Perturbing transient random walk in a random environment with cookies of maximal strength

www.imstat.org/aihp 2013, Vol. 49, No. 3, 638–653

DOI:10.1214/12-AIHP479

© Association des Publications de l’Institut Henri Poincaré, 2013

Perturbing transient random walk in a random environment with cookies of maximal strength

Elisabeth Bauernschubert

Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany.

E-mail:[email protected]

Received 27 October 2011; revised 23 January 2012; accepted 25 January 2012

Abstract. We consider a left-transient random walk in a random environment onZthat will be disturbed by cookies inducing a drift to the right of strength 1. The number of cookies per site is i.i.d. and independent of the environment. Criteria for recurrence and transience of the random walk are obtained. For this purpose we use subcritical branching processes in random environments with immigration and formulate criteria for recurrence and transience for these processes.

Résumé. Nous considérons une marche aléatoire unidimensionnelle en environnement aléatoire qui est transiente à gauche. Cette marche est modifiée par des cookies qui induisent une dérive vers la droite. Le nombre de cookies par site est i.i.d. et indépendant de l’environnement. Des critères pour la récurrence et la transience de la marche sont obtenus. Pour cela, nous utilisons des processus de branchement sous-critiques en environnement aléatoire avec immigration et nous formulons des critères de récurrence et de transience pour ces processus.

MSC:60J80; 60J85; 60K37

Keywords:Excited random walk in a random environment; Cookies of strength 1; Recurrence; Transience; Subcritical branching process in a random environment with immigration

1. Introduction

We investigate random walks with random transition probabilities. Therefore, set Ω :=([0,1]^N)^Z with elements ω=((ω(x, i))_i≥1)_x∈Z. Suppose that elements from Ω are chosen at random according to a probability measureP onΩwith corresponding expectation operatorE. For fixed environmentω∈Ω andz∈Zdefine a nearest-neighbour random walk(S_n)_n≥0on a suitable probability spaceΩwith probability measureP_z,ω, which satisfies:

P_z,ω[S₀=z] =1, P_z,ω

S_n+1=S_n+1|(S_m)_1≤m≤n

=ω

S_n,#{m≤n: S_m=S_n} , P_z,ω

S_n+1=S_n−1|(S_m)_1≤m≤n

=1−ω

S_n,#{m≤n:S_m=S_n} .

In this wayω(x, i)is the transition probability fromxtox+1 upon theith visit at sitex. Furthermore let us denote the so-calledannealedoraveragedprobability measure byP_x[·] :=E[P_x,ω[·]], with corresponding expectation operator E_x. The process(S_n)_n≥0is calledrecurrent(transient) ifS_n=0 infinitely often (limn→∞S_n∈ {±∞})P₀-a.s.

1Supported by ERC Starting Grant 208417-NCIRW.

Fig. 1. Model of the random walk. If there are cookies at his current positionz∈Z, the random walker removes one and makes a step toz+1. If there is no cookie he jumps to the right with probabilitypzand to the left with probabilityqz:=1−pz.

In the case where forP-a.e.ωthere exists a sequence(p_z)_z∈Z∈ [0,1]^Zsuch thatω(z, i)=p_zfor alli≥1,z∈Z, and(pz)_z∈Zi.i.d. underP,(Sn)n≥0is called a one-dimensionalrandom walk in a random environment(RWRE). In general, we refer the reader to [15,17] for results and information about RWRE, but give here the recurrence/transience criteria for RWRE found in [15], Theorem (1.7).

Theorem RWRE (Solomon 1975). Consider an RWRE(Sn)n≥0and assume that0≤pz<1 for allz∈Zor0<

p_z≤1for allz∈Z.LetE[logρ₀]be well defined,whereρ₀:=(1−p₀)p₀⁻¹. (i) IfE[logρ₀]<0,thenlimn→∞S_n= +∞P₀-a.s.

(ii) IfE[logρ0]>0,thenlimn→∞S_n= −∞P0-a.s.

