Random walks in a moderately sparse random environment

A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk ( X n ) n ∈ N ∪{ 0 } in a sparse random environment ( S k , λ k ) k ∈ Z is a nearest neighbor random walk on Z that jumps to the left or to the right with probability 1 / 2 from every point of Z \ { . . . , S − 1 , S 0 = 0 , S 1 , . . . } and jumps to the right (left) with the random probability λ k +1 ( 1 − λ k +1 ) from the point S k , k ∈ Z . Assuming that ( S k − S k − 1 , λ k ) k ∈ Z are independent copies of a random vector ( ξ, λ ) ∈ N × (0 , 1) and the mean E ξ is ﬁnite (moderate sparsity) we obtain stable limit laws for X n , properly normalized and centered, as n → ∞ . While the case ξ ≤ M a.s. for some deterministic M > 0 (weak sparsity) was analyzed by Matzavinos et al., the case E ξ = ∞ (strong sparsity) will be analyzed in a forthcoming paper.


Introduction
Simple random walks on Z (the set of integers) arise in various areas of classical and modern stochastics. However, their intrinsic homogeneity reduces in some situations applicability of the simple random walks. Solomon [36] eliminated this drawback by introducing a random environment which made a modified random walk space inhomogeneous. In the present article we investigate an intermediate model, called random walk in a sparse random environment (RWSRE), in which homogeneity of an environment is only perturbed on a sparse subset of Z. Since RWSRE is a particular case of a random walk in a random environment (RWRE) we proceed by recalling the definition of the latter.
Set Ω = (0, 1) Z and X = Z N . Let F be the Borel σ-algebra of subsets of Ω, P a probability measure on (Ω, F) and G the σ-algebra generated by the cylinder sets in X . A random environment is a random element ω = (ω n ) n∈Z of the measurable space (Ω, F) distributed according to P . A quenched (fixed) environment ω provides us with a probability measure P ω on X whose transition kernel is given by With the initial condition X 0 := 0 the sequence X = (X n ) n∈N0 is a Markov chain on Z (under P ω ) which is called random walk in the random environment ω. Here and hereafter, N 0 := N ∪ {0}. It is natural to investigate RWRE from two viewpoints which are different in many aspects: under the quenched measure P ω for almost all (with respect to P ) ω, that is, for a typical ω or under an annealed measure. Formally, the annealed measure P on (Ω × X , F ⊗ G) is defined as a semi-direct product P = P P ω Random walks in a moderately sparse random environment Note that in general X is no longer a Markov chain under P. Usually one assumes that an environment ω forms a stationary and ergodic sequence or even a sequence of iid (independent and identically distributed) random variables. In this setting RWRE has attracted a fair amount of attention among probabilistic community resulting in quenched and annealed limit theorems [3,11,12,25,26,35,37] and large deviations [5,7,9,15,19,33,34,38,39]. This list of references is far from being complete.
We aim at establishing annealed limit theorems for X (that is, under P) in a so called sparse random environment which corresponds to a particular choice of P which is specified as follows. Let ((ξ k , λ k )) k∈Z be a sequence of independent copies of a random vector (ξ, λ) which satisfies λ ∈ (0, 1) and ξ ∈ N a.s. For n ∈ Z, set The sparse random environment ω = (ω n ) n∈Z is defined by ω n = λ k+1 , if n = S k for some k ∈ Z, The model (with λ k in (1.1) replacing λ k+1 ) was introduced by Matzavinos, Roitershtein and Seol [30]. These authors obtained various results including a recurrence/transience criterion, a strong law of large numbers and limit theorems. However, many results in [30] were proved under quite restrictive conditions including boundedness of ξ, a strong ellipticity condition for the distribution of λ and independence of ξ and λ. In this setting some essential properties of X remain hidden. Our main purpose is to relax the aforementioned assumptions substantially, thereby establishing limit theorems in full generality, and to find out how distributional properties of the vector (ξ, λ) affect the asymptotic behavior of X. It turns out that the asymptotics of X is regulated by the tail behaviors of ξ and ρ := (1 − λ)/λ which determine sparsity of the environment and the local drift of the environment, respectively. In this paper we investigate the case where Eξ < ∞. We call the corresponding environment 'moderately sparse', whereas in the opposite case where Eξ = ∞ we say that the environment is 'strongly sparse'. The analysis of X in a strongly sparse environment requires completely different techniques and will be carried out in a companion paper [6].
The present article is organized as follows. In Section 2 we formulate our limit theorems for X and the first passage times of X. In Section 3.1 we describe our approach and define a branching process Z in a random environment which is used to analyze the random walk X. In Section 3.2 we introduce necessary notation related to the process Z. In Section 4 we explain a heuristic behind our proof and present a number of important estimates and decompositions used throughout the paper. Among other things, we demonstrate in this section how to reduce the initial problem to the asymptotic analysis of sums of certain iid random variables. The tail behavior of these variables is discussed in Section 5. Section 6 is devoted to the analysis of a particular critical Galton-Watson process with immigration which naturally arises in the context of random walks in the sparse random environment. The proofs of the main results are given in Sections 7.1, 7.2 and 7.3. The proofs of auxiliary lemmas can be found in Section 7.4 and the Appendix.

