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Balls-in-bins models with asymmetric feedback and reflection

Balls-in-bins models describe a random sequential allocation of infinitely many balls into a finite number of bins. In these models a ball is placed into a bin with probability proportional to a given function (feedback function), which depends on the number of existing balls in the bin. Typically, the feedback function is the same for all bins (symmetric feedback), and there are no constraints on the number of balls in the bins. In this paper we study versions of BB models with two bins, in which the above assumptions are violated. In the first model of interest the feedback functions can depend on a bin (BB model with asymmetric feedback). In the case when both feedback functions are power law and superlinear, a single bin receives all but finitely many balls almost surely, and we study the probability that this happens for a given bin. In particular, under certain initial conditions we derive the normal approximation for this probability, which generalizes the result in [5] obtained in the case of the symmetric feedback. The main part of the paper concerns the BB model with asymmetric feedback evolving subject to certain constraints on the numbers of allocated balls. The model can be interpreted as a transient reflecting random walk in a curvilinear wedge, and we obtain a complete classification of its long term behavior.


Introduction
A balls-in-bins (BB) model is a classic probabilistic model describing a random sequential allocation of infinitely many balls into a finite number of bins. The probability of allocating a ball into a bin is proportional to a function, which depends on the number of existing balls in the bin. In the standard setup, the feedback function is the same for all bins (i.e. the feedback is symmetric), and there are no constraints on the number of allocated balls. This paper concerns two models of random sequential allocation of balls into bins, which can be regarded as versions of commonly studied BB models with a power law feedback function f (x) = x β , where β > 0. The following results for such s BB model are known (e.g. see [5], [9] and references therein). If β ≤ 1, then, with probability one, all bins receive infinitely many balls. For example, if β = 1 (the classical Pólya urn model), then the distribution of fractions of balls in the bins converges to a Dirichlet distribution. On the other hand, if β > 1, then, with probability one, a single random bin receives all but finitely many balls.
Our first model of interest goes as follows. Suppose that there are two bins labeled 1 and 2 respectively. If x i is the number of existing balls in the bin i = 1, 2, then the next ball is placed in the bin i with probability proportional to x β i i , where β i , i = 1, 2 are given positive constants. If β 1 = β 2 (the symmetric case), then this is the model studied in [5]. Here we focus on the asymmetric case β 1 = β 2 and both β 1 > 1 and β 2 > 1, in which, similar to the symmetric one, all but finitely many balls are placed in a single random bin almost surely, and we study the probability that this will happen in a given bin. The probability depends on initial conditions, and is trivially equal to 1 or 0 (depending on a bin) in most cases. There is, however, a subset of initial conditions, for which this probability is non-trivial, and we derive the normal approximation for it. This generalizes the result in [5], where the normal approximation for this probability is obtained in the symmetric case.
The main results of this paper concern the other model. This model can be thought of as the BB model with asymmetric feedback, where the balls are allocated subject to certain constraints. The model can be naturally interpreted as a reflecting random walk in a curvilinear wedge, and in what follows we call it the BB model with reflection.
We obtain a complete classification of the long term behavior of the BB model with reflection in the case when the boundaries of the wedge are given by power law functions. Briefly this can be described as follows. Let ζ i (t) be the number of balls in the bin i = 1, 2 at time t. Then, with probability one, the process (ζ 1 (t), ζ 2 (t)) eventually confines to a strip of a finite width along the boundary of the wedge. Moreover, we exactly calculate the width of the strip. The width is an integer that depends on the model parameters. This effect is somewhat similar to boundary effects detected in competition processes in [7] and [11].
Our motivation for the models in this paper has arisen from previous studies of growth models that can be thought of reinforced urn models with graph based interaction. In these models bins are labeled by vertices of a graph, and allocation probabilities at a bin depend on the configuration of allocated balls in the neighbourhood of that bin. Several such growth models have been studied in the case where the allocation probability at a bin is proportional to a log-linear function of the local configuration (see [1], [8] and [10]). A strong reinforcement mechanism in those models allowed to characterize precisely their limit behavior. In contrast to the growth models with the log-linear interaction, the growth models with interaction given by power law functions are harder to analyze. The BB model with the power law asymmetric feedback can be regarded as such a growth model without interaction. Secondly, the BB model with reflection can be regarded as a toy model mimicking the behavior of one of these growth models in a very particular situation.
The rest of the paper is organized as follows. In Section 2 we formally define both BB models of interest and state the main results. In Section 3 we collect some known facts about pure birth processes with continuous time. These facts are used in Section 4 to study the long term behavior of the BB model with asymmetric feedback. In Sections 5.1-5.2 we prove a series of lemmas that describe the most likely long term behavior of the BB model with reflection. These statements are used in Section 5.3 to prove the main results (Theorems 2.4 and 2.5 below) for the BB model with reflection. Finally, in Section 6 we state an open problem.

Models and results
Start with some notations and elementary facts. Let N = {1, 2....} and Z + = {0, 1, 2, ...} be sets of natural numbers and non-negative numbers respectively. We assume that all random variables and processes are realised on a probability space (Ω, F , P). The expectation with respect to the probability P is denoted by E. Given a Markov process with values in N 2 we denote by P z the distribution of the process started at z ∈ N 2 . Given quantities a(m) and b(m) depending on m ∈ N, we write a(m) ∼ b(m) to denote the fact that a(m)/b(m) → 1, as m → ∞.
Throughout we deal with many analytical equations involving integer valued quantities. This formally requires using the floor function and the ceiling function. However, most of the equations are of interest for us only asymptotically, when the effect of applying floor/ceiling functions can be neglected. Therefore, to simplify notations, we skip writing these functions.

The model with asymmetric feedback
In this section we define the BB model with asymmetric feedback and state the corresponding results. The model is as follows. Infinitely many balls are sequentially placed in two available bins according to the following rule. If x i is the number of existing balls in the bin i = 1, 2, then the next arriving ball is placed in this bin with probability proportional to x β i i , where β 1 > 0 and β 2 > 0 are given constants. Let X i (t) be the number of balls in the bin i = 1, 2 at time t ∈ Z + . Then the random process X(t) = (X 1 (t), X 2 (t)) ∈ N 2 , t ∈ Z + , is a discrete time Markov chain (DTMC) with the transition probabilities Define events A i = {all but finitely many balls are placed in the bin i} , i = 1, 2.
2) If β 1 ≤ 1 and β 2 > 1, then P x (A 2 ) = 1 for any x ∈ N 2 . Remark 1. In other words, Theorem 2.1 states that, if at least one of β-parameters is greater than one, then the monopolistic regime realizes almost surely. The theorem follows from the well known argument, that consists in embedding the DTMC X(t) into a pair of continuous time pure birth processes (see Section 3 for details).
Remark 2. Note that the curve determines a phase transition in the long term behavior of the process. The curve has a natural probabilistic interpretation explained in Remark 10 below. Theorem 2.3 below deals with the critical case α = α cr , ν = ν cr not covered by Theorem 2.2. Theorem 2.3. Let X(t) = (X 1 (t), X 2 (t)) be the DTMC with transition probabilities (2.1), where 1 < β 1 ≤ β 2 . Assume that x = (x, y) ∈ N 2 is such that y = y(x) = ν cr x αcr + µx δ + o(x δ ), as x → ∞, for some µ ∈ R and δ ∈ [0, α cr ), and let where Φ is the cumulative distribution function of the standard normal distribution.
3) If δ ∈ αcr 2 , α cr and µ = 0, then (a) lim x→∞ P x (A 1 ) = 0 for µ > 0, Remark 3. If β 1 = β 2 = β > 1, and X 1 (0) = a and X 2 (0) = a − ρ √ 2β−1 √ a, then, by Part 1) of Theorem 2.3, the probability of event A 1 is approximated by Φ(ρ) for sufficiently large a. This recovers the result of [5], where they obtained the normal approximation Φ(ρ) for the same probability in the symmetric case assuming that the process starts at (x, y), where The minor difference in initial conditions is just due to our attempt to somehow accommodate both the symmetric and the asymmetric case in a single statement (clearly, other variants of the initial condition are possible).
• We call the Markov chain ζ(t) with transition probabilities given in equations (2.7)-(2.9) the BB model with reflection in the domain Q ϕ,ψ . Curves y = ϕ(x) and y = ψ(x) are called the lower boundary and the upper boundary respectively.
• The one-step transitions of the DTMC ζ(t) described by equations (2.7) are called jumps (jumps to the right and jumps up), and the one-step transitions described by equations (2.8)-(2.9) are called reflections (by the lower and by the upper boundary respectively).
• We say that the process is reflected by the lower (upper) boundary, when the transition described by (2.8) (by (2.9)) occurs.
Remark 4. Note that, by construction, in the interior of the domain Q ϕ,ψ the DTMC ζ(t) evolves in the same way as the DTMC X(t) with transition probabilities (2.1). Equations (2.8)-(2.9) describe a reflection mechanism preventing the process from leaving the domain.
Remark 5. It should be also noted that our results for the DTMC ζ(t) concern only the case, when the boundaries are given by power law functions, that is ϕ(x) = x α and ψ(x) = x γ for some 0 < α < γ. However, it is convenient to have the model domain defined in the general case. First, it is convenient for future references (e.g. Section 6). Also, conditions in both (2.8) and (2.9) look more transparent (at least, as it seems to us) in the general case.
Let ζ(t) be the DTMC with transition probabilities (2.7)-(2.9). Define events for α ∈ (0, 1) and k ∈ N. (2.10) for γ ≥ 1 and k ∈ N. (2.11) (2.12) Remark 6. All events in (2.10)-(2.13) can be easily interpreted geometrically. For example, the event A α,k means that given any arbitrary small ε > 0 the process eventually confines to where it fluctuates by approaching both the lower boundary y = x α and the curve y = x α + k at any arbitrarily small distance. Note also the special case in (2.12), where the event E k means that the process eventually confines to the strip between the bisector y = x and the straight line y = x − k, and visits both of these lines infinitely often. In particular, the event E 1 simply means, that the process eventually follows a "zigzag" trajectory located between the bisector and the line y = x − 1 (see Remark 14). Theorems 2.4 and 2.5 below are the main results concerning the long term behavior of the model with reflection.
where events A α,k 1 and A γ,k 2 are defined in (2.10)-(2.11), k 1 and k 2 are integer solutions of the following inequalities .
Remark 7. In other words, both Theorem 2.4 and Theorem 2.5 state that, with probability one, the DTMC ζ(t) eventually confines to a strip of a finite width along either the upper, or the lower boundary of the process's domain. Given the model parameters, the width of the limit strip can be calculated. Moreover, the process fluctuates within this limit strip "touching" both boundaries of the strip infinitely many times. The main difference between these two theorems is that in the symmetric case (Theorem 2.4) the critical curve coincides with the bisector. In this case events (2.12) and (2.13) are impossible.

Pure birth processes with power law rates
For the reader's convenience we provide in this section some facts about pure birth processes with continuous time. The material of the section will used to obtain Theorems 2.1, 2.2 and 2.3.
We say that a random variable ξ has exponential distribution with parameter λ > 0 and is a continuous time Markov chain (CTMC) on Z + evolving as follows. The process stays at state i for a period of time given by a random variable ξ i ∼ Exp(λ i ) and then jumps to state i + 1 with probability one. Random variables (ξ i , i ≥ 0) are assumed to be independent. The random variable T k = ∞ i=k ξ i is called the time to explosion of the birth process started at Z(0) = k. It is known (e.g. [4]) that P(T 0 < ∞) = 1 if and only if (and, hence, P(T k < ∞) = 1 for k ∈ Z + ) we say that the process is explosive with probability 1. For the explosion time T k we have that Consider a pure birth process birth rates λ k = k β , k ≥ 1, where β > 1. Given the above, such a birth process is explosive with probability 1, and the mean and the variance of the explosion time T k in this case are as follows and, by Chebyshev's inequality, conditions of the Central Limit Theorem. Therefore, the random variable T k − m k s k converges in distribution to the standard normal distribution, as k → ∞. Moreover, since we obtain the following for the pure birth process with polynomial rates Therefore, by Berry-Esseen theorem (e.g. Theorem 2 in [4], Section 5, Chapter XVI), we have that where Φ is the cumulative distribution function of the standard normal distribution.
Remark 9. For comparison, consider a pure birth process with exponential transition rates λ k = e βk , where β > 0. Then, a direct computation gives that s 2 k = Var(T k ) ∼ C 2 e −2βk and r k ≤ C 3 e −3βk . Therefore, r k s 3 k ∼ C, and, hence, the normal approximation does not apply in this case.
Remark 10. The critical curve (2.4) is obtained by equating the mean explosion times of the corresponding continuous time birth processes with polynomial rates. Note that the critical curve lies below the curve y = x β 1 β 2 obtained by equating the probability of the jump up to the probability of the jump to the right.
Given continuous time birth processes (Z i (t), t ∈ R + ), i = 1, 2, we denote by T z,i the explosion time of the process Z i (t) started at z i ∈ N and define V x,y = T x,1 − T y,2 for x, y ∈ N. Theorem 2.1 basically follows from the well known embedding argument known as Rubin's construction ( [2]). We provide the proof just for the sake of completeness.

Proof of Theorem 2.2
The proof of Theorem 2.2 is rather straightforward and is based on the fact that the explosion times can be approximated by their expectations, as it follows from (3.4). We provide main details for the sake of completeness, and prove only Part 1) of the theorem, as Part 2) can be proved similarly.
,2 ) for some δ > 0 and sufficiently large x, and the claim of the theorem in this case follows again from independence of the explosion times and bound (3.4).

Proof of Theorem 2.3
We are going to use notations introduced in Sections 3 and 4.2.
Proof. Using equation (3.3) we obtain that Noting that we get the following approximation Further, by equation (3.3) we have that .
Further, observe that lim x→∞ P A x,y(x) = lim x→∞ P(V x,y(x) < 0) and . The normal approximation for explosion times discussed above yields the normal approximation for the random variable V x,y(x) . In particular, we have (similarly to (3.6)), that Summarizing the above, we obtain the results listed below.

Proofs of results for the model with reflection
In this section we prove a series of lemmas that describe the long term behavior of the DTMC ζ(t) in the case, when the process starts near either the lower, or the upper boundary. The case of the lower boundary is treated in detail in Section 5.1. The case of the upper boundary is considered in Section 5.2. The main results for the model with reflection, i.e. Theorems 2.4 and 2.5, will follow from the series of lemmas and explosiveness of the BB model with asymmetric feedback, see Section 5.3 for details.

The case of the lower boundary
Start with some definitions and auxiliary facts.
Definition 5.2. Let the lower boundary is given by the curve y = x α , for some 0 < α < α cr . Given r > 0 define the set (stripe along the lower boundary) If 0 < α < α cr , then αβ 2 − β 1 < 0, which implies the following approximation Proposition 5.1 below concerns a pair of simple geometric facts, that basically follow from sublinearity of the lower boundary y = x α and the reflection rule defined in (2.8). However, these facts are repeatedly used in subsequent proofs, therefore it is convenient to arrange them in a separate statement.
Proposition 5.1. Consider a trajectory of the DTMC ζ(t) that is reflected by the lower boundary infinitely many times.
In other words, such a trajectory approaches the lower boundary at any arbitrarily small distance.
2) If, in addition, there are at most m ≥ 0 jumps up between consecutive reflections, then such a trajectory eventually confines to the strip U α,m+1+ε for any given ε > 0.
Proof of Lemma 5.1. Denote for short A = {the DTMC ζ(t) jumps to the right, whenever possible}.
It is easy to see that, given an initial state ζ, the event A consists of a single trajectory, such that the component ζ 2 (t) increases only at those moments in time, when the process is reflected by the lower boundary. It is easy to see that, since the function y = x α is monotonically increasing to infinity, the trajectory does not leave the strip U α,C before the first reflection, and then, by Proposition (5.1), it confines to the strip U α,2 . By (5.4) Since α < α 1 = β 1 −1 β 2 ⇐⇒ αβ 2 − β 1 < −1, we get that ∞ k=x k αβ 2 −β 1 → 0, as x → ∞. Therefore, P ζ (A) → 1, as x → ∞, and the lemma is proved.
Proof of Lemma 5.2. Consider the same trajectory as in the proof of Lemma 5.1, i.e. where the component ζ 2 (t) increases only due to reflections at the lower boundary. It is easy to see that this trajectory is reflected by the lower boundary infinitely many times. Consequently, by Part 1) of Proposition 5.1, we have that lim inf t→∞ (ζ 2 (t) − ζ α 1 (t)) = 0. The component ζ 2 (t) increases by 1 at the moment of reflection, therefore, by Part 2) of Proposition 5.1, the trajectory eventually confines to the strip U α,1+ε for any given ε > 0, so that lim sup t→∞ (ζ 2 (t)−ζ α 1 (t)) = 1. Lemma 5.2 follows now from Lemma 5.1.
To proceed further, recall the second item in Definition 2.1 that distinguishes between jumps and reflections of the DTMC ζ(t). Lemma 5.3. Given α ∈ (0, α cr ), let ζ = (x, y) ∈ U α,C 1 and let N = N(x) = C 2 x 1−α for some C 1 > 0 and C 2 > 0. Let ξ N be the number of jumps up of the DTMC ζ(t) on the interval [0, N].
Suppose that x is sufficiently large. Then for any fixed m ≥ 1 the following bounds hold where and constants C 3 > 0 and C 4 > 0 do not depend on x.
Proof of Lemma 5.3. Start with noting that the condition α < α cr is equivalent to the condition αβ 2 − β 1 + 1 − α < 0, which, in particular, implies that λ x → 0, as x → ∞. Also, in this case we have that αβ 2 − β 1 < 0. Therefore, by equation (5.4), Next, assuming that the process does not jump up more than a fixed number of times (not depending on x) on the interval [0, N], we can approximate the probability of the jump up at any time t ∈ [0, N] by the probability of the jump up at time 0, that is, x αβ 2 −β 1 (see (5.9)) for sufficiently large x. This allows to approximate the distribution of the random variable ξ N by the binomial distribution with the mean λ x , which, in turn, can be approximated by the Poisson distribution with the same mean. More precisely, noting first that we obtain that P ζ (ξ N = 0) ∼ e −λx , as x → ∞.
Combining this with equation (5.9) gives that for any fixed i, which, in turn, implies the bound (5.7), as claimed.
Remark 11. Note that due to reflection both ζ 1 (t) → ∞, and ζ 2 (t) → ∞, as t → ∞. Therefore, t r,n < ∞ and t r,n < t r,n+1 for all n. In addition, note that Definition 5.4. Let (t r,n , n > n x ) be the sequence of time moments defined in Definition 5.3 for some ζ = (x, y). Define S r,n as the random variable that is equal to the number of jumps up of the DTMC ζ(t) on the interval [t r,n , t r,n+1 ).
Lemma 5.4. Let α ∈ [α k−1 , α k ) for some k ≥ 2, and let ζ = (x, y) ∈ U α,C for some C > 0. Let (S r,n , n > n x ) be the sequence of random variables defined in Definition 5.4. Then, P ζ (S r,n < k for all n > n x ) → 1, as x → ∞, (5.12) and P ζ (S r,n = k − 1 for infinitely many n) = 1. . Therefore, if r is sufficiently large, and S r,n < k for all n > n x , then {S r,n < k for all n} ⊆ {ζ(t) ∈ U α,2k−1+C for all t ≥ 0}, (5.14) i.e. the process cannot escape the strip U α,2k−1+C . Indeed, it is easy to see that the difference ζ 2 (t) − ζ α 1 (t) cannot be more than 2k − 1 + C. On the other hand, the difference ζ 2 (t) − ζ α 1 (t) can be equal to 2k − 1 + C. Indeed, this can happen, when the process is reflected near the end of the interval [t r,n , t r,n+1 ), immediately jumps up k − 1 times, and, jumps up again k − 1 times at the beginning of the next interval.
Further, by Definition 5. Without loss of generality, we can also assume that the initial position ζ = (x, y) is such that Then, it follows from (5.14)-(5.16) and Lemma 5.3 that P ζ (S r,n ≥ k|S r,nx < k, ..., S r,n−1 < k) ≤ C 3 λ k n , (5.17) and, hence, Note that the assumption α ∈ [α k−1 , α k ) can be rewritten as follows The left inequality in the preceding display implies that ∞ n=nx λ k n → 0, as n x → ∞. Combining this with (5.18) gives that Next, consider the case of an arbitrary r. It is easy to see that in this case time intervals [t r,n , t r,n+1 ) (r-intervals) can be covered by similar intervals with a sufficiently large r ′ > r (r ′ -intervals) in such a manner that each r-interval is covered by an r ′ -interval. Clearly, if it occurs that S r,n ≥ k for infinitely many n, then there are infinitely many r ′ -intervals, such that S r ′ ,n ≥ k. The probability of the latter tends to zero, as we have proved in the case of large r, and, hence, (??) holds for an arbitrary r.
Next, let us show (5.13). To this end observe that where F n−1 is the σ-field generated by ζ(t), t < t r,n . Indeed, it suffices to consider the case when the process starts near the lower boundary (i.e. ζ ∈ U α,C for some C > 0). By Lemma 5.3 and in (5.20), we have that ∞ n=1 λ k−1 n = ∞, and (5.13) follows from the conditional variant of the second Borel-Cantelli lemma (e.g. see Theorem 5.3.2 in [3]).
Proof of Lemma 5.5. We use notations of Lemma 5.4.
First, given a sufficiently large r > 0 consider a trajectory of the process, such that S r,n < k for all n. Comparing the growth of the trajectory with that of the lower boundary (similarly to the proof of Lemma 5.4) gives that such a trajectory is reflected by the lower boundary at least once on each time interval [t r,n , t r,n+1 ). Consequently, the trajectory is reflected infinitely many times. Combining this with Part 1) of Proposition 5.1 and Lemma 5.4 gives that lim x→∞ P ζ lim inf t→∞ (ζ 2 (t) − ζ α 1 (t)) = 0 = 1 for ζ = (x, y) ∈ U α,C . (5.23) Further, given n ≥ 1 let τ n be the time moment, when the process is reflected for the n-th time, and let τ 0 = 0. Let S n be the number of jumps up on the interval [τ n , τ n+1 ) for n ≥ 0. It is easy to see that, if S n < k for all n, then differences τ n+1 − τ n are uniformly bounded, provided that S n < k for all n. Therefore, any of these intervals is covered by some deterministic interval [t r,n , t r,n+1 ) for a sufficiently large r. Consequently, with a little of extra work, it follows from Part 1) of Lemma 5.4 that lim x→∞ P ζ (S n < k for all n) = 1, for ζ = (x, y) ∈ U α,C .