(iii) IfE[logρ0] =0,then−∞ =lim infn→∞Sn<lim sup_n→∞Sn= ∞P0-a.s.

We can see that the RWRE is a special case of a multi-excited (cookie) random walk (ERW) with infinitely many cookies per site. In the ERW-model ω(z, i)is not restricted to be constant inifor allz∈Z P-a.s. Excited random walks go back to the work of Benjamini and Wilson in [6]. Further studies of these processes and an extension to multi- excited random walks have been made by Zerner in [18,19], by Basdevant/Singh in [4,5] and by Kosygina/Zerner in [10]. In [18], Theorem 12, Zerner proves the following recurrence and transience criteria.

Theorem ERW (Zerner 2005). Assume that ω∈([1/2,1]^N)^Z P-a.s.and (ω(x,·))x≥0 is stationary and ergodic underP.Then,(S_n)_n≥0is recurrent if and only ifE[

i≥1(2ω(0, i)−1)] ≤1.

In [10] Kosygina and Zerner discussed an ERW with a bounded number of cookies, i.e.ω(z, i)=1/2 for alli > K for allzP-a.s. for some constantK, and showed the following in [10], Theorem 1.

Theorem ERW (Kosygina, Zerner 2008). LetK∈Nandω(z, i)=1/2for alli > Kfor allzP-a.s.Assume that (ω(z,·))z∈Zis i.i.d.,E[_K

n=1ω(0, n)]>0andE[_K

n=1(1−ω(0, n))]>0.Ifδ:=E[_K

n=1(2ω(0, n)−1)] ∈ [−1,1] then(S_n)_n≥0is recurrent.Ifδ >1 (δ <−1respectively)then(S_n)_n≥0is transient to the right(left respectively).

The recurrence and transience criteria for the RWRE and for the ERW seem to be quite different. So the challenging question arises whether one can find a unifying criterion. In the present paper we begin with a small step in our undertaking and consider the following combination of these processes which we will callexcited random walk in a random environment(ERWRE for short). For an illustration of the model discussed in this paper see Fig.1.

Consider an environment(pz)z∈Z∈(0,1)^Zand putMzcookies on every integerz∈Z. Now a nearest neighbour random walk(S_n)_n≥0is started at 0 with the following transition probabilities. If the random walker comes to sitez and if there is still at least one cookie on this site, he removes one cookie and jumps toz+1. Otherwise he makes a step to the right with probabilitypzand to the left with probabilityqz:=1−pz.

The cookies in our model have maximal strength and induce a drift to the right. On the other hand, an environment (p_z)_z∈Zis assumed that makes an RWRE (i.e. without cookies) tend to−∞. So jump- and cookie-environment cause a drift in opposite directions and the question naturally arises which drift is stronger in certain cases. We found criteria for transience and recurrence of the process.

Let us define the number of cookies of strength 1 at sitex∈Zby Mx:=sup

i≥1:ω(x, j )=1∀1≤j≤i

with the convention sup∅=0. For further discussions we postulate the following.

Assumption A.

A.1 It holdsP-a.s.that for allx∈Zthere existsp_x∈(0,1)such thatω(x, i)=p_xfor alli > M_x. A.2 (p_z)_z∈Zis identically distributed and{p_z, M_z, z∈Z}is independent underP.

A.3 E[|log(ρ0)|]<∞andE[log(ρ0)]>0whereρ_x:=(1−p_x)p⁻_x¹forx∈Z.

A.4 P[Mx= ∞] =0andP[Mx=0]>0for allx∈Z.

Recall thatA.3implies that an RWRE is left-transient, i.e.S_n→ −∞a.s. The goal of this paper is to show the following recurrence and transience criteria.

Theorem 1.1. Let AssumptionAhold and assume that(p_z, M_z)_z∈Zis i.i.d.andE[p⁻₀¹]<∞.

(i) IfE[(logM₀)₊]<∞,thenlimn→∞S_n= −∞P₀-a.s.

(ii) IfE[(logM0)₊] = ∞and iflim sup_t→∞(t·P[logM0> t]) <E[logρ0],thenS_n=0infinitely oftenP0-a.s.