Main results
We focus on the case when X is P-a.s. transient to +∞ and the environment is moderately sparse, that is, Eξ < ∞. Recall the notation ρ = 1 − λ λ .
According to Theorem 3.1 in [30], X is P-a.s. transient to +∞ if E log ρ ∈ [−∞, 0) and E log ξ < ∞. (2.1) The first inequality excludes the degenerate case ρ = 1 a.s. in which X becomes a simple random walk. The second inequality is always true for the moderately sparse environment. We note right away that our standing assumptions E log ρ ∈ [−∞, 0) and Eξ < ∞ hold under the conditions of our main results, Theorems 2.2 and 2.6. The sequence (T n ) n∈Z of the first passage times defined by T n = inf{k ≥ 0 : X k = n}, n ∈ Z is of crucial importance for our arguments. Of course, the observation that the asymptotics of X can be derived from that of (T n ) is not new and has been exploited in many earlier papers in the area of random walks in random environments. Assuming only transience to the right it is shown on p. 12 in [30] that lim n→∞ T Sn n = ET S1 P − a.s.
In Proposition 2.1 we give an explicit formula for v when ξ and λ are allowed to be dependent.

(B2) (ξ dominates)
In what follows, for α ∈ (0, 2), we denote by S α a random variable with an α-stable 2). Note that S α is a positive random variable when α ∈ (0, 1) and it has a spectrally positive α-stable distribution when α ∈ [1, 2). Throughout the paper d −→ and P −→ will mean convergence in probability and convergence in distribution, respectively. In Theorem 2.2 and Corollary 2.4 we treat the case (P 1).

Theorem 2.2.
Assume that one of the following sets of assumptions is satisfied: (A1) (P 1) holds for some α ∈ (0, 2], (Ξ1) holds and E(ρξ) α < ∞; 1 In some cases we also need additional technical assumptions concerning the joint distribution of ρ and ξ, for instance, E(ρξ) α < ∞. These will be stated explicitly in the corresponding theorems.
Then there exist absolute constants A α , B α and C 1 such that the following limit relations hold as n → ∞.
• If α ∈ (0, 1), then Tn • If α ∈ (1, 2), then Tn−Aαn Remark 2.3. See (7.11), (7.12) and (7.14) for explicit forms of the constants A α , B α and C 1 . In Theorem 2.2 we do not specify the constants by two reasons. First, these involve characteristics of random variables that have not been introduced so far. Second, some of these constants are essentially implicit in the sense that these cannot be calculated.
From Theorem 2.2 we deduce the following corollary.
Corollary 2.4. Under the assumptions and notation of Theorem 2.2 the following limit relations hold as k → ∞.
• If α ∈ (1, 2), then Remark 2.5. When α ∈ (0, 1) the distribution of S −α α is called the Mittag-Leffler distribution with parameter α. The term stems from the facts that , u ∈ R and that the right-hand side defines the Mittag-Leffler function with parameter α.
Our next theorem treats weak convergence of T n in cases where ξ plays a dominant role.
Before formulating the corresponding limit theorems for X k we need to introduce more notation. For α ∈ (1/2, 1), denote by c ← α (t) any positive function satisfying exist by Theorem 1.5.12 in [2].
Corollary 2.8. Under the assumptions and notation of Theorem 2.6 the following limit relations hold as k → ∞.
for appropriate sequences s(k) and t(k) which are specified in formula (7.21).
• If α ∈ (1, 2), then The last result of this section is given for completeness only. It can be derived from a general central limit theorem (Theorem 2.2.1 in [40]) for random walk in a stationary and ergodic random environment. Since the sparse random environment is not stationary in general, to apply this theorem one has to pass to a stationary and ergodic environment.

Branching processes in random environment with immigration
The connection between a random walk and a branching process with immigration dates back to Harris [22]. In the context of a random walk in a random environment this connection was successfully used by Kozlov [29] and Kesten, Kozlov and Spitzer [26]. In particular, these authors have shown that the asymptotic behavior of RWRE can be obtained from that of the total progeny of the aforementioned branching process. Since we are going to exploit the same idea we first recall a construction of the latter process. Most of the material in Section 3.1 can be found in [26].

Branching process with immigration
Throughout the paper the fact that X n → ∞ P-a.s. plays a crucial role. Let U (n) i be the number of steps of the process X from i to i − 1 during the time interval [0, T n ), that is, Since X Tn = n and X 0 = 0 we have, for n ∈ N, T n = # of steps during [0, T n ) = # of steps to the right during [0, T n ) + # of steps to the left during [0, T n ) = n + 2 · # of steps to the left during [0, T n ) Recalling that the random walk X is transient to the right we infer In particular, for any γ > 0, Thus, the asymptotics of T n as n → ∞ is regulated by that of n + 2 n i=0 U (n) i .
In what follows, we write Geom(p) for a geometric distribution with success probability p, that is, Claim. Let ω and n be fixed. Then, for 0 ≤ j ≤ n, U is the number of excursions to the left of n − 1 made by X before time T n . Transitivity of X entails that the P ω -distribution of V (n−1) 0 is Geom(ω n−1 ). Finally, for 2 ≤ j ≤ n − 1, we have where V (n−j) 0 denotes the number of excursions to the left from n − j before the first excursion to the left from n − j + 1 (that is, before the time T n−j+1 ) and V (n−j) k denotes the number of excursions to the left from n − j during the kth excursion to the left from n − j + 1. Under P ω , the random variables (V (n−j) k ) k≥0 are iid with distribution Geom(ω n−j ) and also independent of U (n) n−j+1 . The proof of the claim is complete.
Reversing the order of indices leads to a branching process Z = (Z k ) k≥0 in a random environment (BPRE) with one immigrant entering the system in each generation. From the very beginning we stress that immigrants in our model are 'artificial', that is, even though they reproduce, they do not belong to any generation and, as such, they are not counted. The evolution of Z can be described as follows. An immigrant enters the 0th generation which is originally empty, that is, Z 0 = 0. She gives birth to a random number of offspring with P ω -distribution Geom(ω 1 ) which form the first generation. For n ∈ N, an immigrant enters the nth generation. She and the particles of the nth generation, independently of each other and the particles in the previous generations, give birth to random numbers of offspring with P ω distribution Geom(ω n+1 ). The number of these newborn particles which form the (n + 1)st generation is given by is the number of offspring of the (n + 1)st immigrant and, for k ∈ N, G (n) k is the number of offspring of the kth particle in the nth generation (we set G (n) k = 0 if the kth particle in the nth generation does not exist). Observe that, under P ω , for each n ∈ N 0 , the random variables (G (n) k ) k≥0 are iid with distribution Geom(ω n ) and also independent of Z n .
Note that when the random environment is sparse (see (1.1)) and fixed, for the most time, the branching process Z behaves like a critical Galton-Watson process with one immigrant and Geom(1/2) offspring distribution. Only the particles of generation S i − 1 for i ∈ N as well as the immigrants arriving in this generation reproduce according to Geom(λ i ) distribution. Averaging over ω and taking into account the structure of the environment we obtain   where O P (1) is a term which is bounded in probability. Distributional equality (3.3) will prove useful on many occasions.

Notation
Before we explain the strategy of our proof some more notation have to be introduced.
Denote by Z(k, n) the number of progeny residing in the nth generation of the kth immigrant. In particular, Z(k, k) is the number of offspring of this immigrant. Then Z n = n k=1 Z(k, n).
For n ∈ N and 1 ≤ i ≤ n, let Y (i, n) denote the number of progeny in the generations i, i + 1, . . . , n of the ith immigrant, that is, Similarly, for i ∈ N, we denote by Y i the total progeny of the ith immigrant, that is, We also define W n to be the total population size in the first n generations, that is, Motivated by the structure of the environment we shall often divide the population into blocks which include generations 1, . . . , S 1 ; S 1 + 1, . . . , S 2 and so on. As a preparation, we write Z n = Z Sn , n ∈ N for the number of particles in the generation S n , for the total population in the generations S n−1 + 1, . . . , S n and Y n = Sn j=Sn−1+1 Y j , n ∈ N for the total progeny of immigrants arriving in the generations S n−1 , . . . , S n − 1.

Analysis of the environment
The asymptotic behavior of the branching process Z depends heavily upon the environment. At the end of this section we specify qualitatively two aspects of this dependence. A random difference equation which arises naturally in the course of our discussion, as well as in [26] and many other papers on RWRE, plays an important role in the subsequent arguments.
We proceed by recalling the definitions of random difference equations and perpetuities. Let (A n , B n ) n∈N be a sequence of independent copies of an R 2 -valued random vector (A, B). Further, let R 0 be a random variable which is independent of (A n , B n ) n∈N . The sequence (R k ) k∈N0 , recursively defined by the random difference equation  actuarial application. The study of the random difference equations and perpetuities has a long history going back to Kesten [24] and Grincevičius [17]. We refer the reader to the recent monographs [4,23] containing a comprehensive bibliography on the subject.
It is well-known that conditions E log |A| ∈ [−∞, 0) and E log + |B| < ∞ are sufficient for (3.4) and the distributional convergence R k d −→ R * ∞ as k → ∞. There are numerous results in the literature concerning the tail behavior of R * ∞ . The first assertion of this flavor is the celebrated theorem by Kesten [24] (see also Goldie [16] and Grincevičius [18]), to be referred to as the Kesten-Grincevičius-Goldie theorem. It states that the distribution of R * ∞ has a heavy right tail under the assumptions A > 0 a.s., EA s = 1 for some s > 0 and some additional conditions, see formula (7.39) below for more details in the particular case (A, B) = (ρ, ξ). The tail behavior of R * ∞ is also well understood in some other cases, in particular, when P{|B| > x} is regularly varying at ∞ (see, for instance, [18], [20] and [8]). Now we switch attention from the general random difference equations to a particular one which features in the analysis of BPRE Z. Using the branching property one easily obtains the following recurrencē This shows, among others, that the Markov chain (R k ) k∈N0 is an instance of the random difference equation which corresponds to (A, B) = (ρ, ρξ). Asymptotic distributional properties of a particular perpetuity which corresponds to (A, B) = (ρ, ξ) are essentially used in the proof of Lemma 7.2.

Proof strategy
A weak convergence result for T n , properly normalized and centered, will be derived from the corresponding result for T Sn , again properly normalized and centered. In view of (3.3), the latter may in principle be affected by the asymptotic behavior of S n , W Sn or both. Fortunately, the contribution of S n is degenerate in the limit, for it is only regulated by the law of large numbers, fluctuations of S n around its mean do not come into play. Summarizing, analysis of the asymptotics of W Sn is our dominating task.
While dealing with W Sn our main arguments follow the strategy invented by Kesten et al. [26]. Namely, for large n we decompose W Sn as a sum of random variables which are iid under the annealed probability P. For this purpose we define extinction times Let us emphasize that the extinctions of Z are ignored in the generations other than S 1 , S 2 , . . . SetW where τ * n is the number of extinctions of Z in the generations S 0 , . . . , S n , that is, It turns out that the extinctions occur relatively often as the following lemma confirms.
The proof of Lemma 4.1 is given in the Appendix.
Under the assumptions of our main results µ := Eτ 1 < ∞ by Lemma 4.1. The strong law of large numbers for renewal processes makes it plausible that, for large n, the behavior of W Sn is comparable with the behavior of the sum µ −1 n k=1W τ k . The latter, properly centered and normalized, converges in distribution if and only if the distribution ofW τ1 belongs to the domain of attraction of a stable law. To check the latter, for i ∈ N, we divide particles residing in the generations S i−1 + 1, . . . , S i into groups: • P 1,i -the progeny residing in the generations S i−1 + 1, . . . , S i − 1 of the immigrants arriving in the generations S i−1 , . . . , S i − 2, the number of these being • P 2,i -the progeny residing in the generations S i−1 + 1, . . . , S i − 1 of the immigrants arriving in the generations 0, 1, . . . , S i−1 − 1, the number of these being Figure 1: The generations 0 through S 3 of the BPRE Z and the partition of the corresponding population into parts P i,j , i, j = 1, 2, 3. The bold horizontal lines represent particles in the generations S 1 , S 2 and S 3 , that is, those comprising the groups P 3,i , i = 1, 2, 3. By definition, P 2,1 = .
• P 3,i -particles of the generation S i , the number of these being Z i .
The aforementioned partition of the population which is depicted on Figure 1 induces the following decompositions which are of primary importance for what follows.
Depending on the assumptions (P 1), (P 2), (Ξ1) or (Ξ2) the random variables Z i may exhibit different tail behaviors. Often, one of the random variables dominates the others thereby determining the tail behavior of the whole sum W τ1 .

Tail behavior ofW τ 1
In this section we do not assume that Eξ < ∞.
We first analyze the tail behavior of τ1 i=1 W 0 i . Note that by construction (W 0 i ) i∈N are iid and the random variable τ 1 does not depend on the future of the sequence (W 0 i ) i∈N in the sense of the definition given by Denisov, Foss, Korshunov on p. 987 in [10]. The latter means that, for each n ∈ N, the collections of random variables ((W 0 k ) k≤n , 1 {τ1≤n} ) and (W 0 k ) k>n are independent. This observation in combination with Corollary 3 in [10] and Theorem 1 in [28] yields the following lemma which will be used many times throughout the paper.

Lemma 5.2.
Assume that (2.5) holds with some β > 0. Then where ϑ is a random variable with Laplace transform The proof of Lemma 5.2 is given in Section 6. In the next two lemmas we provide moment estimates for the two other summands Assume that E log ρ ∈ [−∞, 0) and that, for some κ ≤ 2, E(ρξ) κ and Eξ κ are finite. Then EZ κ 1 < ∞ and there exists a positive constant C such that, for all n ∈ N, Remark 5.4. Since ξ ≥ 1 a.s., the assumption E(ρξ) κ < ∞ entails Eρ κ < ∞. This explains the absence of the latter condition in Lemma 5.3.
Lemma 5.7 points out the tail behavior ofW τ1 in the situation where the slowly varying factor in (Ξ2) is a constant.
The proofs of Lemmas 5.3 through 5.7 are postponed until Section 7.4. For the ease of reference the tail behavior ofW τ1 is summarized in the following proposition.
Proposition 5.8. The following asymptotic relations hold.
Proof of (C1). Each of Eξ 2α < ∞ and (Ξ2) with β = 2α implies Eξ 3α/2 < ∞. Therefore, in view of Lemma 5.6 it is enough to show that If (Ξ2) holds with β = 2α, then according to Lemma 5.2 This in combination with lim t→∞ (t) = 0 which holds by assumption and (5.1) proves Assuming that Eξ 2α < ∞ we intend to show that which, of course, entails (5.7). The proof of (5.8) utilizes two technical lemmas whose formulations and proofs are postponed until later. Since τ 1 does not depend on the future At the beginning of Section 6 we show that W 0 1 has the same distribution as the total progeny of a critical Galton-Watson process with unit immigration and Geom(1/2) offspring distribution stopped at random time ξ 1 − 1. The conclusion E[W 0 1 ] α < ∞ then follows from Lemma 6.3.

Critical Galton-Watson process with immigration
As has already been mentioned in Section 3, where ξ 1 is assumed independent of (Z crit n ) n∈N0 a critical Galton-Watson process with unit immigration and Geom(1/2) offspring distribution. In this section we collect some known properties of (Z crit n ) n∈N0 and prove several auxiliary results which to our knowledge are not available in the literature. The evolution of (Z crit n ) n∈N0 is the same as that of the BRPE Z with ω n ≡ 1/2 for all n ∈ N, see Section 3.1.
For n ∈ N, let W crit n := n k=1 Z crit k denote the total progeny in the first n generations. Further, for n ∈ N and 1 ≤ k ≤ n, write Z crit (k, n) for the number of the nth generation progeny of the kth immigrant and Y crit (k, n) for the number of progeny of the kth immigrant which reside in generations k through n, that is, Here is the main result of this section of which Lemma 5.2 is an immediate conse- where ξ 1 is assumed independent of (W crit k ) k∈N . Proposition 6.1. Let ς be an integer-valued random variable independent of (W crit n ) n∈N0 and such that for some α > 0 and some slowly varying at ∞. Then where ϑ is a random variable with Laplace transform (5.2).
and the distribution of W crit n inherits an exponential tail from Geom(1/2) offspring distribution. Thus, for ς which has distribution with a heavy tail and is independent of (W crit n ) n∈N it is natural to expect that Proposition 6.1 makes this intuition precise. Lemma 6.3 given next is used in the proof of Proposition 5.8, part (C1). Lemma 6.3. Let ς be an integer-valued random variable independent of (W crit n ) n∈N0 and such that Eς 2α < ∞ for some α > 0.
To prove Proposition 6.1 and Lemma 6.3 we need some auxiliary lemmas. The first one is due to Pakes [32,Theorem 5].
where ϑ is a random variable with Laplace transform (5.2).
In the cited article Pakes investigates Galton-Watson processes with general, not necessarily unit, immigration. One of the standing assumptions of that paper is that the probability of having no immigrants is positive. However, a perusal of the proof of Theorem 5 in [32] reveals that the result still holds without this assumption.
With some additional effort one can prove the convergence of all moments in (6.1).
Proof. Suppose for the moment that we have verified that for some β > 0 and some n 0 ∈ N. Then in view of for all s > 0 and some constant C(s), the Vallée-Poussin criterion for uniform integrability (see e.g. Theorem T22 in [31]) in combination with (6.1) ensures (6.2).
Left with the proof of (6.3) observe that, for fixed k ∈ N, the process initiated by the kth immigrant (Z crit (k, n)) n≥k is a Galton-Watson process with Geom(1/2) offspring distribution. Moreover, the processes started by different immigrants are iid. Therefore, writing we obtain a representation of W crit n as the sum of independent random variables. This formula entails (the case that both sides of (6.4) are infinite for some x > 0 is not excluded), where We have a 0 (x) = 1 for all x ≥ 0 and , j ∈ N.
In particular, for every fixed j ∈ N 0 , a j (x) < ∞ for all x from some right vicinity of the origin.
We are now ready to prove Proposition 6.1 and Lemma 6.3.
Proof of Proposition 6.1. By virtue of (6.1) we infer W crit n → ∞ in probability and then W crit n → ∞ a.s. by monotonicity. Therefore, where h(y) := P{ς ≥ y}. Under the introduced notation, we have to prove that By a standard inversion technique à la Feller (see Theorem 7 in [13]) (6.1) entails We claim that the latter implies further that The simplest way to see it is to pass in (6.8) to versions which converge a.s., that is, and then exploit the fact that (see Theorem 1.5.2 in [2]). This gives because ϑ * > 0 a.s. With (6.9) at hand, relation (6.7) follows if we can show that h(υ x )/h(x 1/2 ) x≥x0 is uniformly integrable for some x 0 > 0. By Potter's bound for regularly varying functions (Theorem 1.5.6 (iii) in [2]), given A > 1 and δ > 0 there exists n 1 ∈ N such that Thus, for uniform integrability of h(υ x )/h(x 1/2 ) x≥x0 it suffices to check two things: for some β > 2α and second for some γ > 1.
From the proof of Lemma 6.5 we know that E exp(sW crit n1 ) < ∞ for some s > 0, whence which proves (6.11).
Now we intend to show that (6.10) holds for all β > 0. We have for x ≥ 4 where the last and penultimate inequalities follow from Lemma 6.5 and Markov's inequality, respectively. The proof of Proposition 6.1 is complete.
Proof of Lemma 6.3. By Lemma 6.5, E[n −2 W crit n ] α ≤ C for all n ∈ N and some C > 0.
This entails The proof of Lemma 6.3 is complete.

Proof of Proposition 2.1
Recalling that v = Eξ/ET S1 it suffices to show that Let us prove the latter convergence in probability. According to Lemma 4.1, we have Left with identifying EW τ1 we recall that, for k ∈ N, Y k denotes the total progeny of immigrants arriving in the generations S k−1 , . . . , S k − 1, that is, where the a.s. convergence of the last series is secured by our assumptions E log ρ ∈ [−∞, 0) and Eξ < ∞. Taking the expectation with respect to P yields The proof of Proposition 2.1 is complete.

Proof of Theorem 2.2 and Corollary 2.4
The assumptions of Theorem 2.2 ensure that Eξ < ∞ and that µ := Eτ 1 and s 2 := Var τ 1 are finite (for the latter use Lemma 4.1). It is also clear that the distribution of τ 1 is nondegenerate, whence s 2 > 0.
From Proposition 5.8 (parts (C1) and (C2)) we know that where C = C 2 (α) in the cases (A1) and (A2) and C = (Eτ 1 )(Eϑ α )C + C 2 (α) in the case (A3). Therefore, the distribution ofW τ1 belongs to the domain of attraction of an α-stable distribution. This means that Our subsequent proof will be based on representation (3.3). In view of this we first analyze the asymptotics of W Sn .
Step 1. Limit theorems for W Sn . We claim that In view of (4.2) relation (7.2) follows once we have checked that (7.1) entails According to the central limit theorem for renewal processes This implies that, for ε > 0 small enough, we can pick z = z(ε) so large that   Motivated by our later needs we have proved this in a slightly extended form with r instead of 2.
To prove the first relation in (7.3) we write, for x ∈ R, Sending n → ∞ in the last inequality and using (7.1) and (7.4) we obtain lim sup Random walks in a moderately sparse random environment The second relation in (7.3) follows in a similar manner.
Step 2. Limit theorems for T Sn .
follows from (7.2)  Case α = 1. Using the weak law of large numbers and (7.2) we arrive at  Step 3. Limit theorem for T n . At this step we are going to deduce limit theorems T n from the corresponding results for T Sn proved at the previous step. Set ν(t) = inf{k ∈ N : S k > t}, t ≥ 0, so that (ν(t)) t≥0 is the first passage time process associated with the random walk (S k ) k∈N0 . The reason for introducing ν(t) is justified by Case α ≥ 1. Fix any r ∈ (1, 2). Then Eξ r < ∞ and thereupon Subcase α = 1. Using (7.9) and (7.10) we obtain, for any x ∈ R and ε > 0, Random walks in a moderately sparse random environment Letting n → ∞ yields, for x ∈ R, having utilized (7.5), (7.7) and (7.10). Arguing similarly we get the converse inequality for the lower limit, thereby proving that Subcase α > 1. An analogous but simpler argument enables us to show that (7.6) entails Case α < 1. The proof given for the case α ≥ 1 does not work in the case (A1) when α ≤ 1/2 because it is then not necessarily true that Eξ r < ∞ for some r > 1. In view of this we use the weak law of large numbers rather than the Marcinkiewicz-Zygmund strong law (7.10).
Another appeal to (7.9) gives, for any x ∈ R and ε > 0, Sending n → ∞ we obtain with the help of (7.8) and (7.13) lim sup Letting ε → 0+ and using continuity of the distribution of S α yields lim sup The converse inequality for the lower limit can be derived analogously. Thus, 14) The proof of Theorem 2.2 is complete.
Proof of Corollary 2.4. The forms of limit relations for T n in our Theorem 2.2 and Theorem on pp. 146-148 in [26] are the same, only the values of constants differ. In view of this the limit relations for X k in our setting are obtained by copying the corresponding limit relations from the aforementioned theorem in [26].

Proof of Theorem 2.6 and Corollary 2.8
The proof goes the same path as that of Theorem 2.2. However, appearance of nontrivial slowly varying factors leads to minor technical complications. We shall only give the weak convergence results explicitly (recall that in the formulation of Theorem 2.6 normalizing and centering functions were not specified). Also, we shall check several claims wherever we feel it is necessary.
To identify them we need more notation. For α ∈ (1/2, 2), let c α (t) be any positive function satisfying lim t→∞ tP{W τ1 > c α (t)} = 1. Further, assuming that α = 2 let r 2 (t) be any positive function satisfying lim t→∞ [0, r2(t)] x 2 dP{W τ1 ≤ x}/(r 2 (t)) 2 = 1. By Lemma 6.1.3 in [23], c α (t) and r 2 (t) are regularly varying at ∞ of indices 1/α and 1/2, respectively. For the latter, the fact is also needed that the function t → [0, r2(t)] x 2 dP{W τ1 ≤ x} is slowly varying at ∞. Observe that the case α = 2 only arises under the assumptions (B1) which then ensure that Eξ 2 = ∞. This in combination with the aforementioned lemma yields lim t→∞ t −1/2 r 2 (t) = ∞. (7.15) Using again Theorem 3 on p. 580 and formula (8.15) on p. 315 in [14] we obtain b(t) = c α (t) and a(t) = 0 if α ∈ (1/2, 1); 2); b(t) = r 2 (t) and a(t) = (EW τ1 )t if α = 2. Case α ∈ (1/2, 1). Repeating verbatim the proof of Theorem 2.2 for the case α ∈ (0, 1)  Case α = 1. We need an analogue of relation (7.5): for r ∈ (1, 2], as n → ∞, The first summand tends to zero in view of two facts: lim n→∞ t n P{W τ1 > c 1 (t n )} = 1 by the definition of c 1 (t) and lim n→∞ c 1 t n + O t 1/r n − c 1 (t n ) /c 1 (t n ) = 0 which is a consequence of regular variation of c 1 (t). The second summand tends to zero because c1(t) 0 P{W τ1 > x}dx is slowly varying at ∞ as a superposition of the slowly varying and regularly varying functions. For Step 2 in the proof of Theorem 2.2 we need the following modified argument. In view of (ξ2) the function P{ξ > t} is regularly varying at ∞ of index −2 and Eξ 2 can be finite or infinite. Therefore, S n satisfies the central limit theorem with normalization sequence which is regularly varying at ∞ of index 1/2. Since c 1 (t) is regularly varying at ∞ of order 1 we infer S n − (Eξ)n c 1 (n) P −→ 0, n → ∞ and thereupon To pass from this limit relation to the final result (7.17) that is, to realize Step 3 in the proof Theorem 2.2, one can mimic the proof of Theorem 2.2.
Case α ∈ (1, 2]. While implementing Step 2 of the previous result in the case α = 2 one uses the fact that according to (7.15) Since the other parts of the proof of Theorem 2.2 do not require essential changes we arrive at (7.18) when α ∈ (1, 2), and 2N (0, 1), n → ∞, (7.19) when α = 2. The proof of Theorem 2.6 is complete.
Proof of Corollary 2.8. Since (T n ) n∈N0 is an 'inverse' sequence for (X k ) k∈N0 we can use a standard inversion technique (see, for instance, the proof of Theorem 7 in [13]) to pass from the distributional convergence of T n , properly centered and normalized, as n → ∞ to that of X k , again properly centered and normalized, as k → ∞. Additional complications arising in the case α = 1 can be handled with the help of arguments given in Section 3 of [1].
is the number of progeny residing in the generation S n − 1 of the jth particle in the generation S n−1 and V (n) is the number of progeny residing in the generation S n − 1 of the immigrants arriving in the generations S n−1 , . . . , S n − 2. For later use, we note that, under P ω , (7.25) where ω is assumed independent of (Z crit k ) k∈N0 a Galton-Watson process with unit immigration and Geom(1/2) offspring distribution.
With the help of (7.24) we now write a standard decomposition for the number of particles in the generation S n over the particles comprising the generation S n−1 and their offspring The two cases κ ∈ (0, 1] and κ ∈ (1, 2] should be treated separately. Random walks in a moderately sparse random environment Case κ ≤ 1. By Jensen's inequality and subadditivity of the function s → s κ on [0, ∞) Taking the expectations we obtain EZ κ n ≤ γEZ κ n−1 + E(ρξ) κ which entails (5.3).
Case κ ∈ (1, 2]. An application of conditional Jensen's inequality yields To estimate the conditional second moment we represent it as follows Appealing now to (7.27) we conclude that E ω Z 2 n Z n−1 ≤ Z 2 n−1 ρ 2 n + Z n−1 E ω V 2 1 + 2Z n−1 ξ n ρ 2 n + E ω V 2 + 2ρ 2 n + ρ n + 2ξ n ρ 2 n . (7.29) Plugging the last inequality into (7.28) and using subadditivity once again we obtain EZ κ n ≤ γEZ κ n−1 + EZ Next, we check that With the help of which is a consequence of (7.25) and (6.12) we infer A similar argument in combination with E ω V = ξ n − 1 leads to the conclusion Left with the proof of finiteness of the first term on the right-hand side we represent V as a sum of independent random variables is the number of progeny residing in the generation S n − 1 of the immigrant arriving in the generation S n − i.
, where ω is assumed independent of (Z crit (i, k)) k≥i . With this at hand, an appeal to (6.12) Here and hereafter, to ease the notation we write V i for V which finishes the proof of (7.31).
Turning to the asymptotic behavior of EZ κ/2 n−1 which appears on the right-hand side of (7.30) we consider yet another two cases.
Since, under P ω , W ξ1−1 where ω is assumed independent of (W crit n ) n∈N0 , an application of Lemma 6.5 yields for a positive constant C. The proof of Lemma 5.3 is complete.
Therefore, we obtain having utilized Jensen's inequality, (7.32) and the fact that ξ i and Z i−1 are independent.

Proof of Lemma 5.6
We follow the method invented by Kesten et al. [26]. While some parts of the proofs given in [26] can be directly transferred to our setting, the others require an additional work. We do not present all the details of the proof focussing instead on the main differences. We begin with a brief overview of the arguments leading to the claim of Lemma 5.6.
Given a large positive constant A, put Thus, we observe the process (Z n ) n∈N0 up to the first time j when it exceeds the level A and then put σ = i for the smallest index i satisfying S i ≥ j. The following decomposition where S σ is the number of particles in the generation S σ plus their total progeny, and, for i ∈ N, Y ↓ i is the total progeny in the generations S i + 1, S i + 2, . . . of the immigrants arriving in the generations S i−1 , . . . , S i − 1.
We intend to prove that the first, second and fourth summands on the right-hand side of this decomposition are negligible in a sense to be made precise, so that In view of the definition of S σ and the fact that Z σ = Z Sσ ≈ A for A as above one can ∞)] is related to a random difference equation whose tail behavior determines that of S σ .
To realize the programme just outlined we need two auxiliary results.
Lemma 7.1. Assume that the assumptions of Lemma 5.6 hold. Then, for any A > 0, as x → ∞, Proof. We only give a proof for the first summand in (7.35). The second summand can be treated along similar lines.
Write, for x > 0, and observe that, in view of (7.36), the first summand on the right-hand side is o(x −3α/2 ) as x → ∞. To estimate the second term we use a decomposition where, for 1 ≤ i ≤ Z τ1−1 , V i is the number of progeny in the generations S τ1−1 + 1, . . . , S τ1 − 1 of the ith particle in the generation S τ1−1 . We claim that EV α 1 < ∞. (7.37) For the proof, note that V 1 (1, n)) n∈N . Consequently, we obtain with the help of Jensen's inequality and the inequality E[Y crit (1, n)] 2 ≤ 3n 3 for n ∈ N which is a consequence of (6.12) where the last inequality is secured by (7.36). With (7.37) at hand, we immediately conclude that The proof of Lemma 7.1 is complete.
Before formulating another auxiliary result we recall from Section 3.2 the notation is the number of progeny residing in the ith generation of the first immigrant, so that Y 1 is the total progeny of the first immigrant.

Lemma 7.2.
Suppose that the assumptions of Lemma 5.6 hold. Let (Y * j ) j∈N be a sequence of P ω -independent copies of Y 1 . Then there exists a constant C > 0 such that Random walks in a moderately sparse random environment Proof. For k ∈ N, put R k = ξ k + ρ k ξ k+1 + ρ k ρ k+1 ξ k+2 + . . . . (7.38) Recall from Section 3.3 that the so defined random variable is called perpetuity. The Kesten-Grincevičius-Goldie theorem says that if (P 1) holds and Eξ α < ∞, then, for all k ∈ N, P{ R k > x} ∼ Cx −α , x → ∞ (7.39) for some positive constant C which does not depend on k. Put Z(1, 0) := 1. For i ∈ N 0 , denote by Z 1 (1, i), Z 2 (1, i), . . . P ω -independent copies of Z(1, i). Recall that S k = S k−1 + ξ k and write Our proof will be based on the following decomposition which holds a.s.
Since k≥1 2 −1 k −2 = π 2 /12 < 1, (7.40) and R k+1 and (Z j (1, S k ), Z j (1, S k−1 ), ρ k ) are independent for each j ∈ N we obtain with the help of (7.39), for x > 0, Here and hereafter, const denote constants which may be different on different appearances. To estimate the last term observe that the equality is the sum of iid centered random variables. In particular, conditioning on the environment, With this at hand an application of conditional Jensen's inequality yields, for k ∈ N, For k ∈ N and 1 ≤ i ≤ Z(1, S k−1 ), take the ith particle among the progeny in the generation S k−1 of the first immigrant and denote by V (k) i the number of progeny residing in the generation S k of the chosen particle. Then and, under P ω , V Observe that, under P ω , 2 , . . . are P ω -independent random variables with Geom(λ k ) distribution, and ω is assumed independent of (Z crit (i, j)) j≥i≥1 . This in combination with Z crit (i, j) d = Z crit (1, j − i + 1) for fixed j ≥ i ≥ 1 and (6.12) gives, for k ∈ N, To obtain the last inequality we have utilized E(ρ α ξ α/2 ) < ∞ which is secured by the assumption E(ρξ) α < ∞ and the inequality Eρ α/2 < 1 which is a consequence of (P 1).

Random walks in a moderately sparse random environment
To estimate U 2 we proceed similarly but use additionally Markov's inequality For k ∈ N and 1 ≤ r ≤ Z(1, S k−1 ), take the rth particle among the progeny in the generation S k−1 of the first immigrant and denote by W (k) r the number of progeny residing in the generations S k−1 , . . . , S k − 1 of the chosen particle. Then . . are independent random variables which are independent of Z(1, S k−1 ) and have the same distribution as Y crit (1, ξ k − 1). Here, as usual, ω is assumed independent of (Y crit (1, n)) n∈N . Invoking (6.12) we infer Var ω (W The proof of Lemma 7.2 is complete.

A Appendix
Lemma A.1 is an important ingredient in the proof of Proposition 5.8, part (C1). In its formulation we use the notion of a random variable which does not depend on the future of a sequence of random variables. The corresponding definition can be found at the beginning of Section 5.
Lemma A.1. Let (θ i ) i∈N be a sequence of iid nonnegative random variables and T a nonnegative integer-valued random variable which does not depend on the future of the sequence (θ i ) i∈N . Assume that Eθ s 1 < ∞ for some s > 0 and that Ee λT < ∞ for some λ > 0. Then E( Proof. Set R 0 := 0 and R i := θ 1 + . . . + θ i for i ∈ N. By assumption, for fixed i ∈ N, θ i is independent of (R i−1 , 1 {T ≥i} ).
The result is trivial when s ∈ (0, 1]. Indeed, we use subadditivity of x → x s on [0, ∞) together with the aforementioned independence to conclude that The remaining part of the Appendix is concerned with the proof of Lemma 4.1. In essence the lemma follows from the arguments presented by Key [27] who considered a model very similar to ours. For n ∈ N and 1 ≤ k ≤ n, set Z(k, n) = S k j=S k−1 +1 Z(j, S n ) and observe that, under P ω , Z(1, n), . . . , Z(n, n) are independent. The following representation holds Z(0) = 0, Z n = n−1 k=1 Z(k, n) + Z(n, n), n ∈ N which shows that (Z n ) n∈N0 is a branching process in a random environment with the random number Z(k, k) of immigrants in the kth generation. The basic observation for what follows is that (Z n ) n≥0 has the structure similar to that of the branching process investigated by Key [27]. The main difference manifests in the term Z(n, n) which is absent in Key's model. It is curious that the branching process in [27] is similar to our (Z n ) n∈N0 in that the immigrants arriving in the generation n only affect the system by their offspring residing in the generation n + 1. In particular, neither Key's process nor our (Z n ) n∈N0 counts immigrants, whereas (Z n ) n∈N0 does.
Even though (Z n ) n≥0 and Key's process are slightly different it is natural to expect that sufficient conditions ensuring finiteness of power and exponential moments of the first extinction time should be similar. While demonstrating that this is indeed the case we shall only point out principal changes with respect to Key's arguments. Assume that E log ρ ∈ [−∞, 0) and E log + ξ < ∞. Then, for k ∈ N 0 , π(k) = lim n→∞ P{Z n = k} exists and defines a probability distribution on N. If additionally P{p(2, 0) > 0, a(2, 0) > 0} > 0, then π(0) > 0.
Sketch of proof. As far as the first claim is concerned, the proofs of Lemmas 2.1, 2.2, 3.1, 3.2 in [27] only require inessential changes concerning the range of summation. The second claim follows after a minor alteration, namely the term q(n, k) appearing in the proof of Theorem 3.3 in [27] should be changed to q(n, k) = P ω {Z n+1 = 0 | Z n = k} = p(n + 1, 0) k a(n + 1, 0), n ∈ N, k ∈ N 0 .
The sequence (q(1, k)) k∈N0 must be positive which justifies condition (A.1). The corresponding condition in [27] is slightly different.
We are ready to prove Lemma 4.1.
Existence of such a δ is justified by assumptions and the Cauchy-Schwarz inequality. In view of h(1, j) ≤ P{Z(1, j) ≥ 1} ≤ E(E ω Z(1, j)) δ = E(ρξ) δ r j−1 we infer that the radius of convergence of H is greater than one. This in combination with H(1) < 1 implies that H(x) < 1 and thereupon V (x) < ∞ for some x > 1.