Combining this result with Part 2) of Proposition 5.1, gives that, with P ζ -probability tending to 1, as x → ∞, a) the DTMC ζ(t) eventually confines to the strip U α,k+ε for any given ε > 0.
On the other hand, b) for any given ε > 0 the DTMC ζ(t) almost surely exits the strip U α,k−ε infinitely many times.
Indeed, choose r > 0 in Part 2) of Lemma 5.4 is such that the lower boundary y = x α grows between points (nr) 1 α and ((n + 1)r) 1 α by not more than an arbitrarily small ε ′ > 0. By Part 2) of Lemma 5.4, almost surely there are infinitely many intervals t r,n , t r,n+1 ), where the process jumps up exactly k − 1 times. This gives b).
Items a) and b) imply that The lemma now follows from (5.23) and (5.24).

The case of the upper boundary
In this section we study the long term behavior of the DTMC ζ(t) in the case, when the process starts near the upper boundary. Recall that 1 < β 1 ≤ β 2 , and the upper boundary is given by the curve y = x γ , where γ > α cr = β 1 −1 β 2 −1 . We distinguish three cases, namely, γ > 1 (superlinear upper boundary), γ = 1 (the case of the bisector) and α cr < γ < 1 (sublinear upper boundary). Note that the last two cases are possible only in the asymmetric case 1 < β 1 < β 2 . The case of the super-linear upper boundary and the case of the bisector are considered in Section 5.2.1, and the case of the sublinear upper boundary is considered in Section 5.2.2.

Upper boundary: the bisector and above
In this section we study the most probable long term behavior of the DTMC ζ(t) in the case when it starts near the upper boundary given by y = x γ , where γ ≥ 1. In this case the corresponding results are similar to the results obtained in Section 5.1 in the case of the lower boundary. Therefore, we only state the results and relate them to their analogues in Section 5.1. However, in the case of a bisector (which is possible only in the asymmetric case 1 < β 1 < β 2 ), the corresponding results can be refined (see Remark 14 and Lemma 5.9 below).
We start with some definitions.
Proof. Part 1) of Lemma 5.6 is an analogue of Lemma 5.1. By (5.27), the probability that the process jumps up whenever possible can be estimated by e − ∞ k=y k β 1 γ −β 2 . Both a) and b) yield, that β 1 γ − β 2 < −1. Therefore, the expression in the exponent above is a tail of a convergent series, so that it tends to 0, and, hence, the exponent tends to 1, as y → ∞.
Part 2) of Lemma 5.6 is the analogue of Lemma 5.2 and can be proven similarly.
Remark 14. Note a special case of Lemma 5.6, when β 2 − β 1 > 1 and γ = 1 (i.e. the upper boundary is given by the bisector). In this case, the process started at a state (x, x) follows the "zigzag" trajectory with P ζ probability tending to 1, as x → ∞.
Definition 5.8. Let (t r,n , n > n y ) be the sequence of time moments defined in Definition 5.7 for some ζ = (x, y). Define S r,n as the random variable that is equal to the number of jumps to the right of the DTMC ζ(t) on the interval [t r,n ,t r,n+1 ).
where events D i , i = 1, 2, 3 are defined below for infinitely many t}.
Proof of Lemma 5.10. Without loss of generality, assume that the initial state ζ = (x, y) ∈ W γ,1 , in which case y = x γ − 1 + ε 0 for some 0 < ε 0 < 1. Given 0 < ε < 1, define the sequence of stopping times t 0 = min(t : ζ γ 1 (t) ≥ x γ + ε 0 ), t n = min(t : ζ γ 1 (t) ≥ x γ + ε + n) for n ≥ 1, t ′ n = min(ζ γ 1 (t) ≥ x γ + 2ε + n) for n ≥ 1. Note that 0 ≤ t 0 < t ′ 1 < t 1 < ... < t ′ n < t n < t ′ n+1 < t n+1 < .... It follows from equation (2.9) and the choice of the initial condition, that the DTMC ζ(t) is reflected by the upper boundary all the time in the interval t ∈ [0, t 0 ). Also, the process can jump up only once in the interval [t 0 , t ′ 1 ). We show that, with P ζ probability tending to 1, as x → ∞, the process behaves in the same manner during the subsequent time intervals. Namely, it jumps up on each time interval [t n , t ′ n+1 ), and, as a result, it is reflected by the upper boundary all the time on each time interval t ∈ [t ′ n , t n ). Clearly, such a behavior implies that the process stays inside the strip W γ,1+ε for all t ≥ t ′ 1 . It is convenient to introduce random variables S n , n ≥ 0, where S n is the number of jumps up of the DTMC ζ(t) in the interval [t n , t ′ n+1 ). In terms of these random variables it suffices to show that lim ζ→∞ P ζ (S n = 1 for all n ≥ 0) = 1, (5.31) to guarantee the described above behavior of the process.

Proofs of Theorems 2.4 and 2.5
Proof of Theorem 2.4. Recall quantities α k , k ≥ 0 defined in (5.1). It follows from (5.2), that for a given α ∈ (0, α cr ) there exists k ≥ 1, such that α ∈ [α k−1 , α k ). It is easy to verify that α ∈ [α k−1 , α k ) is equivalent to (2.15). Similarly, given γ > 1, one can determine a unique k, such that γ ∈ (max(1, γ k ), γ k−1 ], or, equivalently, k satisfying (2.16). It follows from Theorem 2.2 (for the BB model without reflection) that, with probability one, the DTMC ζ(t) hits either the lower, or the upper boundary. If follows then from Lemmas 5.2, 5.5 and 5.8, that, with probability one, either the event A α,k 1 , or the event A γ,k 2 occurs. It is left to note that these events are disjoint, which finishes the proof.
Proof of Theorem 2.5. The proof of Theorem 2.5 is similar to the proof of Theorem 2.4. It only needs to be complemented by using Lemma 5.9, Remark 14 and Lemma 5.10, where appropriate. For example, Lemma 5.10 is needed to deal with the case of the upper boundary given by y = x γ for α cr < γ ≤ 1. We skip further details.

Open problem
In this paper we have given a complete classification of the long term behavior of the BB model with reflection in the domain Q ϕ,ψ (defined in (2.6)), where the lower and the upper boundaries are given by ϕ(x) = x α and ψ(x) = x γ respectively for some 0 < α < α cr < γ. Specifically, we have shown that, for any initial state ζ ∈ Q ϕ,ψ , with probability one, the DTMC ζ(t) eventually (i.e. after a finite time depending on the initial condition) confines either to the strip U α,k 1 (along the lower boundary), or to the strip U γ,k 2 (along the upper boundary), where integers k 1 and k 2 (interpreted as widths of the strips) can be calculated for a given set of the model parameters. We would like to stress that widths of both strips are finite.
We believe in a rather different behavior of the process in the case, when ϕ(x) = ν cr x αcr − b 1 x δ 1 and ψ(x) = ν cr x αcr + b 2 x δ 2 for some b 1 , b 2 > 0 and 0 < δ 1 , δ 2 < α cr . We conjecture that if both 0 < δ 1 ≤ α cr /2 and 0 < δ 1 ≤ α cr /2, then, with probability one, the process will be reflected infinitely many times by both the lower and the upper boundary, i.e. the process "visits" both boundaries infinitely many times. On the other hand, if α cr /2 < δ 1 ≤ α cr and α cr /2 < δ 2 ≤ α cr , then, with probability one, the process eventually sticks to one of the boundaries. At the same time, deviations of the process from the "final" boundary are not uniformly bounded, so that the process eventually confines to an indefinitely expanding strip along one of the boundaries. The open problem of interest is to determine the asymptotic width of the strip.