Positively and negatively excited random walks on integers, with branching processes

We consider excited random walks on the integers with a bounded number of i.i.d. cookies per site which may induce drifts both to the left and to the right. We extend the criteria for recurrence and transience by M. Zerner and for positivity of speed by A.-L. Basdevant and A. Singh to this case and also prove an annealed central limit theorem. The proofs are based on results from the literature concerning branching processes with migration and make use of a certain renewal structure.


Introduction
We consider nearest-neighbor random walks on the one-dimensional integer lattice in an i.i.d. cookie environment with a uniformly bounded number of cookies per site. The uniform bound on the number of cookies per site will be denoted by M ≥ 1, M ∈ N. Informally speaking, a cookie environment is constructed by placing a pile of cookies at each site of the lattice (see Figure 1). The piles of cookies represent the transition probabilities of the random walker: upon each visit to a site the walker consumes the topmost cookie from the pile at that site and makes a unit step to the right or to the left with probabilities prescribed by that cookie. If the cookie pile at the current site is empty the walker makes a unit step to the right or to the left with equal probabilities.
A cookie will be called positive (resp. negative) if its consumption makes the walker to go to the right with probability larger (resp. smaller) than 1/2. A cookie which is neither positive nor negative will be called a placebo. Placebo cookies allow us to assume without loss of generality that each pile originally consists of exactly M cookies. Unless stated otherwise, the random walker always starts at the origin.
The term "excited random walk" was introduced by Benjamini and Wilson in [BW03], where they considered random walks on Z d , d ≥ 1, in an environment of identical cookies, one per each site. Allowing more (or fewer) than one cookie per site and randomizing the environment naturally gave rise to the multi-excited random walk model in random cookie environments. We refer to [Zer05] and [Zer06] for the precise description and first results. It was clear then that this new model exhibits a very interesting behavior for d = 1. We shall mention some of the results for d ≥ 2 in Section 9 and now concentrate on the one-dimensional case.
The studies of excited random walks on integers were continued in [MPV06], [BS08a], and [BS08b]. [AR05] deals with numerical simulations of this model. In all papers mentioned above a (possible) bias introduced by the consumption of a cookie was assumed to be only in one direction, say, positive. The recurrence and transience, strong law of large numbers [Zer05], conditions for positive linear speed [MPV06], [BS08a], and the rates of escape to infinity for transient walks with zero speed [BS08b] are now well understood. Yet some of the methods and facts used in the proofs (for example, comparison with simple symmetric random walks, submartingale property) depend significantly on this "positive bias" assumption. The main novelty of the current paper is in considering cookie environments, which may induce positive or negative drifts at different sites or even at the same site on successive visits. Our main results are the recurrence/transience criterion (Theorem 1), the criterion for positive linear speed (Theorem 2) and an annealed central limit theorem (Theorem 3). The first two theorems are extensions of those for non-negative cookie environments but we believe that this is a purely one-dimensional phenomenon. Moreover, in Section 9 we give an example, which shows that, at least for d ≥ 4, the criteria for recurrence or transience and for positive linear speed can not depend just on a single parameter, the average total drift per site (see (3)). The order of the cookies in the pile should matter as well.
The proofs are based on the connections to branching processes with migration. Branching processes allowing both immigration and emigration were studied by several authors in the late 70-ties through about the middle of the 90-ties, and we use some of the results from the literature (Section 2). See the review paper [VZ93] for more results and an extensive list of references up to about 1990. The connection between one-dimensional random walks and branching processes was observed long time ago. In particular, it was used for the study of random walks in random environments, see e.g. [KKS75]. In the context of excited random walks, this idea was employed recently in [BS08a], [BS08b] (still under the "positive bias" assumption). In the present paper we are using results from the literature about branching processes with migration in a more essential way than [BS08a] and [BS08b]. One of our tasks is to show how to translate statements about excited random walks into statements for a class of branching processes with migration which have been studied in the past.
The purpose of ω(z, i) is to serve as the transition probability from z to z + 1 of a nearest-neighbor ERW upon the i-th visit to a site z. More precisely, for fixed ω ∈ Ω M and x ∈ Z an ERW (X n ) n≥0 starting from x in the cookie environment ω is a process on a suitable probability space with probability measure P x,ω which satisfies: The cookie environment ω may be chosen at random itself according to a probability measure on Ω M , which we shall denote by P, with the corresponding expectation operator E. Unless stated otherwise, we shall make the following assumption on P: (1) The sequence (ω(z, ·)) z∈Z is i.i.d. under P.
Note that assumption (1) does not imply independence between different cookies at the same site but only between cookies at different sites, see also Figure 1. To avoid degenerate cases we shall also make the following mild ellipticity assumption on P: (1 − ω(0, i)) > 0.
After consumption of a cookie ω(z, i) the random walk is displaced on P x,ω -average by 2ω(z, i) − 1. This average displacement, or drift, is positive for positive cookies and negative for negative ones. The consumption of a placebo cookie results in a symmetric random walk step. Averaging the drift over the environment and summing up over all cookies at one site defines the parameter which we shall call the average total drift per site. It plays a key role in the classification of the asymptotic behavior of the walk as shown by the three main theorems of this paper. Our first result extends [Zer05, Theorem 12] about recurrence and transience for non-negative cookies to i.i.d. environments with a bounded number of positive and negative cookies per site.
Our next result extends [MPV06, Theorem 1.1, Theorem 1.3] and [BS08b, Theorem 1.1] about the positivity of speed from spatially uniform deterministic environments of non-negative cookies to i.i.d. environments with positive and negative cookies.
Theorem 2 (Law of large numbers and ballisticity). There is a deterministic v ∈ [−1, 1] such that the excited random walk satisfies for P-a.a. environments ω, and v > 0 for δ > 2.
