Weak convergence of the number of zero increments in the random walk with barrier

We continue the line of research of random walks with barrier initiated by Iksanov and M{\"o}hle (2008). Assuming that the tail of the step of the underlying random walk has a power-like behavior at infinity with exponent $-\alpha$, $\alpha\in(0,1)$, we prove that the number $V_n$ of zero increments in the random walk with barrier, properly centered and normalized, converges weakly to the standard normal law. This refines previously known weak law of large numbers for $V_n$ proved in Iksanov and Negadailov (2008).


Introduction
Let (ξ k ) k∈N be independent copies of a random variable ξ with distribution p k = P{ξ = k}, k ∈ N. The random walk with barrier n ∈ N is a sequence (R Plainly, (R (n) k ) k∈N 0 is a non-decreasing Markov chain which cannot reach the state n. In what follows we always assume that p 1 > 0 which implies that the random walk with barrier n will eventually get absorbed in the state n − 1.
The equalities l +ξ l+1 ≥n} define, respectively, the number of jumps, the absorption time and the number of zero increments before the absorption in the random walk with barrier n.
There is a large number of real life situations where the random walk with barrier appears naturally. Let PTC be a transport company, offering a tour to the national park. The PTC uses buses with total amount of seats n. Various groups of people book seats in order to visit the park. If the size of the group is less than remaining number of vacant seats, the request satisfied, otherwise it is turned down. The quantities of interest are the total number of groups applied T n+1 , the number of accepted groups M n+1 and the number of rejections V n+1 .
Another example is the work of a server. Imagine that a client has bought an internetpackage n Mb in size. Consider the downloading of files with the size being a multiple of 1 Mb: the server receives requests on download, if the size of file is lower than remaining size, then it starts downloading it, else blocks the request. Similarly to the example above, the quantities of interest in this case are the the total number of requests T n+1 , the number of downloaded files M n+1 and the number of blocked requests V n+1 .
In [10] (see also [8] for a particular case) it was shown that, if the law of ξ belongs to the domain of attraction of a stable law, M n , properly normalized and centered, weakly converges. Furthermore, the set of limiting laws is comprised of stable laws and the law of exponential subordinator. In [12] it was checked that the same group of results hold on replacing M n by T n . Finally, in [11] it was proved that: (a) if Eξ < ∞ then V n weakly converges (without normalization); (b) if the law of ξ belongs to the domain of attraction of an α-stable law with α ∈ (0, 1], equivalently if for some ℓ slowly varying at infinity, then V n /EV n P → 1 as n → ∞. To complete the picture, in this paper we give results about the weak convergence of V n . The treatment of V n calls for more delicate argument than that for M n and/or T n . Crudely speaking, while the asymptotics of M n and T n is based on the "first order" arguments, the asymptotics of V n needs the "second order" reasoning. As a result, the approach exploited in [10,11] does not help in the present situation. Moreover, regular variation (1) alone seems not to be enough to ensure the weak convergence of properly scaled and normalized V n and one has to impose more restrictive "second-order" condition on the tail P{ξ ≥ n}. In this work we prove a central limit theorem-type result for V n assuming for some c > 0, α ∈ (0, 1) and ε > 0.
In what follows we reserve notation η for a random variable with the beta (1 − α, α) law, α ∈ (0, 1), i.e., and is the logarithmic derivative of the gamma function.
The main result of this paper is given by the next theorem Theorem 1.1. Assume that (2) holds with α ∈ (0, 1), ε > 0 and c > 0. If α ∈ (0, 1/2] assume additionaly sup n≥1 np n P{ξ > n} < ∞. Then where N (0, 1) is a random variable with the standard normal law. Moreover, there is a convergence of the first absolute moments.
Our approach is based on the analysis of random recursive equation for (V n ). It is shown that the sequence (V n ) can be approximated by a suitable renewal counting process and the error of such an approximation is estimated in terms of an appropriate probability distance. A similar method has already been used in [7] to derive the weak convergence result for the number of collisions in beta coalescents.
The rest of the paper is organized as follows. In Section 2 we define the approximating renewal process and give random recursive equations for related quantities. The proofs are presented in Section 3. An auxiliary lemma is formulated and proved in Appendix.

