Limit theorems for Lévy flights on a 1D Lévy random medium*

We study a random walk on a point process given by an ordered array of points (ωk, k ∈ Z) on the real line. The distances ωk+1 − ωk are i.i.d. random variables in the domain of attraction of a β-stable law, with β ∈ (0, 1) ∪ (1, 2). The random walk has i.i.d. jumps such that the transition probabilities between ωk and ω` depend on `− k and are given by the distribution of a Z-valued random variable in the domain of attraction of an α-stable law, with α ∈ (0, 1) ∪ (1, 2). Since the defining variables, for both the random walk and the point process, are heavy-tailed, we speak of a Lévy flight on a Lévy random medium. For all combinations of the parameters α and β, we prove the annealed functional limit theorem for the suitably rescaled process, relative to the optimal Skorokhod topology in each case. When the limit process is not càdlàg, we prove convergence of the finite-dimensional distributions. When the limit process is deterministic, we also prove a limit theorem for the fluctuations, again relative to the optimal Skorokhod topology.


Introduction
especially in light of the fact that the topic is regrettably less developed than others in the field of random walks, with the exception perhaps of random walks on percolation clusters et similia. For some interesting lines of research see, e.g., [9,10,15,3,28,20] and references therein. A recent paper which we extend with the present work is [17].
In this paper we give annealed limit theorems for Y in all cases α, β ∈ (0, 1) ∪ (1, 2), identifying in each case both the scale n γ whereby Y nt n γ , t ∈ [0, +∞) (1.1) converges to a non-null limit, and the limit process. In all cases we prove the optimal, or at least morally optimal, functional limit theorem, meaning that we show distributional convergence of the process with respect to (w.r.t.) the strongest Skorokhod topology that applies there. There are cases in which there can be no convergence in the J 1 or M 1 topologies: in such cases we prove convergence w.r.t. J 2 . When the limit process is not càdlàg (or càglàd) we show convergence of the finite-dimensional distributions. Finally, in the cases where the limit of (1.1) is deterministic, we prove a functional limit theorem for the corresponding fluctuations, again relative to the optimal topology.
The paper is organized as follows. In Sections 2.1 and 2.2 we describe the model and set the notation for the J 1 and J 2 Skorokhod topologies on spaces of càdlàg/càglàd functions; in Section 2.3 we lay out basic limit theorems for the underlying random walk S and the random medium ω; in Section 2.4 we present our main results. Finally, Section 3 contains all the proofs of the main theorems.

Setup
As mentioned in the introduction, the Lévy flight on random medium that we consider is a random walk performed over the points of a certain random point process. We proceed to define all the necessary constructions.
Random medium Let ζ := (ζ i , i ∈ Z) be a sequence of i.i.d. positive random variables. We assume that the law of ζ i belongs to the normal basin of attraction of a β-stable distribution, with β ∈ (0, 1) ∪ (1, 2). In the case β ∈ (0, 1), this means that, as n → +∞, is a stable variable of index β and skewness parameter 1 (because ζ i > 0). In the case β ∈ (1, 2) we have instead for a stable variableZ (β) 1 of index β. In this case, necessarily, ν is the expectation of ζ i and the skewness parameter is 0.
The random medium associated to (ζ i , i ∈ Z) is defined to be: This determines a point process ω := (ω k , k ∈ Z) on R that we call Lévy random medium to emphasize the fact that the distribution of ζ i has a heavy tail. Each point ω k will be called a target. In other words, the distances between neighboring targets are drawn according to independent random variables ζ i .

Underlying random walk
We consider a Z-valued random walk S := (S n , n ∈ N), with S 0 = 0 and i.i.d. increments ξ i := S i − S i−1 that are independent of ζ (and thus of ω). In other words, S is given by ξ i for n ∈ Z + . (2.4) The law of ξ i belongs to the normal basin of attraction of an α-stable distribution, with α ∈ (0, 1)∪ (1,2). This means that convergences analogous to those given in (2.1) and (2.2) apply to the ξ i , with limit random variables denoted by W (α) 1 and W (α) 1 , respectively. We will refer to S as the underlying random walk.