While Theorems 1 and 2 give necessary and sufficient conditions for recurrence, transience, and the positivity of the speed, the following central limit theorem gives only a sufficient condition. To state it we need to introduce the annealed, or averaged, measure P Theorem 3 (Annealed central limit theorem). Assume that |δ| > 4. Let v be the velocity given by Theorem 2 and define Then (B n t ) t≥0 converges in law under P 0 to a non-degenerate Brownian motion with respect to the Skorohod topology on the space of cadlag functions.
The variance of the Brownian motion in Theorem 3 will be further characterized in Section 6, see (28).
Let us describe how the present article is organized. Section 2 introduces the main tool for the proofs, branching processes with migration, and quotes the relevant results from the literature. In Sections 3 and 4 we describe the relationship between ERW and branching processes with migration and introduce the necessary notation. In Section 5 we use this relationship to translate results from Section 2 about branching processes into results for ERW concerning recurrence and transience, thus proving Theorem 1. In Section 6 we introduce a renewal structure for ERW, similar to the one which appears in the study of random walks in random environments (RWRE), and relate it to branching processes with migration. In Sections 7 and 8 we use this renewal structure to deduce Theorems 2 and 3, respectively, from results stated in Section 2. The final section contains some concluding remarks and open questions.
Throughout the paper we shall denote various constants by c i ∈ (0, ∞), i ≥ 1.

Branching processes with migration -results from the literature
In this section we define a class of branching processes with migration and quote several results from the literature. We chose to give the precise statements of the results that we need since some of the relevant papers are not readily available in English.
Definition 1. Let µ and ν be probability measures on N 0 := N ∪ {0} and Z, respectively, and let ξ (j) i and η k (i, j ≥ 1, k ≥ 0) be independent random variables such that each ξ (j) i has distribution µ and each η k has distribution ν. Then the process (Z k ) k≥0 , recursively defined by Z k +η k , k ≥ 0, is said to be a (µ, ν)-branching process with offspring distribution µ and migration distribution ν. (Here we make an agreement that ξ (k+1) 1 ) An offspring distribution µ which we shall use frequently is the geometric distribution with parameter 1/2 and support N 0 . It is denoted by Geom(1/2) .
Note that any (µ, ν)-branching process is a time homogeneous Markov chain, whose distribution is determined by µ and ν. More precisely, if at time k the size of the population is Z k then (1) η k individuals immigrate or min{Z k , |η k |} individuals emigrate depending on whether η k ≥ 0 or η k < 0 respectively, and (2) the resultant (Z k + η k ) + individuals reproduce independently according to the distribution µ. This determines the size Z k+1 of the population at time k + 1.
In the current paper we are interested in the case when both the immigration and the emigration components are non-trivial and the number of emigrants is bounded from above. This bound will be the same as the bound M on the number of cookies per site. We shall assume that In addition to (5), we shall make the following assumptions on the measures µ and ν: Note that µ = Geom(1/2) satisfies condition (A) on the moment generating function f with b = 1. It also satisfies (B).
Next we state a result from the literature, which relates the limiting behavior of the process (Z k ) k≥0 to the value of the parameter At first, introduce the stopped process ( Z k ) k≥0 . Let Note that the process ( Z k ) k≥0 follows (Z k ) k≥0 until the first time (Z k ) k≥0 returns to 0. Then ( Z k ) k≥0 stays at 0 whereas (Z k ) k≥0 eventually regenerates due to the presence of immigration (see the first inequality in (5)). [FYK90]). Let (Z k ) k≥0 be a (µ, ν)-branching process satisfying (5), (A) and (B). We let describe the tail of the distribution of N (Z) and denote by v n := E n m=0 Z m the expectation of the total progeny of ( Z k ) k≥0 up to time n ∈ N 0 ∪ {∞}. Then the following statements hold.
The above results about the limiting behavior of u n are contained in Theorems 1 and 4 of [FY89], [FYK90]. The proofs are given only in [FYK90]. The behavior of v n is the content of formula (33) in [FYK90]. The statements (i) and (ii) of Theorem A also follow from [YY95, Theorem 2.2] (see also [YMY03, Theorem 2.1]).
Remark 1. We have to point out that we use a slightly different (and more convenient for our purposes) definition of the lifetime, N (Z) , of the stopped process. More precisely, our quantity u n can be obtained from the one in [FYK90] by the shift of the index from n to n − 1 and multiplication by which is positive due to the first inequality in (5) and the fact that µ({0}) < 1 (by the condition f ′ (1) = 1 of assumption (A)). A similar change is needed for the expected total progeny of the stopped process. Clearly, these modifications affect only the values of constants in Theorem A and not their positivity or finiteness.
The papers mentioned above contain other results but we chose to state only those that we need. In fact, we only use the first part of (iv) and the following characterization, which we obtain from Theorem A by a coupling argument.
Proof. Theorem A (i) gives the 'only if'-part of the first statement. To show the 'if' direction we assume that θ ≤ 1, i.e. ν has mean λ ≤ b (see (7)). Then there is another ν ′ with mean b which stochastically dominates ν. Indeed, if X has distribution ν and Y has expectation b − λ and takes values in N 0 then ν ′ can be chosen as the distribution of X + Y . By coupling, the (µ, ν ′ )-branching process stochastically dominates the (µ, ν)-branching process. However, the (µ, ν ′ )branching process dies out a.s. due to Theorem A (ii) since for this process θ = 1. Consequently, the (µ, ν)-branching process must die out, too.
Similarly, Theorem A (iv) gives the 'if'-part of the second statement. The converse direction follows from monotonicity as above and Theorem A (iii).

From ERWs to branching processes with migration
The goal of this mainly expository section is to show how our ERW model can be naturally recast as a branching process with migration. This connection was already observed and used in [BS08a] and [BS08b].
Consider a nearest neighbor random walk path (X n ) n≥0 , which starts at 0 and define Assume for the moment that X 1 = 1 and consider the right excursion, i.e. (X n ) 0≤n<T 0 . The left excursion can be treated by symmetry.
On the set {T 0 < ∞} we can define a bijective path-wise mapping of this right excursion to a finite rooted tree, which corresponds to a realization of a branching process with the extinction time max{X n , 0 ≤ n < T 0 } as illustrated in Figure 2. Moreover, given a tree for a branching process that becomes extinct in finite time, we can reconstruct the right excursion of the random walk. This can be done by making a time diagram of up and down movements of an ant traversing the tree in preorder: the ant starts at the root, always chooses to go up and to the left whenever possible, never returns to an edge that was already crossed in both directions, and finishes the journey at the root ( Figure 2, (III)).
The above path-wise correspondence on {T 0 < ∞} does not depend on the measure associated to the random walk paths. To consider the set {T 0 = ∞} we shall need some of the properties of this measure. The following simple statement leaves only three major possibilities for a long term behavior of an ERW path.
Proof. Let z ∈ Z. If the ERW visits z infinitely many times then it also visits z + 1 infinitely many times due to the second Borel Cantelli lemma, the strong Markov property, and the assumption ω(z, i) = 1/2 for i > M. This implies P 0,ω -a.s. lim sup n X n / ∈ Z. Similarly, P 0,ω -a.s. lim inf n X n / ∈ Z.
Let us now put a measure on the paths and see what kind of measure will be induced on trees. Consider the right excursion of the simple symmetric random walk. Assume without loss of generality that the walk starts at 1. Then the probability that T 0 < ∞ is equal to one and the corresponding measure on trees will be the one for a standard Galton-Watson process with the Geom(1/2) offspring distribution starting from a single particle. More precisely, set U 0 := 1 and let be the number of upcrossings of the edge (k, k + 1) by the walk before it hits 0. Then (U k ) k≥0 has the same distribution as the Galton-Watson process with Geom(1/2) offspring distribution. Therefore, (U k ) k≥0 can also be generated as follows: start with one particle: U 0 = 1. To generate the (k + 1)-st generation from the k-th generation (assuming that the process has not yet died out), the first particle of generation k tosses a fair coin repeatedly and produces one offspring if the coin comes up "heads". It stops the reproduction once the coin comes up "tails". Then the second particle in generation k follows the same procedure independently, then the third one, and so on. Consequently, To construct a branching process corresponding to an ERW with M cookies per site one can use exactly the same procedure except that for the first M coin tosses in the k-th generation the particles should use coins with biases "prescribed" by the cookies located at site k. Since every particle tosses a coin at least once, at most the first M particles in each generation will have a chance to use biased coins. All the remaining particles will toss fair coins only. This can be viewed as a branching process with migration in the following natural way. Before the reproduction starts, the first U k ∧M particles emigrate, taking with them all M biased coins and an infinite supply of fair coins. In exile they reproduce according to the procedure described above. Denote the total number of offspring produced by these particles by η (k+1) U k ∧M . Meanwhile, the remaining particles (if any) reproduce using only fair coins. Finally, the offspring of the emigrants re-immigrate. Therefore, the number of particles in the generation k + 1 can be written as Branching processes of type (10) were considered in [BS08a] (p. 630) and [BS08b] (p. 815), except that they were generated not by the forward but by the backward excursion (see (29) in Section 6). Careful analysis of such processes carried out in these two papers yielded results concerning positive speed and rates of growth at infinity for ERWs with non-negative cookies. However, from a practical point of view, (µ, ν)-branching processes, introduced in Definition 1, seem to be well-known and studied more extensively in the past. In particular, we could find the results we need (see Theorem A) in the literature only for (µ, ν)-branching processes, but not for processes of the form (10). For this reason one of our main tasks will be to relate these two classes of processes in order to translate the results from the literature into results about processes of the form (10).

Coin-toss construction of the ERW and the related (µ, ν)-branching process
In this section we formalize a coin-toss construction of the ERW and introduce auxiliary processes used in the rest of the paper.
Let (Ω, F ) be some measurable space equipped with a family of probability measures P x,ω , x ∈ Z, ω ∈ Ω M , such that for each choice of x ∈ Z and ω ∈ Ω M we have ±1-valued random variables Y (k) i , k ∈ Z, i ≥ 1, which are independent under P x,ω with distribution given by Moreover, we require that there is a random variable X 0 on (Ω, F , P x,ω ) such that P x,ω [X 0 = x] = 1. Then an ERW (X n ) n≥0 , starting at x ∈ Z, in the environment ω can be realized on the probability space (Ω, F , P x,ω ) recursively by: We shall refer to {Y (k) i = 1} as a "success" and to {Y (k) i = −1} as a "failure". Due to (11) every step to the right or to the left of the random walk corresponds to a success or a failure, respectively.
We now describe various branching processes that appear in the proofs. Namely, we introduce processes (V k ) k≥0 , (W k ) k≥0 , and (Z k ) k≥0 . Modifications of the first two processes suitable for left excursions will be defined later when they are needed (we shall keep the same notation though, hoping that this will not lead to confusion). The last process, (Z k ) k≥0 , will belong to the class of processes from Section 2.
For m ∈ N and k ∈ Z let By assumption (2) the walk reaches 1 in one step with positive P 0 -probability.
We shall be interested in the behavior of the process (U k ) k≥0 defined in (9). At first, we shall relate (U k ) k≥0 to (V k ) k≥0 which is recursively defined by Observe that (V k ) k≥0 is a time homogeneous Markov chain, as the sequence of sequences (S Moreover, 0 is an absorbing state for (V k ) k≥0 . We claim that under P 1 , The relation (14) is obvious from the discussion in Section 3 and Figure 2. To show (15) we shall use induction. Recall that U 0 = V 0 = 1 and assume U i ≤ V i for all i ≤ k. From Lemma 5 we know that X n → ∞ as n → ∞ on {T 0 = ∞} a.s. with respect to P 1 . Therefore, the last, U k -th, upcrossing of the edge (k, k + 1) by the walk is not matched by a downcrossing. This implies that U k+1 should be less than or equal to the number of successes in the sequence Y (k+1) i i≥1 prior to the U k -th failure. On the other hand, to get the value of V k+1 one needs to count all successes in this sequence until the V k -th failure. Since U k ≤ V k , we conclude that U k+1 ≤ V k+1 .
Next we introduce the process (W k ) k≥0 by setting Just as (V k ) k≥0 , the process (W k ) k≥0 is a time homogeneous Markov chain on non-negative integers. Moreover, the transition probabilities from i to j of these two processes coincide except for i ∈ {0, 1, . . . , M − 1} and both processes can reach any positive number with positive probability. Therefore, if one of these two processes goes to infinity with positive probability, so does the other: Finally, we decompose the process (W k ) k≥0 into two components as follows.
Proof. By definition, Z 0 = 0 and is defined as the number of successes in Y (k+1) j j≥1 between the (M + i − 1)-th and the (M + i)-th failure, i ≥ 1. Therefore, by definition of Z k and η k , Since ω(k, m) = 1/2 for m > M, the random variables Y Having introduced all necessary processes we can now turn to the proofs of our results.

Recurrence and transience
Definition 2. The ERW is called recurrent from the right if the first excursion to the right of 0, if there is any, is P 0 -a.s. finite. Recurrence from the left is defined analogously.
In the next lemma we shall characterize ERW which are recurrent from the right in terms of branching processes with migration. At first, we shall introduce a relaxation of condition (1), which is needed for the proof of Theorem 1: The sequence (ω(k, ·)) k≥K is i.i.d. under P for some K ∈ N.
Denote the common distribution of η k := S Proof. Since we are interested in the first excursion to the right we may assume without loss of generality that the random walk starts at 1. Then, recalling definition (9), we have {T 0 = ∞} = B means that the two events A and B may differ by a P 1 -null-set only. Indeed, since U k counts only upcrossings of the edge (k, k + 1) prior to T 0 , the inclusion ⊇ is trivial. The reverse relation follows from Lemma 5. This together with (14) and (15) implies that (20) As above (V k ) k≥K is a time homogeneous Markov chain since the sequence of sequences (S (k) m ) m≥0 , k ≥ K, is i.i.d.. For any m the transition probability of this Markov chain from m ∈ N to 0 is equal to which is strictly positive by (19). Since 0 is absorbing for (V k ) k≥0 we get that Next we turn to the process (W k ) k≥0 and recall relation (17). Thus, Finally, we decompose the process (W k ) k≥0 as in Lemma 6 by writing W k+1 = Z k + S = {Z k → ∞}. Together with (21) this shows that the ERW is recurrent from the right iff P 0 [Z k → ∞] = 0. Since (Z k ) k≥K is an irreducible Markov chain this is equivalent to (Z k ) k≥K being recurrent, which is equivalent to recurrence of the state 0 for (Geom(1/2), ν)-branching processes.
Lemma 8. Assume again (1) and (2). If the ERW is recurrent from the right then all excursions to the right of 0 are P 0 -a.s. finite. If the ERW is not recurrent from the right then it will make P 0 -a.s. only a finite number of excursions to the right. The corresponding statements hold for recurrence from the left.
Proof. Let the ERW be recurrent from the right. By Definition 2 the first excursion to the right is a.s. finite. By Lemma 7 the corresponding (Geom(1/2), ν)-branching process dies out a.s.. Let i ≥ 1 and assume that all excursions to the right up to the i-th one have been proven to be P 0 -a.s. finite. If the ERW starts the (i + 1)-st excursion to the right of 0 then it finds itself in an environment which has been modified by the previous i excursions up to a random level R ≥ 1, beyond which the environment has not been touched yet. Therefore, conditioning on the event {R = K}, K ≥ 1, puts us within the assumptions of Lemma 7: the random walk starts the right excursion from 0 in a random cookie environment which satisfies (18). But the corresponding (Geom(1/2), ν)-branching process is still the same and, thus, dies out a.s.. Therefore, this excursion, which is the (i + 1)-st excursion of the walk, is a.s. finite on {R = K}. Since by our induction assumption the events {R = K}, K ≥ 1, form a partition of a set of full measure, we obtain the first statement of the lemma.
For the second statement let D := inf{n ≥ 1 | X n < X 0 } be the first time that the walk backtracks below its starting point. Due to (2), P 0 [X 1 = 1] > 0. Therefore, since the walk is assumed to be not recurrent from the right, Denote by K i the right-most visited site before the end of the i-th excursion and define K i = ∞ if there is no i-th right excursion or if the i-th excursion to the right covers N. Then the number of i ≥ 1 such that K i < K i+1 , is stochastically bounded from above by a geometric distribution with parameter P 0 [D = ∞]. Indeed, each time the walk reaches a level K i + 1 < ∞, which it has never visited before, it has probability P 0 [D = ∞] never to backtrack again below the level K i + 1, independently of its past. Therefore, (K i ) i increases only a finite number of times. Hence P 0 -a.s. R := sup{K i | i ≥ 1, K i < ∞} < ∞. Now, if the walk did an infinite number of excursions to the right, then, P 0 -a.s. sup n X n = R < ∞ and lim sup n X n ≥ 0, which is impossible due to Lemma 5.
Proposition 9. The ERW is recurrent from the right if and only if δ ≤ 1. Similarly, it is recurrent from the left if and only if δ ≥ −1.
For the proof we need the next lemma, which relates the parameter δ of the ERW and the parameter θ of the branching process with migration.
Lemma 10. Let ν be the distribution of S Proof of Proposition 9. Due to Lemma 7 the walk is recurrent from the right iff the (Geom(1/2), ν)-branching process dies out a.s., where ν is the distribution of S Proof of Theorem 1. If δ > 1 then by Proposition 9 the walk is not recurrent from the right but recurrent from the left. If the walk returned infinitely often to 0 then it would also make an infinite number of excursions to the right which is impossible due to Lemma 8. Hence the ERW visits 0 only finitely often. Since any left excursion is finite due to Lemma 8 the last excursion is to the right and is infinite. Consequently, P 0 -a.s. lim inf n X n ≥ 0, and therefore, due to Lemma 5, X n → ∞. Similarly, δ < −1 implies P 0 -a.s. X n → ∞.
In the remaining case δ ∈ [−1, 1] all excursions from 0 are finite due to Proposition 9. Hence, 0 is visited infinitely many times. Remark 2. The equivalence (20) also holds correspondingly for one-dimensional random walks (X n ) n≥0 in i.i.d. random environments (RWRE) and branching processes (V k ) k≥0 in random environments, i.e. whose offspring distribution is geometric with a random parameter. This way the recurrence theorem due to Solomon [So75,Th. (1.7)] for RWRE can be deduced from results by Athreya and Karlin, see [AN72, Chapter VI.5, Corollary 1 and Theorem 3].

A renewal structure for transient ERW
A powerful tool for the study of random walks in random environments (RWRE) is the so-called renewal or regeneration structure. It is already present in [KKS75], [Ke77] and was first used for multi-dimensional RWRE in [SZ99]. It has been mentioned in [Zer05, p. 114, Remark 3] that this renewal structure can be straightforwardly adapted to the setting of directionally transient ERW in i.i.d. environments in order to give a law of large numbers. The proofs of positivity of speed and of a central limit theorem for once-excited random walks in dimension d ≥ 2 in [BR07] were also phrased in terms of this renewal structure. We shall do the same for the present model.
We continue to assume (1) and (2). Let δ > 1, where δ is the average drift defined in (3). This means, due to Theorem 1, that P 0 -a.s. X n → ∞. Moreover, by Proposition 9, the walk is not recurrent from the right, which implies, as we already mentioned, see (22), that P 0 [D = ∞] > 0. Hence there are P 0 -a.s. infinitely many random times n, so-called renewal or regeneration times, with the defining property that X m < X n for all 0 ≤ m < n and X m ≥ X n for all m > n. Call the increasing enumeration of these times (τ k ) k≥1 , see also Figure 3. Then the sequence (X τ 1 , τ 1 ), (X τ k+1 −X τ k , τ k+1 −τ k ) (k ≥ 1) of random vectors is independent under P 0 . Furthermore, the random vectors (X τ k+1 − X τ k , τ k+1 − τ k ), k ≥ 1, have the same distribution under P 0 . For multidimensional RWRE and once-excited random walk the corresponding statement is [ Moreover, the ordinary strong law of large numbers implies that If, moreover, then the result claimed in Theorem 3 holds with Thus, in order to prove Theorems 2 and 3 we need to control the first and the second moment, respectively, of τ 2 − τ 1 . We start by introducing for k ≥ 0 the number (29) D k := # {n | τ 1 < n < τ 2 , X n = X τ 2 − k, X n+1 = X τ 2 − k − 1} of downcrossings of the edge (X τ 2 − k, X τ 2 − k − 1) between the times τ 1 and τ 2 .
Lemma 11. Assume that the ERW is transient to the right and let p ≥ 1. Then the p-th moment of τ 2 − τ 1 under P 0 is finite if and only if the p-th moment of k≥1 D k is finite. Proof. The number of upcrossings between τ 1 and τ 2 is X τ 2 −X τ 1 + k≥1 D k , since X τ 1 < X τ 2 and since each downcrossing needs to be balanced by an upcrossing. Each step is either an upcrossing or a downcrossing, therefore, For every k ∈ {X τ 1 + 1, . . . , X τ 2 − 1} there is a downcrossing of the edge (k, k − 1), otherwise k would be another point of renewal. Hence, X τ 2 − X τ 1 ≤ 1 + k≥1 D k and, by (30), 2 This implies the claim. To interpret (D k ) k≥0 as a branching process (see Figure 4) we define for m ∈ N and k ∈ Z Lemma 12. Assume that the ERW is transient to the right. Then (D k ) k≥0 and ( V k ) k≥0 have the same distribution under P 0 .
Proof. Fix an integer K ≥ 1. For brevity, we set D := (D 1 , . . . , D K ) and V := ( V 1 , . . . , V K ). It suffices to show that Since both processes start from 0 and also stay at 0 once they have returned to 0 for the first time, it is enough to consider vectors i whose entries are strictly positive except for maybe the last one. And, since the process (D k ) k≥0 eventually does reach 0 P 0 -a.s., namely at k = X τ 2 − X τ 1 < ∞, it suffices to consider only i whose last entry is 0. Thus, let i = (i 1 , . . . , i K ) ∈ N K 0 with i 1 , . . . , i K−1 ≥ 1 and i K = 0. At first, we shall show that We start from the partition equation However, on the event { D = i, τ 2 = T m }, we have X τ 1 = m − K by our choice of i. Since X τ 1 ≥ 0, we may start the summation in (37) from m = K. Moreover, comparing the definitions (29) and (36), we see that using that not only on the left but also on the right event we have X τ 1 = m − K. Hence, the right hand side of (37) is equal to Since i K = 0, this is the same as Applying the strong Markov property once more, this time to T m−K , and using the i.i.d. structure of the environment, we get that the above is equal to This proves (35). Now we need to show that The proof is essentially the same as that of Proposition 2.2 of [BS08a]. At first, notice that, given D k . Therefore, the process (D (K) k ) 0≤k≤K is Markov, just as the process ( V k ) 0≤k≤K . Both processes get absorbed after the first return to 0. Then (39) will follow if we show that they have the same transition probabilities and that D see the last line on p. 113 and the first line on p. 114 of [Zer05]. Similarly, by symmetry, if inf n≥0 X n = −∞ a.s. then Now due to Theorem 1 there are only three cases: Either the walk is transient to the right or it is transient to the left or it is recurrent. Consider the case of transience to the right. If u + < ∞ then lim n X n /n = 1/u + follows directly from (40) and (41). If u + = ∞ then lim n X n /n = 0 follows from (40) and inf n X n > −∞. Transience to the left is treated analogously. In the case of recurrence we have a.s. both sup n X n = ∞ and inf n X n = −∞. Hence both u + and u − are infinite due to (41) and (43), respectively. Therefore, by (40) and (42), lim n X n /n = 0. (W k +1)∨M . This is a branching process with migration in the following sense: At each step, it exhibits two types of behavior: 1) if W k ≥ M − 1 then one particle immigrates and then all W k + 1 particles reproduce; 2) if W k < M − 1 then M − W k particles immigrate and then all M particles reproduce.
We shall first establish the equivalence where, as usual, Comparing definitions (32) and (45) we see that V k ≤ W k for all k, which yields the implication ⇐ in (46). For the reverse implication, assume that E 0 [ V ] is finite. Since N (V ) ≤ V + 1 this implies that (V k ) k≥0 is positive recurrent. The following lemma, whose proof is postponed, will help us to compare (V k ) k≥0 and (W k ) k≥0 .
Lemma 15. Let K be the transition matrix of a positive recurrent Markov chain with state space N 0 and invariant distribution π. Assume also that all entries of K are strictly positive. Fix a state j ∈ N 0 and a finite set J ⊂ N 0 \ {j}. Modify a finite number of rows of K by setting Then a Markov chain with the transition matrix K is also positive recurrent and its unique invariant probability distribution π satisfies π(n) ≤ c 6 π(n) for all n ∈ N 0 .
If we let K be the transition matrix of the Markov chain (V k ) k≥0 and set j = M − 1 and J = {0, 1, . . . , M − 2} then K defined in Lemma 15 is the transition matrix of (W k ) k≥0 . Moreover, all entries of this K are strictly positive due to (2). Consequently, we may apply Lemma 15 and get that (W k ) k≥0 is positive recurrent and its invariant probability distribution π is bounded above by a multiple c 6 π of the invariant probability distribution π of (V k ) k≥0 . By Theorem 5.4.3 of [Du05], π and π can be represented as Therefore, also ρ ≤ c 7 ρ. However, and, similarly, . This gives the implication ⇒ in (46). Next, we show that As in Lemma 6 we decompose the process (W k ) k≥0 into two components.
M . Then (Z ′ k ) k≥0 is a (Geom(1/2), ν)branching process, where ν is the common distribution of η k := F (k) The proof of Lemma 16 is almost identical to the one of Lemma 6 and, thus, is omitted.
Since F (k) , which yields the implication ⇒ in (49). For the opposite direction assume that E 0 [ Z] < ∞. Then, as in the proof of (46), (Z ′ k ) k≥0 is positive recurrent and, by the equivalent of (48), its invariant distribution, say π ′ , has a finite mean. Since Z ′ k and F (k) M are independent, it follows from Lemma 16 that the convolution of π ′ and the distribution of F (k) M is invariant for (W k ) k≥0 . This convolution has a finite mean as well, which implies, as in (48), that E 0 [ W ] is finite as well. This concludes the proof of (49). The statement of the lemma now follows from (46) and (49).
Proof of Lemma 15. It suffices to consider the case in which J has only one element, i.e. J = {i} for some i = j. The full statement then follows by induction, changing one row at a time. Let (ζ k ) k≥0 and (ζ k ) k≥0 be Markov chains with transition matrices K and K, respectively. Their initial point will be denoted by a subscript of P and E. Since all the entries of K are strictly positive, K is irreducible. It is recurrent, since its state i is recurrent. Indeed, Moreover, since (ζ k ) k≥0 and (ζ k ) k≥0 are indistinguishable as long as they do not touch i, i.e. since K(s, ·) = K(s, ·) for all s = i, we can switch from the process (ζ k ) k≥0 to (ζ k ) k≥0 and obtain that P i [∃k ≥ 1 : Similarly, one can show that (ζ k ) k≥0 is also positive recurrent. Define the hitting time σ := inf{k ≥ 1 | ζ k = i} for (ζ k ) k≥0 and analogously σ for (ζ k ) k≥0 . Then, again by [Du05,Theorem 5.4.3], ρ and ρ, defined by are invariant measures for K and K, respectively. Using the relations between K and K as above, we have for all s ∈ N 0 , On the other hand, for all s ∈ N 0 , Since (ζ k ) k≥0 is positive recurrent, ρ's total mass, E i [σ], is finite. Consequently, by the above and since K(i, j) > 0, ρ's total mass, E i [σ], is finite as well. Therefore, (ζ k ) k≥0 is positive recurrent and its invariant measure π satisfies π ≤ c 6 π with The following lemma is the counterpart of Lemma 10.
Lemma 17. Let ν be the distribution of F Proof of Theorem 2. The first statement of the theorem, the existence of the velocity v, is just Proposition 13, or (25) in the transient case. If δ ∈ [−1, 1] then the walk is recurrent by Theorem 1 and therefore v = 0. Now let |δ| > 1. Without loss of generality we may assume δ > 1. Then the walk is transient to the right by Theorem 1. By (26), v > 0 iff E 0 [τ 2 − τ 1 ] < ∞.
By Lemma 11 with p = 1 this is the case iff E 0 k≥0 D k < ∞. By Lemma 12 this holds iff E 0 k≥0 V k < ∞. Due to Lemma 14 this is true iff E 0 k≥0 Z k is finite. By the second statement in Corollary 4 this holds iff θ < −1. Thus, by Lemma 17, v > 0 iff δ > 2.

Central limit theorem
Lemma 18. Let ( V k ) k≥0 be defined by (33). Then δ > 4 implies that the random variable V := k≥0 V k has a finite second moment.
Proof. Let (W k ) k≥0 and W be defined by (45) and (47), respectively, using the same sequences (F . We shall prove that the latter is finite. By Minkowski's inequality we have ) 2 ). Combining this with the fact that (a+b) 1/2 ≤ a 1/2 +b 1/2 for all a, b ≥ 0, we get Applying Hölder's inequality with 1/α + 1/α ′ = 1, α > 1, we obtain We are going to show that (i) for every ε ∈ (0, δ − 4) there is a constant c 8 (ε, δ) such that (ii) for each ℓ ∈ N there is a constant c 9 (ℓ) such that E 0 Z ℓ k ≤ c 9 (ℓ)k ℓ for all k ∈ N 0 . Let us assume (i) and (ii) for the moment and see that both series in the right hand side of (51) are finite. Choose α ′ ∈ (1, δ/4) so that α = α ′ /(α ′ − 1) is an integer and let ε = (δ − 4α ′ )/2. Then by (i) and (ii) for all k ≥ 1, Since δ/(4α ′ ) > 1, the first series in the right hand side of (51) converges. It is obvious now that for the same choice of α ′ and ε the second series in the right hand side of (51) also converges. Therefore we only need to prove (i) and (ii).
Proof of (i). Observe that W k is zero if and only if both Z k−1 and F (k−1) M are equal to zero. Set N 0 := 0 and consider the times N i := inf{k > N i−1 | Z k = 0}, i ∈ N, when the process (Z k ) k≥1 dies out. Due to Lemma 17 and δ > 4 the parameter θ for the process (Z k ) k≥0 satisfies In particular, Corollary 4 implies that the process (Z k ) k≥0 is positive recurrent. Therefore, all N i , i ∈ N, are a.s. finite and (N i − N i−1 ) i∈N is i.i.d.. We are interested in the sequence (F M n≥0 is i.i.d. and, by con- where β ∈ (1, δ/(δ − ε)), β ′ = β/(β − 1). Denote (δ − ε)β by γ. Writing N m as m i=1 (N i − N i−1 ) and using Minkowski's inequality we get From part (iv) of Theorem A and (52) we know that P 0 [N 1 > k] ∼ c 4 k −δ . Therefore, and since γ < δ by assumption, P 0 [N γ 1 > k] is summable in k. Consequently, E 0 [N γ 1 ] = c 12 (ε, δ) < ∞. This implies (i). Proof of (ii). The proof can be easily done by induction in ℓ. The statement is trivial for ℓ = 0. (Here 0 0 = 1.) Assume now ℓ ≥ 1 and that for each j ∈ {0, 1, 2, . . . , ℓ − 1} there is a constant c 9 (j) such that E 0 Z j k ≤ c 9 (j)k j for all k ∈ N 0 . Using that (Z k ) k is a (Geom(1/2), ν)-branching process we have for all k ∈ N 0 and n ∈ N, Observe that the expectation in (54) is bounded by a constant c 13 (ℓ). To control the series in (55) we use the following lemma, whose proof is postponed until after the end of the present proof.
Lemma 19. Let (ξ i ) i∈N be non-negative i.i.d. random variables such that E[ξ 1 ] = 1 and E ξ ℓ 1 < ∞ for some positive integer ℓ. Then there is a constant c 14 such that for all n > ℓ, E (ξ 1 + ξ 2 + · · · + ξ n ) ℓ ≤ n ℓ + c 14 Applying this lemma to the series in (55) we obtain By the induction hypothesis we conclude that E 0 Z ℓ k+1 − Z ℓ k ≤ c 16 k ℓ−1 . Summation over k implies (ii) and finishes the proof of Lemma 18.
Proof of Lemma 19. By expanding and using independence, E 0 [ k≥0 Z k ] < ∞. By the above argument, the distribution of ( Z k ) k≥0 remains unchanged under permutations.
Remark 4 (Higher dimensions). Multi-dimensional ERW with cookies that induce a bias with a non-negative projection in some fixed direction were considered in [BW03], [Ko03], [Ko05], [Zer06], [HH06] and [BR07]. A special non-stationary cookie environment with two types of cookies pointing into opposite directions was studied in [ABK08]. However, to the best of our knowledge so far there are no criteria for recurrence, transience or ballistic behavior of ERWs in i.i.d. environments of "positive" and "negative" cookies in higher dimensions. To us, it is not even clear how such criteria could look like. In the following example we shall indicate that the situation cannot be as simple as in one dimension. We shall show that, unlike for d = 1 (see Remark 3), permuting cookies in higher dimensions may change the sign of the velocity.
Example. Let d ≥ 4. Denote by ω(x, e, i) the probability for the ERW to jump from x ∈ Z d to the nearest neighbor x + e upon the i-th visit of x. Fix 0 < ε < 1/(2d) and consider the two deterministic cookie environments ω k , k = 1, 2, defined by where {e 1 , . . . , e d } is the canonical basis of Z d . In both environments, ω 1 and ω 2 , there are M = 2 cookies per site. In the environment ω 1 the walk experiences a drift into direction e 1 upon the first visit to a site and an equal drift into the opposite direction −e 1 upon the second visit. Otherwise, it behaves just like a simple symmetric random walk. We shall show: (57) There is some v > 0 such that lim n→∞ X n n = ve 1 P 0,ω 1 -a.s..
The environment ω 2 is obtained from ω 1 by permuting the two cookies at each site. By symmetry, we obtain from (57) that for the same v > 0, we have P 0,ω 2a.s. lim n X n /n = −ve 1 . Thus, in this example, permuting two cookies reverses the direction of the speed. For the proof of (57) denote by R n,1 := {X m | m < n} the range of the walk before time n ∈ N and by R n,2 := {x ∈ Z d | ∃k < m < n X k = x = X m } the set of vertices which have been visited at least twice before time n. It is easy to see that both #R n,1 and #R n,2 tend to ∞ as n → ∞. Coupling (X n ) n≥0 in the natural way to a simple symmetric random walk (S n ) n≥0 on Z d yields that (X n ) n≥0 can be represented as X n = S n + 2e 1 a n − 2e 1 b n , where a n = #R n,1 i=1 Y i,1 and b n = #R n,2 i=1 Y i,2 and Y i,j (i ≥ 1, j = 1, 2) are independent and Bernoulli distributed with parameter ε. Consequently, X n · e 1 n = S n · e 1 n + 2a n #R n,1 #R n,1 n − 2b n #R n,2 #R n,2 n = S n · e 1 n + 2a n #R n,1 #R n,1 − #R n,2 n + 2a n #R n,1 − 2b n #R n,2 #R n,2 n .
The first term on the right hand side of (58) tends to zero P 0,ω 1 -a.s.. The same holds for the last term in (58) since by the strong law of large numbers both a n /#R n,1 and b n /#R n,2 tend to ε as n → ∞ and #R n,2 /n is bounded. Therefore, lim inf n→∞ X n · e 1 n ≥ 2ε lim inf n→∞ #R n,1 − #R n,2 n To bound the right hand side of (59) from below we introduce the projection π : Z d → Z d−1 defined by π(x 1 , x 2 , . . . , x d ) := (x 2 , . . . , x d ) onto the subspace spanned by e 2 , . . . , e d and consider the process (X ′ n ) n≥0 defined by X ′ n := π(X n ). Under P 0,ω 1 , (X ′ n ) n≥0 is a simple symmetric random walk on Z d−1 with holding times which are i.i.d. and geometrically distributed with parameter 1 − 1/d. As above, denote by R ′ n,1 := {X ′ m | m < n} the range of this walk before time n ∈ N and by R ′ n,2 := {x ∈ Z d−1 | ∃k < m < n X ′ k = x = X ′ m } the set of vertices which have been visited at least twice before time n by (X ′ n ) n . Returning to the right hand side of (59) we first note that #R n,1 − #R n,2 = #(R n,1 \R n,2 ) is the number of vertices which have been visited exactly once by (X n ) n before time n. This number is greater than or equal to the number #(R ′ n,1 \R ′ n,2 ) of vertices which have been visited exactly once by the projected walk (X ′ n ) n before time n. According to [Pi74] this last number satisfies a strong law of large numbers and grows like p 2 n, where p is the probability that (X ′ n ) n never returns to its starting point. Since d ≥ 4 and simple symmetric random walk in three or more dimensions is transient, we have p > 0. Therefore, by (59), lim inf n→∞ X n · e 1 n ≥ 2εp 2 > 0.
In particular, (X n · e 1 ) n is transient to the right. Using a renewal structure like in [SZ99] for RWRE and in [BR07] for once-ERW and as outlined in Section 6 this implies that (X n · e 1 ) n satisfies a strong law of large numbers under P 0,ω 1 , i.e. P 0,ω 1 -a.s., (X n · e 1 )/n → v for some v ≥ 0. Due to (60) we even have v > 0. Since obviously P 0,ω 1 -a.s. X ′ n /n → 0 , this yields the statement (57). Open questions. We have already mentioned that not much is known about positively and negatively ERW in d ≥ 2. Below we shall discuss d = 1. It might be also interesting to consider ERW on strips Z × {0, 1, . . . , L}, L ≥ 1.
(a) Recurrent regime. For the case of non-negative cookies and δ < 1, D. Dolgopyat has shown, [Do08], (the case of strips was also considered) that, under some assumptions, X [nt] √ n converges in law to the unique pathwise solution W (t) (see [CD99]) of the equation where B(t) is the Brownian motion with variance t. Can this result be extended to positively and negatively excited random walks? What happens in the case |δ| = 1?
(b) Transient regime with zero linear speed. A.-L. Basdevant and A. Singh obtained in [BS08b] for non-negative (deterministic) cookie environments and 1 < δ ≤ 2 the following results.
1. If δ ∈ (1, 2) then Xn n δ/2 converges in law to a random variable S −δ/2 , where S is a positive strictly stable random variable with index δ/2, i.e. with Laplace transform E[e −λS ] = e −cλ δ/2 for some c > 0. 2. If δ = 2 then Xn n/ log n converges in probability to a positive constant. Their proof is based on the study of branching processes with migration but uses the assumption that all cookies are non-negative. The same result with essentially the same proof might hold in the more general setting studied in the current paper. Is there a result for δ = 2 similar to (ii) of [KKS75]?
(c) Transient regime with positive linear speed. We do not know whether our condition δ > 4 for the validity of the central limit theorem is optimal. How does the process scale for δ ∈ (2, 4]? Is the behavior similar to (iii) and (iv) of [KKS75]?