(iii) Iflim inft→∞(t·P[logM0> t]) >E[logρ0],thenlimn→∞Sn= +∞P0-a.s.

In [4,5,10,18,19] the random walker steps to the right or to the left with equal probability at sites without cookies.

Furthermore, the number of cookies per site is bounded in [4,5,10]. In the case of ERWs in two and higher dimen- sions, Holmes investigates in [8] the velocity, allowing infinitely many cookies, and discusses a question of “which drift wins?” for a certain subclass of models, so-called “excited against the tide” walks. Schapira considers in his unpublished paper [13] a model similar to the one used in this paper, where the cookies induce a positive drift and the walker gets a negative drift on sites without cookies. But the number of cookies per site is still bounded. The novelty in our model is that the number of cookies per site is not necessarily bounded and that we allow a random environment for the transition probabilities on sites without cookies. However, cookies of maximal strength are considered only.

For the proof of Theorem1.1a well-known relationship between branching processes and random walks is used.

This method also has been employed in [4,5,10]. In our case we have to deal with a subcritical branching process in a random environment with immigration and we also intend to prove a recurrence/transience criterion for this process.

As can be seen in our further discussion there is some similarity to random difference equations.

Since the late 1960s several authors worked on branching processes in random environments (BPRE for short), for example Athreya, Karlin, Smith and Wilkinson in [1,2,14]. In [12] Pakes proved some recurrence/transience criteria for subcritical branching processes with immigration, but without a random environment. The results presented in our paper differ from the ones in [12], mainly because of the extension to a random environment, but also if we assumed fixed environments.

Let us now give the structure of this paper in a few words. Section2is dedicated to subcritical branching processes in random environments with immigration. In Section3we consider the model with cookies on the positive integers only. The process with cookies on the negative integers only will be discussed in Section4. Finally, Theorem1.1is proven in Section5and examples are given for the different cases of this theorem.

2. Transience and recurrence of subcritical branching processes in a random environment with immigration For information about branching processes in general we refer to [3]. In our paper we deal with the following Smith–

Wilkinson model extended by immigration, see also [2] and [3], VI.5 and VI.7.1:

Definition 2.1. LetΛdenote the set of probability generating functions(p.g.f.)that is isomorphic to the set of prob- ability distributions onN0 and let ϑ be a probability distribution on N0.Let{ξ_i^{(j )}, M_n, i, j, n∈N}be a family of N0-valued random variables on a suitable probability space with sample spaceΛ equipped with a set of measures {Q_ϕ, ϕ∈Λ^N}such that for every fixedenvironmentϕ=(ϕ_j)_j∈N∈Λ^Nthe family{ξ_i^{(j )}, M_n, i, j, n∈N}is indepen- dent,(ξ_i^{(j )})i∈Nis identically distributed with p.g.f.ϕj forj∈Nand(Mn)n∈Nis identicallyϑ-distributed underQϕ. We call(Z_n)_n≥0,defined byZ₀=1and

Z_n:=ξ₁⁽ⁿ⁾+ · · · +ξ_Z⁽ⁿ⁾

n−1+M_n abranching process with immigration.

Letν denote a probability measure onΛandϕbe randomly chosen according toQ˜ := ⊗_Nν onΛ^N with corre- sponding expectation operatorE.˜ Now we define the annealed measure byQ[·] := ˜E[Q_ϕ[·]],byE_Qits expectation operator and call(Z_n)_n≥0aBPRE with immigrationunderQ.

In the above definitionξ_i⁽ⁿ⁾gives the number of progeny of theith member of generationn−1, andM_nthe number of immigrants in generationn. All members in a generation reproduce according to the same offspring distribution.

Due to our further discussion, especially in Section3, we use the same notation for immigrants as for cookies. Note that(ϕ_n, M_n)_n∈Nis identically distributed and{ϕ_n, M_n, n∈N}is independent underQ. Furthermore, we remark that (Z_n)_n≥0is a time-homogeneous Markov chain under Q. In order to speak about recurrence and transience of the process it is assumed that the BPRE with immigration is irreducible underQ.

