Var ω T n converges to 1, hence (4.11) yields