Random walk on the random medium
The random walk on the random medium Y := (Y n , n ∈ N) is defined to be: Y n := ω Sn , n ∈ N. (2.5) In other words, Y performs the same jumps as S, but on the points of ω; see Figure 1 for a hands-on explanation. In the following we will focus on the derivation of the asymptotic law of Y , under suitable scaling, with respect to the probability measure P governing the entire system (medium and dynamics). This is sometimes referred to as the annealed or averaged law of Y . Figure 1: Top: A realization of the underlying random walk S on Z. Middle: A realization of the random medium ω, with inter-distances given by ζ i . Bottom: The corresponding process Y jumps between the targets ω according to the walk S.
Before recalling certain basic facts about the processes ω and S, and stating our main results on the process Y , let us fix the notation for spaces of càdlàg functions endowed with certain Skorokhod topologies. EJP 26 (2021), paper 57.

Càdlàg functions and Skorokhod topologies
Given I, an interval or a half-line contained in R + := [0, +∞), we denote by D(I) ≡ D(I; R) the space of all càdlàg functions f : I −→ R, where we recall that these are right-continuous functions with left limits at all points of their domain. If I is an interval or a half-line intersecting (−∞, 0), or I = R, we consider a less customary function space: D(I) is the space of all functions f : I −→ R such that s → f (s) is càdlàg for s ≥ 0 and s → f (−s) is càdlàg for s ≥ 0 (in other words, the restriction of f to I ∩ (−∞, 0] is càglàd). Notice that this implies that f is continuous at 0. We also use the abbreviations D + ≡ D(R + ) and D ≡ D(R). Lastly, we denote by D 0 and D + 0 the subspaces of nondecreasing functions of D and D + , respectively.
In this section we introduce two notions of distance/topology that turn out to be crucial in the following. A complete treatment of these topologies can be found, e.g., in [ where the infimum is taken over all increasing homeomorphisms λ : I −→ I. This defines a distance on D(I), which we refer to as the J 1 or J 1 (I) distance.
This metric induces a topology and a notion of limit in D(I) which can be reformulated as follows: given (f n ) n∈N and f in D(I), the sequence f n is said to converge to f in the J 1 topology, and we write f n → f in (D(I), J 1 ), as n → ∞, if there exists a sequence of increasing homeomorphisms λ n : The analogous definition is given for I = (a, +∞) or I = (−∞, a], etc. If I = R, we say that f n → f in (D, J 1 ) if, for all T > 0 such that f is continuous at T and −T , The above definition defines a J 1 topology on D(I), in all cases where I is a half-line or the entire R. It is easy to write a metric that generates the J 1 (I) topology (see [24, Section 2]).

Remark 2.4.
In this paper the only two cases in which we work with I intersecting (−∞, 0) are I = [−M, M ] and I = R. In both cases we only deal with functions f such that f (0) = 0. It is easy to see that, under such additional condition, it is no loss of generality to require that the homemorphism λ fixes 0, i.e., λ(0) = 0. This makes it clear that, in such cases, If we think of f n as describing the spatial motion of some particle, the function λ : I −→ I of (2.6) is sometimes called the time change. Requiring the time change to be a homeomorphism is occasionally too strong a condition. One has a weaker topology if they only require that λ be a (possibly discontinuous) bijection: Definition 2.5. If I is a bounded interval and f, g ∈ D(I), the J 2 or J 2 (I) distance d J2,I (f, g) is defined as in the r.h.s. of (2.6), but with the infimum taken over all bijections λ : I −→ I. The notions of J 2 -convergence in all cases of I are derived as seen earlier for J 1 .
Remark 2.6. It is a known and easy-to-prove fact that, if I 0 ⊆ I is an interval at positive distance from the discontinuities of f , and f n → f in (D(I), J i ), for either i = 1 or i = 2, then sup t∈I0 |f n (t) − f (t)| → 0.
Remark 2.7. The definition of limit in (D([a, +∞)), J i ) (i = 1, 2) amounts to checking that f n → f in (D([a, T ]), J i ), for all T > a such that f is continuous at T , see (2.10). With the help of the previous remark, it is easy to see that this is tantamount to checking that f n → f in (D([a, T )), J i ), for all T > a such that f is continuous at T . In the remainder (see for example Section 3.3) we will liberally switch between the two conditions, as is more convenient.

Limit processes for ω and S
We now recall some elementary functional limit theorems for suitable rescalings of the processes ω and S, cf. (2.3) and (2.4).
By definition, for all k ∈ Z, ω k is a sum of |k| i.i.d. random variables ζ i in the normal domain of attraction of a β-stable distribution. We first deal with the case β ∈ (0, 1). For every s ∈ R we defineω ± (s), s ≥ 0) be two i.i.d. càdlàg Lévy β-stable processes such that Z (β) 1 , as introduced in (2.1) (these two conditions uniquely determine the common distribution of the processes). Set (2.14) Then (see, e.g., [25,Section 4.5.3]), as n → ∞, Then, as n → ∞,ω where the processZ (β) is defined similarly to Z (β) , cf. (2.14), but withZ 1 , introduced in (2.2). Analogous limit theorems hold for the continuous-time rescaled versions of the underlying random walk S. By definition, S n is a sum of n i.i.d. random variables ξ i in the normal domain of attraction of an α-stable distribution. We distinguish two regimes, depending on the values of α and µ := E[ξ i ] (when applicable).

Results
We now present our convergence results for the Lévy flight Y which, as we shall see, strongly depend on the values of α and β. All theorems are stated using the notation established in the previous section.
Theorem 2.8 is rather weak, in that it only proves convergence of the finite-dimensional distributions of the processŶ (n) defined in (2.26). Observe, however, that the limit process Z (β) • W (α) has trajectories that are not càdlàg with positive probability (see for example the explanation around (2.9) of [5]). Therefore, a functional limit theorem w.r.t. a Skorokhod topology is not the natural result to expect. On the other hand, when α ∈ (1, 2) and µ = 0, the assertion can be strengthened as follows.
− , one could put either process in the r.h.s. of (2.29), irrespectively of the sign of µ.
Remark 2.11. The convergence (2.29) fails in the topology J 1 , or even M 1 [25, Section 3.3]. The topology J 2 is thus the strongest among the classical Skorokhod topologies with respect to which the convergence holds. To justify the claim, observe that, in general, S (n) is a wildly oscillating function around µid, and Z (β) is almost surely discontinuous. More in detail, assume that µ > 0 and let s ∈ R be a discontinuity point of Z (β) + with a jump, say, of order 1 in n. Since, for n → ∞,ω (n) is very close to Z (β) + in J 1 , there exists a discontinuity point s n ofω (n) , very close to s, with a jump of order 1. Now, if we exclude the case where the underlying random walk S is deterministic,S (n) (t) is a non-monotonic function of t ∈ I, for every interval I ⊂ R + and n large enough, depending on I (this is an elementary Brownian-bridge result). So one can find a small interval I such that, as t runs through I,S (n) (t) oscillates many times around s n . Thereforeω (n) •S (n) (t) has many back-and-forth jumps of order 1. This prevents convergence both in J 1 and in M 1 , cf. [25, Figure 11.2]. What allows for J 2 -convergence is that the fluctuations ofS (n) around µid vanish, as n → ∞. This means that the oscillations ofS (n) (t) around s n , and therefore the large oscillations ofω (n) •S (n) (t), occur only in a vanishing interval I n ⊂ I. Therefore one can find a non-continuous change of the coordinate t, say ρ n : [0, T ) −→ [0, T ), which is globally close to the identity and "reorders" the points in I n in the sense that ω (n) •S (n) • ρ n only has one jump of order 1. The problem thus reduces to the much easier problem of showing the J 1 -convergence of the latter process. See the proof of Theorem 2.9 for the rigorous arguments. Lastly, we observe that all the results presented in this paper involving the J 2 topology could in fact be stated for a stronger Skorokhod-type topology. We refer the interested reader to Remark A.2 of the Appendix.
As stated in point 2 above, when α ∈ (1, 2) and µ = 0, the sequence of processesȲ (n) converges to a multiple of the identity function. The next theorem gives the explicit asymptotics of the fluctuations ofȲ (n) around its deterministic limit.
+ and W (α) be two independent α-stable processes, as previously defined.

Proof of Theorem 2.8: convergence of finite-dimensional distributions
We establish the assertion by extending the proof of [5, Theorem 2.2]. We first prove the following: Then, when n → ∞, where J 1 ⊗ J 1 denotes the product topology on the product space D × D + .
Proof. From (2.15) and (2. By virtue of the Skorokhod Representation Theorem, we may assume that the convergence in the statement of Lemma 3.1 holds almost everywhere. If this is not the case, there exists a probability space where it does, and since the specifics of the probability space are irrelevant for the next discussion, we avoid here to change the notation for the processes in the new space. Notice also that since Z (β) is a β-stable process, it is almost surely continuous at s, for any s ∈ R, and similarly W (α) is almost surely continuous at t, for any t ∈ R + . In particular, by the independence of the two processes, the event that W (α) is continuous at t and Z (β) is continuous at W (α) (t) has probability 1, for any t ∈ R + . Therefore the hypotheses of the next lemma hold almost surely. Lemma 3.2. Fix t > 0 and consider a realization (ω, S) of the random medium and of the underlying random walk such that W (α) is continuous in t and Z (β) is continuous at Proof. Let ε ∈ (0, 1) and η ∈ (0, ε) be such that Hence, using (3.6) and (3.4) we get EJP 26 (2021), paper 57. since |ϕ n (t) − t| < ς/2. Assume moreover that n is large enough so that where the notation in the l.h.s. of (3.9) was introduced in Remark 2.4. Then there exists (3.11) Note also that (3.7) ensures thatŜ (n) (t) ∈ [−|W (α) (t)| − 1, |W (α) (t)| + 1], so that by (3.10) and (3.7), and from (3.11) we get (3.14) This shows (3.2).

Proof of Theorem 2.12: limit theorems for β ∈ (1, 2)
Although Theorem 2.12 was stated after Theorem 2.9, we give the proof of the former first, because it is simpler and somehow preliminary to the proof of the latter. As a matter of fact, we only prove assertion 1. Assertion 2 is carried out similarly with no additional effort. Proof of Lemma 3.3. As the measurability of h is easy, we concentrate on the continuity statement. Assume that, as n → ∞, (w n , s n ) → (w, s) in (D 0 ×D + , J 1 ⊗J 1 ), with w ∈ C ∩D 0 and s ∈ D + . This means that, for all M, T > 0, In particular, if we fix T > 0, there exists a sequence (λ n ) n∈N of homeomorphisms of Proof of assertion 1 of Theorem 2.12. As defined in (2.30),Ŷ (n) :=ω (n) •Ŝ (n) . Denoting by h the composition map as in the previous lemma, we set out to prove that To apply the theorem and obtain (3.21) it remains to prove that the probability that (ν id, W (α) ) hits a discontinuity of h is zero. But (ν id, W (α) ) ∈ (C ∩ D 0 ) × D + , where h is continuous by Lemma 3.3.

Proof of Theorem 2.9: limit theorems for β ∈ (0, 1)
In comparison with the proof of Theorem 2.12, the main technical hurdle here is that in the compositionŶ (n) =ω (n) •S (n) , cf. (2.28), the inner function (also referred to as random time change) is not increasing and one cannot use [6, Theorem 5.1]. We shall only prove Theorem 2.9 in the case µ > 0, as the other case is all but identical. In view of Remark 2.7, we need to show that, for any T > 0, which we consider fixed throughout this proof, the restriction ofŶ (n) to [0, T ) converges in (D([0, T )), J 2 ) to the restriction of W (α) • µid to [0, T ). By a double use of the Skorokhod Representation Theorem, there exist two probability spaces (Ω 1 , P 1 ) and (Ω 2 , P 2 ), and processeŝ Sinceω (n) andS (n) are independent, we regard the processes (3.23) as defined on (Ω 1 × Ω 2 , P 1 × P 2 ), so all the joint distributions of processes in boldface type are the same as for the corresponding processes in regular type. Also, in the interest of readability and confident there will be no confusion, we slightly abuse the notation and write the boldface processes in regular type.

p(i) nT
property that, for all u ∈ B i , S u ≤ S p(i) , see Figure 3. Observe that, for small values of p(i), B i might be empty. These considerations show that On the other hand, since S p(i) is the (i + 1)-th smallest value of the set {S j } nT −1 j=0 , we know that |L i | = i + 1. From the above inequality, then, (3.43) We proceed analogously to produce a lower bound for |U i |. Set A i := [a i,n , nT ), where a i,n = p(i) + 2C η n 1/α /µ. Figure 3 shows that S t ≥ S p(i) for all t ∈ A i , whence |A i ∩ U i | = nT − a i,n + 1. On the other hand, i ≤ a i,n + 1 = p(i) + 2 C η n 1/α µ + 1,  as n → ∞. This is in fact a consequence of the following uniform convergence: But the restrictions of Z (β) cadlag and Z (β) to [0, µT ) coincide, so will obtain (3.33) when we prove thatω (D([0, T )), J 1 ), as n → ∞. We will show more, namely that, for some C > 0, where the numbers ζ j , for j ∈ Z − , are fixed, as the realization γ 1 ∈ Ω 1 of the medium is fixed. Now, the realization γ 2 ∈ B η of the underlying random walk is also fixed. Since the drift µ is positive, S n < 0 occurs only for a finite number of times n. The values of these excursions below zero and their times are contained in this chain of inequalities We are left to prove that τ n = ρ n • λ n satisfies (3.36). We do so with the help of which converges to 0 as n → ∞ by (3.34) and (3.39). This finally shows thatω (n) •S (n) → Z (β) • µid in (D([0, T )), J 2 ) for all (γ 1 , γ 2 ) ∈ Ω 1 × B η , concluding the proof of Theorem 2.9.
Case α = β Except for certain complications, the proof of this case will follow the same ideas as that of Theorem 2.9 in Section 3.3. We will detail the parts that need a new argument and describe quickly those that are proved exactly as done earlier.
In view of (3.53), we rewrite our process of interest as where h(x, y) := x • y is the composition map from D × D + to D + and (x, y) := x + y is the addition map from D + × D + to D + . Also δ n := νµ( nid − nid)/n 1/α is a negligible term, as n → ∞, in any relevant distance. As was done in Section 3.3, we use the Skorokhod Representation Theorem twice to obtain two probability spaces (Ω 1 , P 1 ) and (Ω 2 , P 2 ), and processesω with respectively the same distribution asω (n) ,Z (α) ,S (n) ,S (n) and W (α) , and such that Once again, since the processes relative to the medium and those relative to the dynamics are independent, it is correct to regard all boldface processes as defined on (Ω 1 × Ω 2 , P 1 × P 2 ). Again we simplify the notation and use the regular typeset for all processes (3.59). Let us define These are full-measure sets in their respective spaces. Notice that (essentially by the definition ofS (n) )S (n) [γ 2 ] → µid, for all γ 2 ∈ Ω 2 . Now let us fix T > 0. We already know that for any η ∈ (0, 1), there exist C η > 0 andn η ∈ N (both numbers depending on T as well) such that the set has measure P 2 (B η ) > 1 − η. Now one proceeds as in the proof of Theorem 2.9, using ω (n) andZ (α) in place ofω (n) and Z (β) , respectively. The fact that now β = α ∈ (1, 2) causes no breaks in the proof. One obtains that, for all (γ 1 , for all realizations (γ 1 , γ 2 ) ∈ Ω 1 × B η . Since η ∈ (0, 1) is arbitrary, the above limit extends to an almost sure limit in (Ω 1 × Ω 2 , P 1 × P 2 ). Passing to distributional convergence and (freely) varying the choice of T , we finally achieve (2.39) for the case µ > 0.  The latter assertion amounts to the claim that d J2,I (x n , x) → 0 and d J2,I (y n , y) → 0, as n → ∞, imply d J2,I (x n + y n , x + y) → 0.
Proof. We follow the same line of arguments as in the proof of [24,Theorem 4.1]. In fact, the measurability of is proved exactly as in the referenced theorem. As for the continuity claim, we fix I := [a, b), which is the case needed in Section 3.4. The other three cases, I = [a, b], I = (a, b] or I = (a, b), are proved exactly in the same way. Without loss of generality, we also assume to work with càdlàg functions (as opposed to functions that have a càdlàg and a càglàd restriction). This case happens, e.g., if 0 ≤ a < b. For a fixed ε > 0, we must show that there existn ∈ Z + and, for all n ≥n, bijections From this construction we have that the discontinuity points of x (respectively y) with jump size bigger than ε/8 are contained in P x (respectively P y ). By hypothesis these two sets of points are disjoint. Moreover, we can select the other points of P x and P y so that P x ∩ P y = {a, b}. Let 4δ be the distance between the closest pair of points of P := P x ∪ P y . For i = 1, . . . , n and j = 1, . . . , m, we construct closed intervals J This implies in particular that these intervals are pairwise disjoint. Now let us assume that there existn ∈ Z + and bijections µ n , ν n : for all i = 1, . . . , n and j = 1, . . . , m. The first and second conditions in both (A.10) and (A.11) can be satisfied by the hypotheses x n → x, y n → y in (D([a, b)), J 2 ). We postpone for a moment the proof that µ n , ν n can be found to satisfy the third conditions as well.
by (A.10), and (A.14) In the first term of the final estimate of (A.14) we have renamed u := µ −1 n • ν n (t) and used (A.10). For the second term we have observed that, by (A.10)-(A.11), the bijection µ −1 n • ν n is closer to the identity than 2δ. Since t ∈ J (y) j this implies, by (A.9), that t and µ −1 n • ν n (t) belong to the same interval [t i−1 , t i ), for some i. where we have used the same arguments as for (A.14): observe in fact that if t ∈ J then, by (A.8)-(A.9), t is at distance larger than δ from P \ {a, b}. By (A.10) |µ −1 n (t) − t| < δ and so t and µ −1 n (t) belong to the same interval [t i−1 , t i ), for some i, triggering (A.6). From (A.13)-(A.15) we have that sup t∈[a,b) |x n • λ n (t) − x(t)| < ε/2, and the same obviously holds for y, whence We proceed by explicitly constructing µ n , as the construction of ν n is completely analogous.  where A i ∩ (a i , a i + η) has the cardinality of the continuum because, by the first inequality of (A.22), there exists σ > 0 such that (a i , a i + σ) ⊂ A i ∩ (a i , a i + η). By reasons of cardinality, then, there exists a bijection φ − i : (a i , a i ) −→ A i ∩ (a i , a i + η). By construction, since a i = a i − η, sup t∈(a i ,ai) |φ − i (t) − t| ≤ 2η < min{ε, δ}.