Let us setE_ϕfor the expectation operator corresponding toQ_ϕandμ_n(ϕ):=E_ϕ[ξ₁⁽ⁿ⁾]for the expected number of offspring produced by a single member of generationn−1. Analogously to the literature, we call(Zn)n≥0subcritical, critical or supercritical, if E_Q[logμ1]<0,=or >0 respectively. This is due to the fact that under the first two assumptions, a BPRE without immigration is mortal, whereas under the third assumption this process can explode, see [14], Theorem 3.1 or [3], VI. 5.5. Note the analogy to TheoremRWRE(i)–(iii), see also [10], Remark 2.

Let us now give criteria for recurrence and transience of a subcritical BPRE with immigration in Theorems2.2 and2.3. We remark that these criteria are similar to the ones for random difference equations found in [9], Theo- rem 3.1.

Theorem 2.2. Let(Z_n)_n≥0be a BPRE with immigration according to Definition2.1.Suppose thatE_Q[|log(μ1)|]<

∞,E_Q[log(μ1)]<0,E_Q[μ⁻₁^θ]<∞for someθ >0andE_Q[Varϕ(ξ₁⁽¹⁾)·μ⁻₁²]<∞. Iflim inft→∞(t·Q[logM₁> t]) > E_Q[log(μ⁻₁¹)],then(Z_n)_n≥0is transient.

Proof. The proof will be divided into two parts. First, we discuss a random difference equationX_nforn∈N0, and show that Q[

n≥1{Xn> n²}]>0. The second part consists in coupling (Xn)n≥0 and the subcritical BPRE with immigration in order to show thatQϕ[limn→∞Zn= ∞]>0 forQ-a.e.ϕ.

First, let us define some constants. We write x:=E_Q

log μ⁻₁¹

>0

and choosec1<∞,c2>1 andc3>1 such that lim inf

t→∞

t·Q[logM₁> t]

> c₁>logc₂+c₃x >0.

Furthermore, letε∈(0, c1−(logc2+c3x))and define

γ:= c₁

ε+logc₂+c₃x >1.

We choose 0< l <1 such that(1−l)γ >1.

In order to make use of a convergence property, set T :=inf

k∈N: ∀n≥k: 1 n

n

m=1

logμm≥ −c3x

.

Then,T <∞Q-a.s. by the law of large numbers. A result about large deviations under the assumptionE_Q[μ⁻₁^θ]<∞, see [7], p. 71, gives

Q _n

m=1

logμ⁻_m¹> nc3x

≤e^−n·c

for somec >0. This implies E_Q

T^{1/ l}

=

n≥1

Q[T =n] ·n^{1/ l}≤

n≥1

Q _n−1

m=1

logμ⁻_m¹> (n−1)c3x

·n^{1/ l}<∞. (1)

We begin now with a random difference equation defined byX0:=0 and Xn+1:=αn+1Xn+Mn+1,

withα_n:=μ_nc⁻₂¹< μ_n. Induction yields

X_n=α_n· · ·α₂M₁+α_n· · ·α₃M₂+ · · · +α_nM_n−1+M_n. We follow [9] in setting

W_n:=M₁+α₁M₂+α₁α₂M₃+ · · · +α₁· · ·α_n−1M_n

forn∈N. Since(ϕ₁, M₁), . . . , (ϕ_n, M_n)are exchangeable, the law ofX_nand the law ofW_nare the same, in particular Q

X_n≤n²

=Q

W_n≤n²

for alln∈N. (2)

Our first goal is to show that Q

lim inf

n→∞

X_n> n² =1. (3)

Therefore, let us start with a discussion ofW_n: Q

W_n≤n²

=Q

W_n≤n², T ≥n^l +Q

W_n≤n², T < n^l

≤Q T ≥n^l

+Q

W_n≤n², T < n^l

, (4)

and by definition ofα_i andT: Q

W_n≤n², T < n^l

≤Q

n^l≤i<n

M_i+1≤n²(α1· · ·α_i)⁻¹ , T < n^l

≤Q

n^l≤i<n

M_i+1≤n²

c₂·e^c3xi

, T < n^l

≤Q

n^l≤i<n

Mi+1≤n²

c2·e^c3xi

=

n^l≤i<n

Q

M₁≤n²

c₂·e^c3xi

=exp

n^l≤i<n

log 1−Q

M₁> n²

c₂e^c3xi

≤exp

−

n^l≤i<n

Q

M₁> n²

c₂e^c3xi

. (5)

According to lim inft→∞(t·Q[logM1> t]) > c1, we get fornlarge enough

n^l≤i<n

Q

M₁> n²

c₂e^c3xi

>

n^l≤i<n

c₁

2 logn+i(logc2+c3·x)≥

n^l≤i<n

c₁

iε+i(logc2+c3·x)

≥γ

log(n)− log

n^l +1

. (6)

Thus, combining (5) with (6) yields fornlarge enough Q

W_n≤n², T < n^l

≤e^{−γ·log(n1−l)}·e^γ ≤n^(l−1)·γ·e^γ. And finally we get by (2) and (4)

n≥1

Q

X_n≤n²

≤

n≥1

Q

T^{1/ l}≥n +Q

W_n≤n², T < n^l

<∞

since

n≥1n^(l−1)·γ<∞and

n≥1Q[T^{1/ l}≥n] ≤E_Q[T^{1/ l}]<∞, see (1).

Thus, by the Borel–Cantelli Lemma, we obtain (3). This implies

Q_ϕ

n≥1

X_n> n² >0 forQ-a.e.ϕ, (7)

since forQ-a.e.ϕ there existn0∈Nandx0> n²₀such thatXn₀ =x0andXn> n²for eachn≥n0+1 with positive probability underQ_ϕ. On the other hand, the assumptions of the theorem give thatX_j> j²for each 1≤j < n₀and X_n0≥x₀with positive probability.

Our next objective is to couple (X_n)_n≥0 and BPRE with immigration. Therefore, recall Definition 2.1for the branching process(Zn)n≥0. If we couple these processes, the increments of the difference equation correspond to the immigrants in the BPRE and the multiplication withαnto the expected number of descents of one individual in generationn−1, multiplied byc⁻₂¹. Our goal is to show thatQϕ[

n≥1{Zn≥Xn> n²}]>0 forQ-a.e.ϕ.

We define forn∈N0

B_n:=

n

j=1

{Z_j≥X_j} ∩

k≥1

X_k> k² .

Then, we get forQ-a.e.ϕ,

Q_ϕ[Z_n< X_n, B_n−1] =

k>(n−1)²

Q_ϕ[Z_n< X_n, Z_n−1=k, B_n−1]

≤

k>(n−1)²

Qϕ

μn·k−

k

i=1

ξ_i⁽ⁿ⁾> (μn−αn)·k, Zn−1=k, Bn−1

=

k>(n−1)²

Q_ϕ

μ_n·k− k

i=1

ξ_i⁽ⁿ⁾>

1−c⁻₂¹ μ_n·k

·Q_ϕ[Z_n−1=k, B_n−1].

Now, Chebyshev inequality implies Q_ϕ

μ_n·k−

k

i=1

ξ_i⁽ⁿ⁾>

1−c⁻₂¹ μ_n·k

≤ Varϕ(_k

i=1ξ_i⁽ⁿ⁾) (1−c⁻₂¹)²·μ_n(ϕ)²·k²

≤ Varϕ(ξ₁⁽ⁿ⁾) (1−c⁻₂¹)²·μ_n(ϕ)²·k. Thus, we have

Q_ϕ[Z_n< X_n, B_n−1]< Varϕ(ξ₁⁽ⁿ⁾)

(1−c⁻₂¹)²·μ_n(ϕ)²·(n−1)²·Q_ϕ[B_n−1]

and therefore

n∈N

Q_ϕ[Z_n< X_n|B_n−1]<∞ Q-a.s. (8)

according to assumptionE_Q[Varϕ(ξ₁⁽¹⁾)·μ⁻₁²]<∞and [11], Theorem 4.2.1. It is easy to see that for alln≥1 and Q-a.e.ϕ Q_ϕ[Z_n< X_n|B_n−1]<1.Now we conclude from (7) and (8) that

Q_ϕ

n∈N

{Z_n≥X_n}

k∈N

X_k> k² =

n∈N

Q_ϕ[Z_n≥X_n|B_n−1]>0.

Combining this with (7) we can assert that forQ-a.e.ϕ Q_ϕ

nlim→∞Z_n= ∞

≥Q_ϕ

n∈N

{Z_n≥X_n}

k∈N

X_k> k² ·Q_ϕ

k∈N

X_k> k² >0.

Therefore, Q

nlim→∞Z_n= ∞

=E_Q

Q_ϕ

nlim→∞Z_n= ∞

>0

which completes the proof.

The following theorem gives a criterion for recurrence.

Theorem 2.3. Let (Z_n)_n≥0 be a BPRE with immigration according to Definition 2.1 with Q[M_n =0]>0 and ϕ₁(0) >0Q-a.s.Suppose thatE_Q[|log(μ⁻₁¹)|]<∞andE_Q[log(μ⁻₁¹)]>0.

Iflim sup_t→∞(t·Q[logM1> t]) < EQ[log(μ⁻₁¹)],then(Zn)n≥0is recurrent.

Proof. We will show that

n≥0Q[Z_n=0] = ∞, which is equivalent to the recurrence of the process since(Z_n)_n∈N0 is a time-homogeneous Markov chain underQ. For the proof we use a second well-known definition for a BPRE with immigration, illustrated in Fig.2, and employ the exchangeability of(ϕ_k, M_k)_1≤k≤nunderQforn∈N.

Equivalently to Definition2.1the BPRE with immigration can also be defined by Z_n=

n

j=0

Z_n−j(j ), n∈N0,

where forj∈N0,(Zn(j ))_n∈N0 is an ordinary BPRE characterized by offspring p.g.f.(ϕn+j)_n∈NwithZ0(0)=1 and Z₀(j )=M_j forj ≥1. Furthermore, the BPREs have to be independent under conditioning onϕ, see [3], p. 250. For an explanation see Fig.2.

For the moment let us fixn∈N. We exchange(ϕ₁, M₁)by(ϕ_n, M_n),(ϕ₂, M₂)by(ϕ_n−1, M_n−1), . . . , (ϕ_n, M_n)by (ϕ₁, M₁)and defineZ₀(0)=1 and

Z_n = n

j=0

Z_n−j(j ),

where Z_n−j(j ) uses the exchanged random vectors (ϕ_i, M_i)1≤i≤n, i.e. the distribution of Z_n−j(j ) is given by ϕ_n−j(ϕ_n−j−1(· · ·ϕ₁(s)· · ·))^Mn−j+1 – whereas the distribution of Z_n−j(j ) is given by ϕ_j+1(ϕ_j+2(· · ·ϕ_n(s)· · ·))^Mj – andZ₀(j )=M_n−j+1. Exchangeability of(ϕ_k, M_k)_1≤k≤nthus implies

Q[Zn=0] =Q Z_n=0

for alln∈N.

Fig. 2. BPRE with immigration. The figure shows a realization of the first generations of the BPREs(Zn(j ))n∈N0,j∈ {0, . . . ,3}. The p.g.f.ϕi for reproduction is written on the left side. Summation over thenth row givesZn.

The task is now to show that

n≥0

Q Z_n =0

= ∞.

Therefore let us examineQϕ[Z_n=0]. The strategy is the following:

In the first generations a high value of immigration is allowed – knowing that the size of the population becomes small, when the time passes by, since we consider a subcritical branching process – whereas later, when we come closer to the generation to be examined, there are only few people permitted to immigrate.

Choosec >0 such that lim sup_t→∞(t·Q[logM1> t]) < c < EQ[log(μ⁻₁¹)]. Thus, we obtain for larget∈R, Q[M₁> t]< c

logt <1. (9)

We consider someγ >1 such thatγ·c < E_Q[log(μ⁻₁¹)]and define fork∈N c_k:=e^{γ ck}>1.

Note that the lines of descent are independent under conditioning onϕand that Q_ϕ

Z_j(n−j )=0, Z₀(n−j )≤c_j

≥ϕ_j ϕ_j−1

· · ·ϕ₁(0)· · ·c_j

·Q_ϕ

Z₀(n−j )≤c_j for 1≤j≤n. Hence, we get for everyn∈N,

Q_ϕ Z_n=0

≥ Q_{ϕ n}

j=0

Z_n−j(j )=0,

n−1

j=0

Z₀(j )≤c_n−j

=Q_ϕ

Z₀(n)=0

·

n−1

j=0

Q_ϕ

Z_n−j(j )=0, Z₀(j )≤c_n−j

≥ Q[M1=0] · n

j=1

ϕj

ϕj−1

· · ·ϕ1(0)· · ·cj·

n−1

j=1

Q[Mj+1≤cj]

=:an. (10)

Sincea_n>0Q-a.s., we are able to consider forn≥2 an

a_n−1=ϕ_n ϕ_n−1

· · ·ϕ₁(0)· · ·c_n

·Q[M_n≤c_n−1]. (11)

In the case of subcritical BPREs, convexity implies, see e.g. [1], p. 1853, ϕ_n

ϕ_n−1

· · ·ϕ₁(0)

≥1−μ₁· · ·μ_n.

Chooseε >0 such thatγ·c+E_Q[logμ₁] +ε <0 and let T :=inf

k∈N: ∀n≥k: 1 n

1≤m≤n

logμm≤EQ[logμ1] +ε

.

By the law of large numbers,T <∞Q-a.s. For everyN∈Nwe obtain on{T ≤N}forn≥N, ϕ_n

ϕ_n−1

· · ·ϕ₁(0)· · ·c_n

≥(1−μ_n· · ·μ₁)^cn

≥

1−e^{n(EQ[logμ1]+ε)}c_n

≥1−e^{n(γ·c+EQ[logμ1]+ε)}. (12) Applying (9) yields

Q[M_n≤c_n−1] ≥1− 1

γ (n−1) fornlarge enough. (13)

Thus, we get from (11), (12) and (13) _a^an

n−1 ≥1−_n−¹₁ fornlarge enough and finally, by a criterion of Raabe and (10),

n≥0

Q_ϕ Z_n=0

= ∞ on{T ≤N}

for anyN∈Nand henceQ-a.s. Therefore, we can conclude

n≥0

Q[Z_n=0] =

n≥0

Q Z_n =0

=

n≥0

E_Q Q_ϕ

Z_n =0

= ∞.

3. Random walk in a random environment with cookies on the positive integers

We now return to the random walk(Sn)n≥0. This section deals with the case where cookies are only allowed on the positive integers. For its discussion a well-known connection between branching processes with migration and excited random walks is used. This idea was also employed in [4,5,10] for a simple symmetric random walk disturbed by cookies. For detailed explanations we refer to [10].

In our case, an analogous relation between a subcritical BPRE with immigration and the ERWRE is obtained. The purpose of this connection is to prove a recurrence and transience criterion for(Sn)n≥0.

Proposition 3.1. Let AssumptionAhold.AssumeP[M_z=0] =1for allz∈ −N0,(p_n, M_n)_n∈Ni.i.d.andE[p₀⁻¹]<

∞.

(i) Iflim inft→∞(t·P[logM₁> t]) >E[logρ₀],thenP₀[limn→∞S_n= +∞]>0.

(ii) Iflim sup_t→∞(t·P[logM₁> t]) <E[logρ₀],thenP₀[limn→∞S_n= +∞] =0.

Let us first derive the connection to BPRE with immigration and state some useful results. We denote the hitting time ofk∈Zby

T_k:=inf{n∈N: S_n=k}.

Lemma 3.2. Let Assumption A hold. Assume P[Mz =0] =1 for all z∈ −N0 and (pn, Mn)_n∈N i.i.d. Then P0[limn→∞Sn∈ {±∞}] =1.

Proof. We first show thatP_0,ω[−∞<lim sup_n→∞S_n<+∞] =0 forP-a.e.ω∈Ωby using the strong Markov prop- erty for the processH_n:=(S_m)_0≤m≤n and Borel–Cantelli. Assume that there existsz∈Zwith lim sup_n→∞S_n=z.

Thenzis visited infinitely many times. But so isz+1 sincep_z(ω) >0. This is a contradiction.

Similar considerations show thatP0,ω[−∞<lim infn→∞Sn<+∞] =0 forP-a.e.ω.

Finally, we claim that P0,ω[lim infn→∞Sn= −∞,lim sup_n→∞Sn= +∞] =0 forP-a.e.ω. This holds since – under the assumptions of the lemma – the environments on sitesz≤0 satisfy case (ii) of TheoremRWRE. Hence, we

haveP_0,ω[T₁= ∞]>0.

A relation between an RWRE and a BPRE without immigration has already been remarked in [10], Remark 2. We show that our model of an ERWRE can be associated with a BPRE with immigration. Considerk∈N. Then, it holds forP-a.e.ω∈Ωthat on the event{T_k< T₀<∞}the random walk has to doM_ksteps fromktok+1 after its first visit inkuntil all cookies are removed and until it gets a chance to return tok−1. Further steps fromktok+1 can be regarded as the number of successes in coin tossing prior to the first failure with an unfair coin of probabilitypkto succeed.

LetX_i^{(j )},i∈N,j∈Z, be a family of independent±1-valued random variables onΩ, such that P_z,ω

X_i^{(j )}=1

=ω(j, i) and P_z,ω

X^{(j )}_i = −1

=1−ω(j, i).

Following [10], the ERWRE can be realized recursively by Sn+1=Sn+X_#^(S_{mⁿ⁾_≤n:S

m=Sn} forn≥0.

We call the events{X_i^{(j )}=1}a success and{X_i^{(j )}= −1}a failure and define ξ_j^(k):=#

successes in X^(k)_i

i>M_k between the(j−1)th and thejth failure , V0:=1,

Vk:=ξ₁^(k)+ · · · +ξ_V^(k)

k−1+Mk.

Then, if AssumptionAand(p_n, M_n)_n∈Ni.i.d. hold, we obtain by Definition2.1that(V_k)_k≥0is a BPRE with immi- gration underP₁, with immigrants(M_k)_k≥1and progeny given by(ξ_i^{(j )})_i,j∈N, whereξ_i^{(j )}has geometric distribution geo_N0(1−pj), i.e.P1,ω[ξ_i^{(j )}=n] =pj(ω)ⁿ·(1−pj(ω))forP-a.e.ω.

For the formalization of the connection let us define analogously to [10]

U₀:=1 and U_k:=#{n≥0:n < T₀, S_n=k, S_n+1=k+1},

whereUkdenotes the number of upcrossings fromktok+1 before the return to 0 fork∈N. Analogously to Eqs (14) and (15) in [10] the following connection betweenUk andVk can be formulated and proven.

Lemma 3.3. Let AssumptionAhold and assume(p_n, M_n)_n∈Ni.i.d.It holds a.s.underP₁: U_k=V_k·1_{Vj>0∀0≤j <k} for allk≥1on{T₀<∞},

Uk≤Vk for allk≥1on{T0= ∞}.

The indicator function ensures the equality after the first return to 0 of the random walker or, equivalently, the first time the population became extinct.

Let us use the same definition of recurrence from the right as in [10], p. 1962.

Definition 3.4. The ERWRE is calledrecurrent from the right,if the first excursion to the right of0,if there is any,is P₀-a.s.finite,i.e.P₁[T₀<∞] =1.

Now the connection between ERWRE and BPRE with immigration can be formulated in the following way.