Renewal process and recursion with random indicies
Given the sequence (ξ n ) n∈N , define a zero-delayed random walk S 0 = 0, S n = ξ 1 + . . . + ξ n , n ∈ N, and the first passage process The random variable Y n := n − S Nn−1 is called undershot. It was shown 1 in [11] that the sequence (V n ) n∈N satisfies the following recursion with random index where The recursion (5) can be slightly simplified by setting X n := V n + 1 {n>1} , then where likewise X ′ k d = X k for all k ∈ N and (X ′ k ) k∈N and Y n are independent. Clearly, the asymptotic behavior of X n is the same as of V n .
It is a classical observation due to Dynkin [3] that under the assumption (1) where η has density (3). Let (η k ) k∈N be iid copies of η. Define a zero-delayed random walk the corresponding renewal counting process where W t d = W ′ t for every t > 0 and (W ′ t ) t≥0 and η are independent. Comparing recursions (6) and (8) and in view of (7) we may expect that the weak asymptotic behavior of X n is the same as of W n . We will show, assuming (2), that this heuristic can be made rigorous and leads to the desired result on the asymptotic of V n .

Proofs
We start with a refinement of (7) by estimating the speed of convergence of Y n /n to η in terms of so-called minimal L 1 -distance. Let us recall its definition. Let D 1 be the set of probability laws on R with finite first absolute moment. The L 1 -minimal (or Wasserstein) distance on D 1 is defined by where the infimum is taken over all couplings ( X, Y ) such that X d = X and Y d = Y . For ease of reference we summarize the properties of d 1 to be used in this work in the following proposition.
Proposition 3.1. Let X, Y be random variables with finite first absolute moments. The distance d 1 has the following properties: (Int) d 1 (X, Y ) has an integral representation: (Rep) d 1 (X, Y ) has a dual representation: (Conv) For X, X n ∈ D 1 convergence d 1 (X n , X) → 0, n → ∞, is equivalent to X n d → X and E|X n | → E|X|, n → ∞.
We refer the reader to Chapter 1 in [13] for an introduction to the theory of probability metrics, in particular for the proofs of the aforementioned properties of d 1 .
In view of (Conv) characterization of d 1 the next lemma is indeed a refinement of (7).
Proof. The first equality follows from (Lin) property of d 1 . Using (Rep) we have From the distributional identity Substituting this into (10) and using the triangle inequality gives Letξ be independent ofη andξ d = ξ,η d = η. The first term can be written as where we have utilized (Lin) property of d 1 in the second equality.

Proof of Theorem 1.1
It is enough to prove Theorem 1.1 for V n replaced by X n . In view of (Conv) property of d 1 , in order to prove Theorem 1.1 we need to check Using the triangle inequality yields for n ≥ 2, The second term converges to zero in view of the CLT for the renewal process with finite variance (see Chapter XI.5 in [4]) as well as the convergence of first absolute moments (see Proposition A.1 in [9]). From (Lin) property of d 1 we see that it is enough to prove Using the recursions for X n and W n we have, in view of (Lin) property of d 1 , Passing to infimum over all such pairs in both sides of inequality leads to In order to estimate c n we proceed as follows. Let (Ŷ n ,η) be a coupling of Y n and η such that d 1 (log Y n , log(nη)) = E| logŶ n −log(nη)|. Let (ν t ) t∈R be a copy of (ν t ) t∈R independent of (Ŷ n ,η). We have where the penultimate inequality follows from the definition of d 1 , since (Ŷ n ,η, (ν(t))) is a particular coupling. There exists ρ > 0 such that the last two summands are O(n −ρ ).
It remains to apply Lemma A.1 from [6] with φ n ≡ 1 to (12) to conclude that The proof of Theorem 1.1 is complete.

Appendix
The next lemma is the main ingredient in the proof of Proposition 3.2.
Lemma 4.1. Assume that θ is a random variable on [1, +∞) such that for some c > 0, α ∈ (0, 1) and ε > 0 Let η be a random variable with density (3) independent of θ. Then for every β > 0 there exists δ > 0 such that Proof. Denote the left-hand side of (16) by s θ (x, β). In view of relations and ), x → ∞, it is enough to prove the result for c = 1. Fix β for the rest of the proof. Using representation (Int) from Proposition 3.1 we have Integrating by parts the first probability in the integrand, we obtain for z ∈ [0, 1) and x > 1 + β, Let θ α be a random variable independent of η and with distribution By the same reasoning as above, Subtracting the corresponding equations and using (15) we have for z ∈ [0, 1) and x > 1 + β, for some K > 0 which does not depend on x and z. Therefore, Firstly we calculate I 2 (x) explicitly as follows: Pick ε ′ ∈ (0, ε] such that α + ε ′ < 1. The third summand I 3 (x) is estimated using the Fubini's theorem: It remains to bound the first integral. To this end, note that for every z ∈ [0, 1) and x ≥ 1 + β, and therefore Putting this into I 1 (x) yields The second term is O(x −α−1 log x) by the same argument as was used in the estimation of I 2 (x). Using simple algebra we obtain that the first term is equal to (1 − y −1 z) −α P{η ∈ dy} − P{η ≤ β −1 xz} z −1 dz. (17) The first summand, again by the Fubini's theorem, is calculated easily: