A fluid cluster Poisson input process can look like a fractional Brownian motion even in the slow growth aggregation regime
Vicky Fasen∗ and Gennady Samorodnitsky †
May 17, 2009
Abstract
We show that, contrary to the common wisdom, the cumulative input process in a fluid queue with cluster Poisson arrivals can converge, in the slow growth regime, to a fractional Brownian motion, and not to a Lévy stable motion. This emphasizes the lack of robustness of Lévy stable motions as ”birds-eye” descriptions of the traffic in communication networks.
AMS 2000 Subject Classifications: primary: 90B22 secondary: 60F17
Keywords: cluster Poisson process, fluid queue, fractional Brownian motion, input model, slow growth regime, scaling limit, workload process
1 Introduction
This paper concerns the asymptotic behavior of certain fluid random streams of the type that have often been taken as natural models of input to fluid queues and queuing networks. We are specifically interested in the effect of heavy tails on such asymptotic behavior. We are not considering the actual queues in this paper, in the sense that we only investigate the potential input process to a queue, and not what happens when the service starts. However, our task, which lies in understanding how the input process deviates from its completely regular and linear average behavior, will help understanding how an actual queue with such input behaves. Indeed, it is precisely the deviations from the average behavior that build the queue!
∗Center for Mathematical Sciences, Technische Universität München, D-85747 Garching, Germany, email: [email protected]. Parts of the paper were written while the first author was visiting the Department of Operations Research and Information Engineering at Cornell University. She takes pleasure in thanking the colleagues there for their pleasant hospitality. Financial support from the Deutsche Forschungsgemeinschaft through a research grant is gratefully acknowledged. †School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, email: [email protected]. Samorodnitsky’s research was partially supported by an NSA grant MSPF-05G-049 and an ARO grant W911NF-07-1-0078 at Cornell University.
1 The motivation for our interest in deviations of input processes from their average (as well as the motivation in the many other papers on this subject) lies in the fact that networks with heavy tailed inputs are difficult to analyze, since they are not well suited to Brownian or Poisson approximations. Nonetheless, it is believed that heavy tails cause unusual (and often negative) effects. For example, it is believed that infinite variance in the distribution of the file sizes or bandwidth requests in communication networks causes long range dependence and self-similar structure in the network (see e.g. Park and Willinger (2000)). Since queues with heavy tailed input are difficult to analyze directly, the hope has been to get insight into their behavior by approximating the input by something standard, specifically by the average, linear, stream plus a certain deviation from that average. In the influential paper of Mikosch et al. (2002) they showed that for the so-called ON/OFF model and the infinite source Poisson model, the properly compensated and normalized cumulative input in a fluid queue looks like a fractional Brownian motion in the fast growth regime and like a Lévy stable motion in the slow growth regime. This result was later extended to networks of fluid queues by D’Auria and Samorodnitsky (2005). A random field version of such results is in Kaj et al. (2007). The terms fast growth regime and slow growth regime are important. They refer to the relative magnitude of the time scale at which the input process is considered and the number of independent streams the input consists of. We will return to this important point shortly. Boundary regimes have been discovered as well; see e.g. Gaigalas and Kaj (2003). These previous results appear to indicate that the deviations from the average in a heavy tailed input process could look, in the limit, as either a fractional Brownian motion or a Lévy stable motion. These are two very different stochastic processes, one with light tails but strongly dependent increments, while the other with independent increments but heavy tails. One expects very different performance in a queue with such different inputs. What was needed, therefore, was a study of robustness of these two possible limits under departures from the very specific model assumptions of Mikosch et al. (2002). Such study was undertaken by Mikosch and Samorodnitsky (2007) in a general setup, described below. Consider a stationary marked point process
((Tm, Zm))m∈Z, where a possible interpretation in the language of communication systems views ... < T−1 <
T0 < 0 < T1 <... as the arrival times of data packets at a server and Zm as the file size of the data packet transmitted at Tm. Each arrival corresponds to a ”source”, and it transmits its data at a unit rate. Thus, Zm denotes also the transmission period. The number of active sources at time t is given by the process
1 U(t)= {Tm≤t t A(t)= U(y) dy = [Zm ∧ (t −Tm) − Zm ∧ (−Tm)], t ≥ 0, (1.2) 0 Z Z mX∈ 2 which has continuous sample paths and thus, reflects the fluid queue. Under the assumption that the marks (Zm) have, under the Palm measure, a finite mean, A(t) has a finite mean with E(A(t)) = µt where µ> 0 is the expected amount of data arriving at the server in [0, 1]. Let (Ai)i∈N be iid copies of the process A. We view each (Ai) as the input process of data generated by the ith “user”, with different users having nothing to do with each other, hence the independence assumption. With n such independent input processes and at a time scale M, the deviation of the cumulative input process from its mean is the stochastic process n Dn,M (t)= (Ai(tM) − µtM) for t ≥ 0. (1.3) Xi=1 One is interested in the limits of the sequence of processes (Dn,M ) as n and M grow to infinity. It is here where the idea of “fast growth” and “slow growth” appears. The terms fast growth regime and slow growth regime were introduced in Mikosch et al. (2002) for the ON/OFF and infinite source Poisson model, and they describe the relative rates at which n and M in (1.3) grow to infinity. Intuitively, the fast growth regime is the situation where the number n of the input processes is relatively large in comparison to the time scale M, while the slow growth regime means the opposite situation. In fact, the paper Mikosch et al. (2002) used a very specific boundary, n(M) ↑∞ as M ↑∞ such that the fast growth regime meant n ≫ n(M) while the slow growth regime meant n ≪ n(M). There are substantial reasons to separate the regimes. Indeed, if the number of sources n is very large, then the process Dn,M (t), t ≥ 0 in (1.3) is the sum of a very large number of iid terms that change relatively slowly. If the number of active sources in (1.1) has a finite variance (as is the case in most systems considered in literature), the same would be the case for the input process in (1.2). Then one would expect a Gaussian limit for the deviation from the mean of the cumulative input process as in a classical central limit theorem. On the other hand, if the time scale M is very large, then the main phenomenon in (1.3) is, actually, in the deviations of the individual input processes from their means, Ai(t) − µt, for large t. A priori, there is no reason to expect these latter deviations to be Gaussian-looking, unless very specific assumptions are imposed on the generic input process A(t), t ≥ 0). Those are assumptions that are not of the kind that is usually imposed in the literature on the input to communication systems. Correspondingly, Mikosch et al. (2002) discovered that, in the ON/OFF and infinite source Poisson model, the deviation from the mean of the cumulative input process have a stable limit in the slow growth regime. We take a related, but somewhat more general point of view on the notions of fast growth regime and slow growth regime, which was already used in Mikosch and Samorodnitsky (2007). Specifically, if there is a function n(M) ↑ ∞ as M ↑ ∞ such that a limit theorem holds when n ≫ n(M), we say that this limit theorem holds in the fast growth regime because the main effect in (1.3) is the averaging over the many independent input streams. In fact, this regime typically allows an iterated limiting procedure: first let n →∞ and then let M →∞. Similarly, if there is a function n(M) ↑ ∞ as M ↑ ∞ such that a limit theorem holds when n ≪ n(M), we say that this limit theorem holds in the slow growth regime because the main 3 effect in (1.3) are the deviations of the individual input processes from their means, and the limit will typically hold if we let first M →∞ and then n →∞. It has become a part of the folklore that in the former scenario a fractional Gaussian limit is likely to arise, while in the latter scenario a Lévy stable limit can be expected. What Mikosch and Samorodnitsky (2007) discovered was that the fractional Brownian limits of Mikosch et al. (2002) in the fast growth regime were very robust, and held under very general assumptions on the underlying stationary marked point process. On the other hand, the Lévy stable limits turned out to be non-robust, and very special conditions were needed to ensure such limits. One of the conclusions of Mikosch and Samorodnitsky (2007) was, in certain circumstances of a very irregular arrival process, a fractional Brownian limit was possible even in the slow growth regime. They provided a somewhat artificial example of such situation, and conjectured that the same was true in the important case of a cluster Poisson arrival process. It is the purpose of this paper to consider that case and establish the fractional Brownian limit. Once this is accomplished, we understand that the appearance of a fractional Brownian limit in the slow growth regime is not exotic but, in fact, can be possible under very natural and common assumptions. This emphasizes how robust the fractional Brownian motion limiting behavior is. In a related work, a reflected version of the fractional Brownian limit was established in Delgado (2007) for the workload in fluid queuing networks in a heavy traffic regime. We would like to mention, at this point, that for certain input point processes, changing the number n of independent input streams as above is equivalent to changing the intensity λ0 of an underlying Poisson process. This is true for the M/G/∞ model of Mikosch et al. (2002), and it is also true for the model considered in the present paper. For such point processes, it is possible (and natural) to distinguish between different situations according to the relative rates at which the Poisson intensity and the time scale grow to infinity. Accordingly, if there exists a function λ(M) ↑∞ such that a limit theorem holds if λ0 ≫ λ(M), one will say that the limit holds in the fast growth regime, and if a limit theorem holds under the assumption λ0 ≪ λ(M), one will say that the limit holds in the slow growth regime. In this paper we will use, nonetheless a description via a discrete number n of independent input streams, because this is a language in which a key result of Mikosch and Samorodnitsky (2007), which we use in this paper, is stated (see Theorem 3.3 below). We will use, correspondingly, the fast and slow growth regime terminology introduced above, that compares the rates of growth of the number n and time scale M. This paper is arranged as follows. The arrival cluster Poisson model we are working with is formally described in Section 2. The main result of the paper is stated and discussed in Section 3. The arguments required to prove the main result uses a number of renewal theoretical and extreme value results, some of which may be of independent interest. These appear in Section 4. Section 5 presents the proof of the main theorem. Finally, Section 6 contains additional lemmas and other technical results needed for the proof of the main theorem. 4 2 The cluster Poisson model We assume that the data files sizes (Zm)m∈Z form an iid sequence independent of the arrival process (Tm)m∈Z. Let the number of sources arriving at the server in the interval (s,t] be described by 1 N (s,t]= {s (i) initial cluster points, denoted by ... < Γ−1 < 0 < Γ1 < Γ2 < ... form a homogeneous Poisson process N with rate λ0; (ii) at each initial clustere center Γm an independent copy of a randomly stopped renewal point process Nc starts. A generic point process Nc has the form Nc[0,t]= N0[0,t] ∧ (K + 1), where N0 is a renewal point process with arrival times 0= T0 < T1 <..., and K is a nonnegative integer valued random variable independent of N0. The interarrival times Xk = Tk − Tk−1 for k ≥ 1 are iid random variables, with a common distribution F , and the cluster size K has distribution FK . The cluster with the initial point Γm will have the points Γm,k =Γm + Tk,m for k = 0,...,Km where (Tk,m)k∈N0 are independent copies of (Tk)k∈N0 , independent of (Γm)m∈Z. The within-cluster interarrival times and the cluster sizes are assumed to satisfy the following conditions. Assumption A (a) The interarrival distribution function satisfies F ∈ R−1/β with β > 1. (b) The cluster size distribution function satisfies F K ∈ R−α with 1 <α< min(2, β). (c) The marks (Zm) form a sequence of iid random variables, independent of the underlying 2 point process. Further, we will assume that E|Zm| < ∞. Notice that Assumption A(a) assures that the within-cluster interarrival times have infinite mean; it also makes the arrival process sufficiently irregular for our result. The Assumption A(b) makes sure that the data files transmitted within each cluster have infinite variance. Note that the intensity of N is λ = λ0(1 + E(K)). (2.1) 5 For our main result, Theorem 3.1 below, we will introduce an additional assumption on the interarrival distribution function F , as follows. Assumption B Assume that either 1. β < 2 and F (x) − F (x + 1) lim sup x < ∞, (2.2) x→∞ F (x) or 2. F is arithmetic, with step size ∆ > 0, and F {n∆} lim sup n < ∞ . (2.3) n≥0 F (n∆) Remark 2.1 We need the technical Assumption B above to obtain a local renewal theorem; see Lemma 4.3 below or Theorem 3 in Doney (1997). In fact, if the local renewal theorem is known to hold (if only in the form of an upper bound), then Assumption B is unnecessary. We conjecture that the local renewal theorem holds under (2.2) for any β > 1, regardless of whether or not F is arithmetic. We denote by ← h(u)= F (1/u)= uβl(u) for u> 1 (2.4) the generalized tail inverse function of the within-cluster interarrival time distribution (see Resnick (2006), Section 2.1.2). Here, l is a slowly varying function. One implication of Assumption A(a) is the weak convergence T⌊nt⌋/h(n) t≥0 =⇒ (S1/β(t))t≥0 (2.5) in D [0, ∞) as n →∞; see Kallenberg (2002), Theorem 16.14. Here (S1/β(t))t≥0 is an 1/β-stable subordinator. We will use the notation I(u) = inf{t ≥ 0 : S1/β(t) > u} for u> 0, (2.6) for its inverse process. Then the weak convergence F (r)N0 (0, ru] u≥0 =⇒ (I(u))u≥0 (2.7) in D [0, ∞) as r →∞ holds. In particular, the process (I(u))u≥0 is self-similar of index 1/β; cf. Meerschaert and Scheffler (2004). P We will continue using the notation =⇒ for weak convergence, −→ for convergence in proba- fidi bility, =ν⇒ for vague convergence, and =⇒ for weak convergence of the finite dimensional distri- d butions. For x ∈ R we write x+ = max(0,x). For two random variables X,Y the symbol X = Y means that X has the same distribution as Y . We will also adopt the following convention. We will use the notation α1, α2, β1 and β2 for positive numbers satisfying α1 <α<α2 and β1 <β<β2, in the sense that the statements in the text where this notation appears hold for any choice of numbers satisfying the above conditions with, perhaps, different multiplicative constants. 6 3 The Main Result Below is the main result of this paper. It describes a slow growth regime under which the properly normalized deviations from the mean process (1.3) converge to a fractional Brownian motion. For a positive sequence Mn ↑ ∞ serving as the time scale for a system with n input processes we define −2 −1 bn = nMnF (Mn) P(K > F (Mn) ) for n ≥ 1 . (3.1) q The sequence (bn)n∈N turns out to be the right normalization for process (1.3). Theorem 3.1 Let the Poisson cluster model satisfy Assumption A and B. Further, let Mn be a sequence of positive constants such that Mn ↑∞ and such that bn in (3.1) satisfies α−1 − β +ρ lim nbn = 0 (3.2) n→∞ −1 for some ρ> 0. Then the cumulative input process Sn(t)= bn Dn,Mn (t), n ≥ 1, t ≥ 0, satisfies fidi (Sn(t))t≥0 =⇒ (E(Z)BH (t))t≥0 as n →∞, and the limiting process BH is a fractional Brownian motion with 2+ β − α H = ∈ (0.5, 1) (3.3) 2β and 2λ ∞ Var(B (1)) = 0 y−(2+β−α)/βP(S (1) ≤ y) dy+ H 2+ β − α 1/β Z0 ∞ 2 2 2 +λ E I(w + 1)2−α + I(w)I(w + 1)1−α − I(w)2−α dw 0 2 − α α − 1 (2 − α)(α − 1) Z0 where (S1/β(t))t≥0 and (I(w))w≥0 are as in (2.5) and (2.6), respectively. Remark 3.2 Note for any ǫ> 0 there exist C > 1 such that −1 1 H−ǫ 1 H+ǫ C n 2 Mn ≤ bn ≤ Cn 2 Mn for n ≥ 1 , with H given by (3.3). Hence, a necessary and sufficient condition for (3.2) is that for some ǫ> 0 2β−α+1 +ǫ Mn ≫ n 2H(α−1) . This identifies (3.2) as a slow growth condition and explains the appearance of this term in the title of the paper. We will prove Theorem 3.1 by showing that the assumptions of Theorem 5.9 in Mikosch and Samorodnitsky (2007) are satisfied. For convenience, we state that theorem below, in a form simplified for the situation where the marks are independent of the arrival process. 7 Theorem 3.3 (Mikosch and Samorodnitsky (2007)) Consider a marked stationary point process, where the marks (Zm) are independent of the arrival process N (whose intensity is λ), and have a finite first moment. Let Mn be a sequence of positive constants with Mn ↑∞. Suppose that there exists a sequence bn ↑∞ such that the following conditions are satisfied: (a) Let Ni be iid copies of N. Then n −1 fidi bn (Ni (0, Mnt] − λMnt) =⇒ (ξ(t))t≥0, ! Xi=1 t≥0 where (ξ(t)) is some non-degenerate at zero stochastic process. (i) (b) Let (Zm )m∈Z for i ∈ N be iid copies of (Zm)m∈Z. Then n ⌊Mn⌋ −1 (i) E P bn (Zm − (Z)) −→ 0 as n →∞. Xi=1 mX=1 ∗ (c) Let Ii (0) be the total amount of data, of the ith input process, in the session arriving by time 0 which are not finished by that time. Then n −1 ∗ P bn Ii (0) −→ 0 as n →∞. Xi=1 −1 N Under these conditions the normalized process Sn(t)= bn Dn,Mn (t), n ∈ , t ≥ 0, satisfies fidi (Sn(t))t≥0 =⇒ (E(Z)ξ(t))t≥0 as n →∞. As we will see, the slow growth condition (3.2) is needed only for verification of condition (c) in Theorem 3.3. 4 Some Renewal and Extreme Value Theory Our first proposition in this section deals with the tails of randomly stopped random sums when both the individual terms and the number of terms have infinite means. It complements the existing results dealing with the situations where at least one of these means is finite; see e.g. Faÿ et al. (2006). Proposition 4.1 Let (Xk) be iid random variables independent of the positive integer-valued K random variable K with distribution function FK , and let TK = k=1 Xk with distribution function F . Let G be the distribution function of |X |. Assume that TK 1 P F K ∈ R−κ for some 0 <κ< 1 (4.1) and P(X1 >x) G ∈ R−γ for some 0 <γ< 1, lim = p ∈ (0, 1] . (4.2) x→∞ G(x) 8 Then P(TK >x) γκ lim = E (Sγ) , (4.3) x→∞ P K > G(x)−1 + where Sγ is a strictly γ-stable random variable such that −γ −γ P(Sγ >x) ∼ px and P(Sγ < −x) ∼ (1 − p)x as x →∞. In particular, F TK ∈ R−κγ. ← Proof. For k ≥ 1, let ak := G (1/k), and note that 1 k→∞ X1 + . . . + Xk =⇒ Sγ (4.4) ak (cf. (2.5)). For large M > 1 we write −1 −1 −1 P(TK >x) = P TK >x,K>MG(x) + P TK >x, K ≤ M G(x) −1 −1 −1 + P TK >x, M G(x) < K ≤ MG(x) (4.5) =: E1,M (x)+ E2,M (x)+ E3,M (x). Note that, as x →∞, −1 −κ −1 E1,M (x) ≤ P K > MG(x) ∼ M P K > G(x) , and so E (x) lim lim sup 1,M = 0 . (4.6) M→∞ x→∞ P K > G(x)−1 We claim, further, that for any κ < κ1 < 1, for all M large enough, E (x) lim sup 2,M ≤ M −(1−κ1) . (4.7) x→∞ P K > G(x)−1 Indeed, suppose that (4.7) fails for some κ < κ1 < 1. Then there is a sequence xj ↑∞ such that jG(xj) → 0 as j →∞ and 1 E (x ) ≥ j−(1−κ1)P K > G(x )−1 (4.8) 2,j j 2 j for j ∈ N. Let pk := P(K = k) for k ≥ 1. Note that −1 −1 ⌊j G(xj ) ⌋ E2,j(xj)= pkP(X1 + . . . + Xk >xj) . Xk=1 Theorem 9.1 in Denisov et al. (2008) shows that P(X1 + . . . + Xk >xj) ∼ kpG(xj) −1 −1 as j →∞ uniformly in k ≤ j G(xj) . Therefore, for large j, by Karamata’s theorem, ⌊j−1G(x )−1⌋ j 4p E (x ) ≤ 2 kp pG(x ) ≤ j−1P K > j−1G(x )−1 , 2,j j k j 1 − κ j k=1 X 9 and by Potters’s inequalities (cf. Resnick (2006), p. 36), for any κ < κ2 < κ1 there is C1 > 0 such that for large j −(1−κ2) −1 E2,j(xj) ≤ C1j P K > G(xj) . This, clearly, contradicts (4.8), and so (4.7) has to hold. We conclude that E (x) lim lim sup 2,M = 0 . (4.9) M→∞ x→∞ P K > G(x)−1 −1 −1 −1 We now consider the term E3,M (x) in (4.5). For M G(x) < k ≤ MG(x) we denote −1 −1 −1 −1 r = xak . Since G(ak) ∼ k as k →∞, we see that, for all x large enough, and M G(x) < k ≤ MG(x)−1, −1 −1 −1 −1 (2M) G(x) ≤ G(ak) ≤ 2MG(x) , −1 γ which implies that for the same range of x and k, (4M) ≤ r ≤ 4M. In particular, P(Sγ > r) is bounded away from 0. Since by (4.4) 1 P(X + . . . + X >x)= P (X + . . . + X ) > r −→ P(S > r) as k →∞ 1 k a 1 k γ k (if r is kept fixed), we conclude that P(X + . . . + X >x) lim sup 1 k − 1 = 0. x→∞ P(S > x/a ) M −1G(x)−1 Therefore, ⌊MG(x)−1⌋ E3,M (x) ∼ pkP Sγ > x/ak as x →∞. −1 −1 k=⌊M XG(x) ⌋+1 If f denotes the density of Sγ , this statement translates by Fubini into −1 ∞ ⌊MG(x) ⌋ 1 E3,M (x) ∼ pk {ak>x/y} f(y) dy 0 Z k=⌊M −1XG(x)−1⌋+1 −1 ∞ ⌊MG(x) ⌋ 1 ∼ pk {k>G(x/y)−1} f(y) dy 0 Z k=⌊M −1XG(x)−1⌋+1 ∞ = P max M −1G(x)−1, G(x/y)−1 < K ≤ MG(x)−1 f(y) dy . Z0 Now, for every y > 0, as x →∞, P max M −1G(x)−1, G(x/y)−1 < K ≤ MG(x)−1 −→ min M κ,yγκ − M −κ , P K > G(x)−1 + 10 while the same ratio on the left hand side is bounded from above, for large x uniformly in y > 0 by P K > M −1G(x)−1 ≤ 2M κ . P −1 K > G(x) Therefore, by the dominated convergence theorem, E (x) ∞ lim 3,M = min M κ,yγκ − M −κ f(y) dy. (4.10) x→∞ P K > G(x)−1 Z0 + E + γκ As M → ∞, the right hand side of (4.10) converges to (Sγ ) , and so the statement of the proposition follows from (4.5), (4.6), (4.9) and (4.10). The next two results are renewal theorems needed in the proof of the main theorem. Proposition 4.2 Let (Xk) be an iid sequence of positive random variables with distribution j N function F , such that F ∈ R−1/β, 0 < 1/β < 1. Let Tj = k=1 Xk, j ∈ 0. Suppose that (c(t))t≥0 is a non-negative eventually non-increasing function,P regularly varying of index −η in infinity, 1 <η< 2. Then ∞ 1 c(j)P(T >x) ∼ C F (x)−1c(F (x)−1) ∈ R as x →∞, j η − 1 η,β (1−η)/β Xj=0 E (η−1)/β where Cη,β = ((S1/β(1)) ), and (S1/β(t))t≥0 is the positive strictly 1/β-stable stochastic process in (2.5). Proof. Let Hβ be the distribution function of S1/β(1). Then by the weak convergence in (2.5) lim sup |Hβ(r) − P(Tn ≤ anr)| = 0, n→∞ r∈R ← where an = F (1/n) (cf. Petrov (1975), Theorem 11, p.15, and Theorem 10, p. 88). Thus, there exist a positive sequence (ǫj)j≥0 with ǫj ↓ 0 as j →∞ such that for any r> 0 P −1 (Tj > r) ≤ Hβ(aj r)+ ǫj. Let δ1, δ2 > 0, δ1 < δ2 and δ = (δ1, δ2). Then −1 −1 −1 ⌊δ2F (r) ⌋ ⌊δ2F (r) ⌋ ⌊δ2F (r) ⌋ P −1 c(j) (Tj > r) ≤ c(j)H β(aj r)+ c(j)ǫj =: J1(δ, r)+ J2(δ, r). −1 −1 −1 j=⌈δ1XF (r) ⌉ j=⌈δ1XF (r) ⌉ j=⌈δ1XF (r) ⌉ (r) First, we study the first summand. Let xj := jF (r) and ℓ be a slowly varying function such that F (x)= ℓ(x)x−1/β. Then, as n →∞, −1/β nF (an)= nℓ(an)an −→ 1. (4.11) 11 −1 −1 Since δ1F (r) ≤ j ≤ δ2F (r) we have for some C1,C2 > 0, for all r large enough, ← C1r ≤ F (j)= aj ≤ C2r. By Theorem 1.5.2 of Bingham et al. (1987) we obtain ℓ(aj) ∼ ℓ(r) as r → ∞ uniformly for −1 −1 −1 1/β −1 δ1F (r) ≤ j ≤ δ2F (r) . Thus, (4.11) gives ℓ(r) ∼ j aj as r →∞ uniformly for δ1F (r) ≤ −1 j ≤ δ2F (r) , and −β (r) −β −1/β −1 (xj ) = jℓ(r)r ∼ aj r as r →∞. Hence, as r →∞, −1 −1 ⌊δ2F (r) ⌋ ⌊δ2F (r) ⌋ −1 (r) −1 (r) −β c(j)H β(aj r) ∼ c xj F (r) Hβ((xj ) ) −1 −1 j=⌈δ1XF (r) ⌉ j=⌈δ1XF (r) ⌉ −1 ⌊δ2F (r) ⌋ −1 (r) (r) (r) −1 (r) −β = F (r) (xj+1 − xj )c xj F (r) Hβ((xj ) ). −1 j=⌈δ1XF (r) ⌉ Since c ∈ R−η we obtain by Theorem 1.5.2 of Bingham et al. (1987) as r →∞, −1 −1 ⌊δ2F (r) ⌋ ⌊δ2F (r) ⌋ 1 −1 (r) (r) (r) −η (r) −β c(j)H β(a r) ∼ (x − x )(x ) Hβ((x ) ) F (r)−1c(F (r)−1) j j+1 j j j −1 −1 j=⌈δ1XF (r) ⌉ j=⌈δ1XF (r) ⌉ δ2 −η −β ∼ y Hβ(y ) dy, Zδ1 and so δ2 −1 −1 −η −β J1(δ, r) ∼ F (r) c(F (r) ) y Hβ(y ) dy as r →∞. Zδ1 On the other hand, J (δ, r) ≤ ǫ −1 c(j) 2 δ1F (r) −1 j≥δ1XF (r) −1 −1 −1 ∼ ǫ −1 (η − 1) δ F (r) c(δ F (r) ) δ1F (r) 1 1 −1 1−η −1 −1 ∼ ǫ −1 (η − 1) δ F (r) c(F (r) ) as r →∞ (4.12) δ1F (r) 1 by Bingham et al. (1987), Proposition 1.5.10. Since δ is arbitrary and ǫ −1 → 0 as r → ∞ 1 δ1F (r) we obtain −1 −1 1 (η−1)/β lim lim lim F (r)c(F (r) ) (J1(δ, r)+ J2(δ, r)) = E((S1/β(1)) ). (4.13) δ2→∞ δ1↓0 r→∞ η − 1 Next, Proposition 1.5.8 in Bingham et al. (1987) and Lemma 6.4 result in c(j)P(Tj > r) ≤ C3 c(j)jF (r) −1 −1 j≤δ1XF (r) j≤δ1XF (r) 2−η −1 −1 ∼ C4δ1 F (r) c(F (r) ) 12 as r →∞ for some C3,C4 > 0. Hence, −1 −1 lim lim F (r)c(F (r) ) c(j)P(Tj > r) = 0. (4.14) δ1↓0 r→∞ −1 j≤δ1XF (r) Also, by Bingham et al. (1987), Proposition 1.5.10, −1 −1 −1 −1 F (r)c(F (r) ) c(j)P(Tj > r) ≤ F (r)c(F (r) ) c(j) −1 −1 j≥δ2XF (r) j≥δ2XF (r) 1 δ →∞ ∼ δ1−η −→2 0. (4.15) η − 1 2 By (4.13), (4.14) and (4.15) the result follows. The following result is a local renewal theorem. Lemma 4.3 Let the conditions of Proposition 4.2 hold, and assume additionally Assumption B. Then ∞ 1 c(j) [P(T >x) − P(T >x + 1)] ∼ C x−1F (x)−1c(F (x)−1) ∈ R as x →∞. j j β η,β (1−η)/β−1 Xj=0 Proof. Under the first scenario of Assumption B, the proof, using Proposition 4.2, is the same as the proof of Theorem 2 in Anderson and Athreya (1988), which in particular requires β < 2. Under the second scenario of Assumption B, the statement is Theorem 3 of Doney (1997). 5 Verification of the conditions of Theorem 3.3 The main result of this paper, Theorem 3.1, is proved in this section via verifying the conditions of Theorem 3.3. 5.1 Verification of condition (a) of Theorem 3.3 Proposition 5.1 Let the Poisson cluster model satisfy Assumption A and B. Further, let Mn, bn be a sequence of positive constants such that Mn ↑ ∞ and bn ↑ ∞, respectively. Let Ni be iid copies of N. Then n −1 fidi bn (Ni (0, Mnt] − λMnt) =⇒ (BH (t))t≥0, ! Xi=1 t≥0 with (BH (t))t≥0 as given in Theorem 3.1. Proof. We can write n n −1 −1 (0,Mnt] E (0,Mnt] bn [Ni (0, Mnt] − λMnt] = bn Ni (0, Mnt] − (Ni (0, Mnt]) i=1 i=1 h i X Xn −1 (−∞,0] E (−∞,0] +bn Ni (0, Mnt] − (Ni (0, Mnt]) i=1 h i + X − =: ξn (t)+ ξn (t), 13 A (i) Z (i) (i) (i) (i) (i) where Ni (B)=#{Γm,k : m ∈ , k ∈ {0,...,Km }, Γm,k = Γm + Tm,k ∈ B and Γm ∈ A}. We will show in Lemma 5.2 and Lemma 5.3 that + fidi + − fidi − (ξn (t))t≥0 =⇒ (BH (t))t≥0 and (ξn (t))t≥0 =⇒ (BH (t))t≥0, (5.1) + − where (BH (t))t≥0 and (BH (t))t≥0 are independent fractional Brownian motions of index H with time 1 variances 2λ ∞ σ2 = 0 y−(2+β−α)/βP(S (1) ≤ y) dy (5.2) + 2+ β − α 1/β Z0 and ∞ 2 E 2 2−α 2 1−α 2 2−α σ− = λ0 I(w + 1) + I(w)I(w + 1) − I(w) dw. 0 2 − α α − 1 (2 − α)(α − 1) Z (5.3) + − By the independence of (ξn (t)) and (ξn (t)), (5.1) implies that + − fidi + − (ξn(t))t≥0 := (ξn (t)+ ξn (t))t≥0 =⇒ (BH (t)+ BH (t))t≥0 =: (BH (t))t≥0, (5.4) 2 2 2 where (BH (t))t≥0 is a fractional Brownian motion with time 1 variance σ = σ+ + σ−. + − In order to prove (5.1) we notice that ξn (t) and ξn (t) are infinitely divisible random variables whose characteristic function can be written in the form ∞ E ± iθx ± (exp(iθξn (t))) = exp e − 1 − iθx νn,t(dx) , 0 Z ± where νn,t are the corresponding Lévy measures. These can be represented in the form ± P −1 νn,t = nλ0( 1 × Leb) ◦ ζ± , with the following notation. Let (Ω1, F1, P1) be a probability space on which a generic cluster process (Nc [0, u])u≥0 is defined. The maps ζ+ and ζ− are defined as follows: ζ+ :Ω1 ×(0, Mnt] → [0, ∞) is given by ζ+(ω1, u)= Nc [0, u] (ω1)/bn, and ζ− :Ω1 ×R+ → [0, ∞) is given by ζ−(ω1, u)= (0,Mnt] (−∞,0] Nc (u, u + Mnt] (ω1)/bn. To see this write Ni (0, Mnt] and Ni (0, Mnt] as integrals with respect to a Poisson random measure and use, for example, Lemma 12.2 (i) in Kallenberg (2002) (cf. proof of Proposition 3.5 in Faÿ et al. (2006)). For the notational simplicity below we often drop the subscript in P1 and, hence, write for A ∈B(R) Mnt ∞ + P − P νn,t(A)= nλ0 (Nc [0, u] /bn ∈ A) du, νn,t(A)= nλ0 (Nc (u, u + Mnt] /bn ∈ A) du. Z0 Z0 Since the Lévy measures are concentrated on the positive half line, we can apply standard results for the weak convergence of infinitely divisible distributions, see e. g. Theorem 15.14 in Kallenberg (2002). Without loss of generality we will assume λ0 = 1 in the following. Lemma 5.2 Let Assumption A hold and let t ≥ 0, ǫ> 0. Then + ν (a) νn,t =⇒ 0 on (0, ∞] as n →∞. 14 2 + 2H 2 2 (b) limn→∞ {|x|≤ǫ} x νn,t(dx)= t σ+ with σ+ as in (5.2) and H as in (3.3). R + (c) limn→∞ {|x|>ǫ} xνn,t(dx) = 0. In particular, R + fidi + (ξn (t))t≥0 =⇒ (BH (t))t≥0 + 2 where (BH (t))t≥0 is a fractional Brownian motion with Hurst index H and time 1 variance σ+. Proof. We use the decomposition 2 + x νn,t(dx) Z{|x|≤ǫ} Mnt Mnt∧T n n ⌊ǫbn−1⌋ = E 1 N [0, u]2 du + E 1 N [0, u]2 du b2 {K+1≤ǫbn} c b2 {K+1>ǫbn} c n Z0 n Z0 =: I1,1(n)+ I1,2(n), (5.5) and Mnt Mnt ∞ + P P xνn,t(dx) = nǫ (Nc [0, u] > ǫbn) du + n (Nc [0, u] > xbn) dx du Z{|x|>ǫ} Z0 Z0 Zǫ =: I2,1(n)+ I2,2(n). (5.6) The claim (b) now follows from Lemma 6.5 and Lemma 6.6, while the claim (c) follows from Lemma 6.7 and Lemma 6.8. Then (a) is a conclusion of + −1 + 0 ≤ lim sup νn,t(ǫ, ∞) ≤ lim sup ǫ xνn,t(dx) = 0 ∀ ǫ> 0. n→∞ n→∞ Z{|x|>ǫ} Hence, (a)-(c) and Theorem 15.14 in Kallenberg (2002) result in + + ξn (t)=⇒ BH (t) as n →∞, ∀ t ≥ 0. (5.7) Applying Lemma 4.8 of Kallenberg (2002) we see that for every k ≥ 1 and 0 ≤ t1 + + (ξn (t1),...,ξn (tk))n∈N, (5.8) are tight. Let BH = (BH (t1),..., BH (tk)) be a weak subsequential limit of this family, i.e. there exist a subsequence (ni) such that e e e + + B (ξni (t1),...,ξni (tk)) =⇒ H as i →∞. On the one hand, BH is infinitely divisible (becausee of the Poisson arrivals of clusters). On the B d + other hand, the one-dimensional marginal distributions of H are Gaussian with BH (ti) = BH (ti) e by (5.7). Hence, BH is zero mean multivariate Gaussian. We will compute now its covariance e e e 15 + + matrix. The stationarity of the Ni ’s and, hence, that of the ξn ’s imply by (5.7) that for 1 ≤ j ≤ i ≤ k, + + + ξn (ti) − ξn (tj)=⇒ BH (ti − tj) as n →∞. d + Thus, BH (ti) − BH (tj) = BH (ti − tj), and so 1 2 2 2 Cov(e BH (tei), BH (tj)) = E(BH (ti) )+ E(BH (tj) ) − E((BH (ti) − BH (tj)) ) 2 1 e e = E(Be+(t )2)+ E(Be+(t )2) − E(Be+(t − t )e2) 2 H i H j H i j σ2 = + (t2H + t2H − (t − t )2H ). 2 i j i j This implies that the random vectors in (5.8) converge weakly to the corresponding finite di- mensional distributions of the appropriate fractional Brownian motion, and this verifies the statement. Lemma 5.3 Let Assumption A and B hold and let t ≥ 0, ǫ> 0. Then − ν (a) νn,t =⇒ 0 on (0, ∞] as n →∞. 2 − 2H 2 2 (b) limn→∞ {|x|≤ǫ} x νn,t(dx)= t σ− with σ− as in (5.3) and H as in (3.3). R − (c) limn→∞ {|x|>ǫ} xνn,t(dx) = 0. In particular, R − fidi − (ξn (t))t≥0 =⇒ (BH (t))t≥0 − 2 where (BH (t))t≥0 is a fractional Brownian motion with Hurst index H and time 1 variance σ−. Proof. (b) The argument is similar to that of Lemma 5.2, but somewhat more involved tech- nically. We start with introducing some notation. Let (1) E 2 1 1 Hn (w) := Nc (Mnw, Mn(w + t)] {Nc(Mnw,Mn(w+t)]≤ǫbn} {K>N0(0,Mn(w+t)]} , (5.9) (2) E 2 1 1 Hn (w) := Nc (Mnw, Mn(w + t)] {Nc(Mnw,Mn(w+t)]≤ǫbn} {N0(0,Mnw] Therefore, (c) follows by Lemma 6.12. The remainder of the proof is the same as in Lemma 5.2. 5.2 Verification of condition (b) of Theorem 3.3 This is an immediate consequence of the Chebyshev inequality and (6.2). 5.3 Verification of condition (c) of Theorem 3.3 It is in this part of the argument that the slow growth condition (3.2) plays a role. Condition (c) ∗ of Theorem 3.3 is a direct conclusion of (3.2) and the next lemma since then limn→∞ nP(I (0) > bn) = 0. Lemma 5.4 Let the Poisson cluster model satisfy Assumption A. Then there exists a constant C > 0 such that α −1 ∗ − 1 P(I (0) > z) ≤ Cz β2 ∀ z > 0. Proof. The initial step is to show that we may, without loss of generality, assume that the interarrival times of the cluster process N0 are bounded from below by a positive number. To this end we modify the renewal point process N0 into a different renewal point process, N0, as follows. Let δ > 0 be such that P(X ≥ δ) > 0. U1−1 e Let T1 := min{Tj : Tj ≥ δ}. Define Z0 := Z0 + i=1 (Zi + δ), where U1 := min{j : Tj ≥ δ}. We view Z0 as the amount of data in the single arrival at time T0 := 0. e e P In general, given Tm and Um, we define the next arrival by Tm+1 := min{Tj : Tj − Tm ≥ δ}, e e Um+1−1 and the amount of data brought in by the arrival at time Tm as Zm := ZUm + i=U +1(Zi + δ), e e m e where Um+1 := min{j : Tj − Tm ≥ δ}. P e e Note that with this (sample path) modification, every arrival point of the original process e N0 will arrive, in the new process, not later than before (but it may be aggregated with other points of N0 into a single new arrival), and its transmission will last in the new process for at least as long as in the original process. We will still take K of the new aggregated arrivals, so this modification can only increase the random variable I∗(0). For the new process the random amount of data brought in with any arrival has the repre- U1−1 sentation Z0 = Z0 + i=1 (Zi + δ), and since U1 is stochastically dominated by a geometric E 2 random variable, we seeP that (Z0 ) < ∞. Furthermore, the interarrival times of the new process e e 17 satisfy Xm ≥ δ a. s. and P(X1 > x) ≤ P(Xm > x) ≤ P(X1 + δ>x) ∼ P(X1 > x) as x → ∞. Hence, P(X1 >x) ∼ P(X1 >x) as x →∞. e e Therefore, for the purpose of obtaining an upper bound, we may work with the new renewal e process, and we will simply assume that the original renewal process N0 has interarrival times that are bounded from below by a positive constant. We observe that I∗(0) is an infinitely divisible random variable with Lévy measure given by ∞ (c) µ(B)= λ0 P(A (x) ∈ B) dx for B ∈B(R), Z0 where A(c)(x) is the total amount of data in a session belonging to a single cluster, initiated (c) Nc(0,x] at zero, that does not finish by time x > 0, i.e. A (x) = j=1 (Tj + Zj − x)+. To see this ∗ write I (0) with respect to a Poisson random measure andP use, for example, Lemma 2.2 (i) in Kallenberg (2002). Without loss of generality let λ0 = 1. We have, therefore, the decomposition ∞ c µ(z, ∞)= P(A (x) > z) dx ≤ I4,0 + I4,1 + I4,2 + I4,3, (5.11) Z0 where z (c) I4,0 = P(A (x) > z) dx, 0 Z ∞ P I4,1 = (ZNc(0,x] > z + (x − TNc(0,x]), TK > z) dx, Zz ∞ Nc(0,x]−1 I = E 1 P {Z >x − T } F dx , 4,2 {TK >z} j j Zz j=0 [ ∞ K I = E 1 P {Z >x − T } F dx , 4,3 {TK ≤z} j j Zz j=0 [ where F is the σ-field generated by the cluster point process N c. Let z ≥ 1. Then Proposition 4.1 in Faÿ et al. (2006) gives us K I ≤ zP Z > z ≤ C z1−α1 . (5.12) 4,0 j 0 Xj=0 Next, K−1 Tk+1 E 1 P I4,1 ≤ {TK >z} (Zk > z + (x − Tk)|F) dx Tk k=NX0(0,z] Z ∞ E 1 P + {TK >z} (ZK > z + (x − TK)|F) dx . ZTK 18 By Markov’s inequality we obtain K−1 Tk+1 E 1 −2 I4,1 ≤ C1 {TK >z} (z + (x − Tk)) dx Tk k=NX0(0,z] Z ∞ E 1 −2 +C2 {TK >z} (z + (x − TK)) dx ZTK K−1 = C E 1 [z−1 − (z + X )−1] + C E 1 z−1 1 {TK >z} k+1 2 {TK >z} k=N (0,z] X0 −1E 1 ≤ C3z ( {TK >z} K). (5.13) Note that ∞ E 1 E P P E P ( {TK >z} K)= (K) K = k X1 + . . . + Xk > z = (K) (TK > z) , k=1 X e e where K is a positive integer valued random variable with P K = k = kP(K = k)/E(K), k ∈ N. Further, by Karamata’s Theorem e e 1 α P K>n ∼ nP(K>n) as n →∞. E(K) α − 1 Hence, by Proposition 4.1e − α1−1 E 1 −1P −1 β2 ( {TK >z} K) ∼ C4F (z) K > F (z) ≤ C5z , (5.14) and thus, α1−1 α1−1 −1 − β − β I4,1 ≤ C6z z 2 ≤ C7z 2 . (5.15) Next, we decompose I4,2 into I4,2 = I4,2,1 + I4,2,2 (5.16) where N0(0,x]−1 TK+1 I = E 1 P {Z >x − T } F dx , 4,2,1 {TK >z} j j Zz j=0 [ ∞ K I = E 1 P {Z >x − T } F dx . 4,2,2 {TK >z} j K ZTK+1 j=0 [ Then N0(0,x]−1 TK+1 E 1 P I4,2,1 = {TK >z} (Zj >x − Tj|F) dx z Z Xj=0 K Tk+1 k−1 E 1 P ≤ {TK >z} (Zj >x − Tj|F) dx . Tk k=NX0(0,z] Z Xj=0 19 Again applying Markov’s inequality and (5.14) lead to K k−1 Tk+1 E 1 −2 I4,2,1 ≤ C8 {TK >z} (x − Tj) dx Tk k=NX0(0,z] Xj=0 Z K−1 = C E 1 (T − T )−1 − (T − T )−1 8 {TK >z} N0(0,z]∨(j+1) j K+1 j j=0 X K ≤ C E 1 X−1 9 {TK >z} j Xj=1 −1E 1 ≤ C9δ ( {TK >z} K) α1−1 − β ∼ C10z 2 (5.17) as z →∞. Further, by Markov’s inequality, as above and (5.14) ∞ − α1−1 E 2 E 1 −2 E 1 β2 I4,2,2 ≤ (Z ) {TK >z} K (x − TK ) dx ≤ C11 ( {TK >z} K) ≤ C12z (5.18) ZTK +1 for z large enough. We decompose I4,3 into z+1 K I = E 1 P {Z >x − T } F dx 4,3 {z/2≤TK ≤z} j j Zz j=0 [ ∞ K +E 1 P {Z >x − T } F dx {z/2≤TK ≤z} j j Zz+1 j=0 [ ∞ K +E 1 P {Z >x − T } F dx {TK On the one hand, by Proposition 4.1 in Faÿ et al. (2006), 1 − β I4,3,1 ≤ P(TK > z/2) ≤ C13z 2 . (5.20) On the other hand, by Markov’s inequality and (5.14) we obtain ∞ − α1−1 E 1 −2 E 1 β2 I4,3,2 ≤ C14 {z/2≤TK ≤z} K (x − TK) dx ≤ C15 ( {TK >z/2} K) ≤ C16z . (5.21) Zz+1 Finally, another application of Markov’s inequality gives us ∞ E 1 −2 E −1 I4,3,3 ≤ {TK α1−1 1 α1−1 1−α1 − β − β −1 − β µ(z, ∞) ≤ C18z + C19z 2 + C20z 2 + C21z ≤ C22z 2 . Hence, a stochastic domination argument and the fact that the tail of a regularly varying Lévy measure is equivalent to the tail of its distribution function give the result. 20 6 Auxiliary Results A number of lemmas and other auxiliary results are collected in this section. We start with a lemma that clarifies the behavior of the normalizing sequence (bn) in Theorem 3.1. Lemma 6.1 Let Assumption A hold and (bn) be defined as in (3.1). Then −1 lim (F (Mn)bn) = 0, (6.1) n→∞ −2 lim nMnb = 0. (6.2) n→∞ n Proof. For n large we have by Potter’s theorem 1 1 1 α2 1 1 1 α2 1 − 2 − 2 2 2β1 F (Mn)bn = n 2 Mn P(K > F (Mn)) 2 ≥ n 2 Mn F (Mn) 2 ≥ n 2 Mn Mn . Since (β1 − α2)/(2β1) > 0 and Mn →∞ as n →∞ we obtain β1−α2 1 − −1 − β1 n→∞ (F (Mn)bn) ≤ n 2 Mn −→ 0. Finally, (6.2) results from n→∞ −2 2P −1 −1 2−α2 nMnbn = F (Mn) (K > F (Mn) ) ≤ F (Mn) −→ 0. The next result is a simple consequence of the strong Markov property which is useful in various places in our arguments. Lemma 6.2 Let f, g be measurable functions and f be increasing. Suppose N0 is a renewal process. Then for w,δ > 0 E 1 (f(N0 (w, w + δ])g(N0 (0, w]) {N0(0,w]6=N0(0,w+δ]}) E E 1 ≤ (f(1 + N0 (0, δ])) (g(N0 (0, w]) {N0(0,w]6=N0(0,w+δ]}). Proof. Condition on the time and the number of the first arrival after w and use the iid assump- tion of the interarrival times. The next lemma gives a simple estimate on the probability of having ”too many” arrivals within a time interval. Lemma 6.3 Let (Xk) be an iid sequence of positive random variables with distribution function F , such that F ∈ R−1/β, 0 < 1/β < 1 and let h be the generalized tail inverse function (2.4). m N Let Tm = k=1 Xk, m ∈ . For any δ > 0 such that F (δ) > 0 and m ≥ 1, (i) we haveP m −mF (δ) P Tm ≤ δ ≤ F (δ) ≤ e ; (6.3) 21 (ii) if x ≥ δ/h(m), then for any β1 <β<β2 we have − 1 − 1 −C min(x β1 ,x β2 ) P Tm ≤ h(m)x ≤ e (6.4) for some C = C(δ, β1, β2); Proof. Trivially, for δ > 0, m P Tm ≤ δ ≤ 1 − F (δ) . Now (6.3) follows from the fact that (1−a−1)a ≤e−1 for a≥ 1, and Potter’s bounds (cf. Resnick (2006), p. 36) give (6.4). The following simple result on convolution tails of random variables with infinite mean is often useful. Lemma 6.4 Let (Xk) be an iid sequence of positive random variables with distribution function N F , F ∈ R−1/β and 0 < 1/β < 1. Then there exist K > 0 and n0 ∈ such that for any x > 0 and n ≥ n0, F n∗(x) ≤ KnF (x). (6.5) Proof. Suppose that the statement is not true. Then for each j ≥ 1 there exist a nj ≥ j and a xj > 0 such that n ∗ F j (xj) ≥ jnjF (xj). (6.6) Let h be the generalized tail inverse function (2.4). Assume first that there is a sequence jk ↑∞ as k →∞ such that x lim jk = ∞. k→∞ h(njk ) This implies limk→∞ njk F (xjk ) = 0. Therefore, by Theorem 9.1 in Denisov et al. (2008) we obtain n ∗ F jk (x ) lim jk − 1 = 0, k→∞ njk F (xjk ) which contradicts (6.6). Next, we suppose that there is M > 0 such that xj ≤ Mh(nj) for all j ∈ N. (6.7) Then j→∞ −1/β njF (xj) ≥ njF (Mh(nj )) −→ M by the regular variation of F . Thus, (6.6) results in n ∗ j→∞ F j (xj) ≥ jnjF (xj) −→ ∞. Since F nj ∗ is bounded by 1, this is impossible. Hence, the claim follows. 22 6.1 Auxiliary Results for the Proof of Lemma 5.2 + The next series of lemmas provides estimates needed to prove the convergence of ξn in Lemma 5.2. We are using the same notation. Lemma 6.5 Let Assumption A hold and let I1,1(n) be as in (5.5). Then ∞ 2 2+β−α − 2+β−α β β P lim I1,1(n)= t y (S1/β(1) ≤ y) dy. n→∞ 2+ β − α Z0 Proof. We have by the independence of K and N0 2 2 ǫ bn E 2 1 P 2 2 1 (Nc [0, u] {K+1≤ǫbn}) = ((N0 [0, u] ∧ (K + 1) ) {K+1≤ǫbn} >x) dx Z0 ǫbn = 2 yP(N0 [0, u] >y)P(y < K + 1 ≤ ǫbn) dy. Z0 Hence, n Mnt ǫbn I (n) = 2 yP(N [0, u] >y)P(y < K + 1 ≤ ǫb ) dy du 1,1 b2 0 n n Z0 Z0 n ǫbn Mnt = 2 yP(y < K + 1 ≤ ǫb ) P(T ≤ u) du dy b2 n ⌊y⌋ n Z0 Z0 n ǫbn = 2 yP(y < K + 1 ≤ ǫb )E(M t − T ) dy b2 n n ⌊y⌋ + n Z0 ǫbnF (Mn) n −2 −1 = 2 F (Mn) zP(zF (Mn) < K + 1 ≤ ǫbn)E(Mnt − T −1 )+ dz b2 ⌊zF (Mn) ⌋ n Z0 ǫ −1 P(zF (M ) < K + 1 ≤ ǫb ) T⌊zF (M )−1⌋ = 2 z n n E t − n dz P(K > F (M )−1) Mn Z0 n + ǫbnF (Mn) −1 P(zF (M ) < K + 1 ≤ ǫb ) T⌊zF (M )−1⌋ +2 z n n E t − n dz P(K > F (M )−1) Mn Zǫ n + =: J1(n,ǫ)+ J2(n,ǫ). (6.8) By Karamata’s theorem ǫ 2t −1 J1(n,ǫ) ≤ zP(K + 1 > zF (Mn) ) dz P(K > F (M )−1) n Z0 −1 P(K + 1 > F (M )−1) 2t ǫF (Mn) = n zP(K + 1 > z) dz P(K > F (M )−1) F (M )−2P(K + 1 > F (M )−1) n n n Z0 2tα n−→→∞ ǫ2−α, (6.9) 2 − α and we conclude that limǫ↓0 limn→∞ J1(n,ǫ) = 0. We estimate J2(n,ǫ) as follows. By Potter’s inequality there exists C1 > 0 such that for z ≥ ǫ and n large, P(K + 1 > zF (M )−1) n ≤ C z−α1 . −1 1 P(K > F (Mn) ) 23 Similarly, Potter’s inequality leads to h(⌊zF (M )−1⌋) h(zF (M )−1 − 1) n ≥ n ≥ C zβ1 for z ≥ ǫ. −1 2 Mn h(F (Mn) + 1) −1 If we define mn = ⌊zF (Mn) ⌋, then for δ > 0 such that F (δ) < 1, T⌊zF (M )−1⌋ T h(m ) E t − n = E t − mn n M h(m ) M n + n n + T ≤ E t − mn C zβ1 h(m ) 2 n + δ/h(m ) C−1tz−β1 n 2 T = C zβ1 + P mn ≤ x dx 2 h(m ) "Z0 Zδ/h(mn) # n β1 =: C2z [V1(n, z)+ V2(n, z)]. We have by (6.3) for large n, P −mnF (δ) −1 −C3z V1(n, z) ≤ δ/h(mn) (Tmn ≤ δ) ≤ δe ≤ C3 e for some C3 > 0, since mn ≥ z for n large. Further, by (6.4) C−1tz−β1 2 −1/β −1/β −C4 min(x 1 ,x 2 ) −1 −β1 −C5z −1 −C6z V2(n, z) ≤ e dx ≤ C5 tz e ≤ C6 e Z0 for some C4,C5,C6 > 0. Hence, we have T⌊zF (M )−1⌋ E t − n ≤ C zβ1 [V (n, z)+ V (n, z)] ≤ C−1zβ1 e−C7z, M 2 1 2 7 n + and so by the dominated convergence theorem, (2.5) and the regular variation of F K , ∞ 1−αE β lim J2(n,ǫ) = 2 z (t − z S1/β(1))+ dz. n→∞ Zǫ Therefore, by (6.8), ∞ 1−α E β lim I1,1(n) = 2 z (t − z S1/β(1))+ dz n→∞ Z0 2+β−α ∞ 2+β−α 2 − −1 = t β x β E(1 − x S1/β(1))+ dx β 0 Z ∞ x 2 2+β−α 2+β−α β − β −1 = t x P(S1/β(1) ≤ z) dz dx β 0 0 Z ∞ Z 2 2+β−α − 2+β−α = t β z β P(S (1) ≤ z) dz. 2+ β − α 1/β Z0 Lemma 6.6 Let Assumption A hold and let I1,2(n) be as in (5.5). Then lim I1,2(n) = 0. n→∞ 24 Proof. By the independence of K and N0 we have n Mnt n Mnt I (n) ≤ E 1 N [0, u]2 du = P(K + 1 > ǫb ) E(N [0, u]2) du. 1,2 b2 {K+1>ǫbn} 0 b2 n 0 n Z0 n Z0 Thus, n Mnt ∞ I (n) ≤ P(K + 1 > ǫb ) P(N [0, u]2 >x) dx du 1,2 b2 n 0 n Z0 Z0 n ∞ ≤ 2 M tP(K + 1 > ǫb ) zP(T ≤ M t) dz. b2 n n ⌊z⌋ n n Z0 By (6.3), Potter’s inequality and (6.1), n ∞ I (n) ≤ C M tP(K + 1 > ǫb ) ze−⌊z⌋F (Mnt) dz 1,2 1 b2 n n n Z0 n P −2 ≤ C2 2 Mnt (K + 1 > ǫbn)F (Mnt) bn P(K >ǫb − 1) n→∞ = C t1−α1 n −→ 0. 3 −1 P(K > F (Mn) ) Lemma 6.7 Let Assumption A hold and let I2,1(n) be as in (5.6). Then lim I2,1(n) = 0. n→∞ Proof. Suppose ǫ = 1. The independence of K and N0 results in Mnt Mnt P P P P I2,1(n) = n (K + 1 > bn) (N0 [0, u] > bn) du = n (K + 1 > bn) (T⌊bn⌋ ≤ u) du. Z0 Z0 As in (6.3) we obtain Mn −⌊bn⌋F (u) I2,1(n) ≤ nP(K + 1 > bn) e du Z0 −(bn−1)F (Mn) ≤ nP(K + 1 > bn)Mne P(K + 1 > b ) n→∞ = n (F (M )b )2e−(bn−1)F (Mn) −→ 0, −1 n n P(K > F (Mn) ) n→∞ since bnF (Mn) −→ ∞ by Lemma 6.1. Lemma 6.8 Let Assumption A hold and let I2,2(n) be as in (5.6). Then lim I2,2(n) = 0. n→∞ Proof. Suppose ǫ = 1 and t = 1. Then ∞ Mn I2,2(n) = n P(K + 1 > bnx) P(N0 [0, u] > bnx) du dx Z1 Z0 ∞ Mn P P = n (K + 1 > bnx) (T⌊bnx⌋ ≤ u) du dx. Z1 Z0 25 Again as in (6.3) we obtain by (6.1), ∞ Mn −⌊bnx⌋F (u) I2,2(n) ≤ n P(K + 1 > bnx) e du dx 1 0 Z ∞ Z −(bnx−1)F (Mn) ≤ nMn P(K + 1 > bnx)e dx 1 Z ∞ 1 −bnxF (Mn) ≤ e nMn e dx Z1 1 −1 −bnF (Mn) ≤ e nMn(bnF (Mn)) e 1 1 H− 1 1 β max{2, } −1 −bnF (Mn) ≤ e (n 2 Mn ) H−1/β (bnF (Mn)) e 1 max{2, }−1 −bnF (Mn) n→∞ ≤ C1(bnF (Mn)) H−1/β e −→ 0, which is the result. 6.2 Auxiliary Results for the Proof of Lemma 5.3 − The next several results deal with the convergence of ξn in Lemma 5.3. (1) Lemma 6.9 Let Assumption A hold and let Hn (w) be as in (5.9). Then nMn (1) E 2 −α lim 2 Hn (w)= ((I(w + t) − I(w)) I(w + t) ). n→∞ bn (1) Proof. We divide Hn in three parts and define An,w = {N0 (Mnw, Mn(w + t)] ≤ ǫbn,K >N0 (0, Mn(w + t)]}. For M > 0 let (1,1,M) 2 H (w)= E N (M w, M (w + t)] 1 −1 1 , n 0 n n {M ≤F (Mn)N0(0,Mn(w+t)]≤M} An,w (1,2,M) 2 H (w)= E N (M w, M (w + t)] 1 −1 1 , n 0 n n {F (Mn)N0(0,Mn(w+t)] (1) (1,1,M) (1,2,M) (1,3,M) Hn (w)= Hn (w)+ Hn (w)+ Hn (w). (6.10) Regularly varying functions converge uniformly on compact sets (cf. Bingham et al. (1987), Theorem 1.5.2). Thus, (2.7) gives 2 nMn (1,1,M) N0 (Mnw, Mn(w + t)] H (w) = E 1 −1 2 n −2 {N0(Mnw,Mn(w+t)]≤ǫbn,M ≤F (Mn)N0(0,Mn(w+t)]≤M} bn F (Mn) P(K > N (0, M (w + t)] |F ) × 0 n 0 P(K > F (M )−1) n n→∞ E 2 1 −α −→ ((I(w + t) − I(w)) {M −1≤I(w+t)≤M} I(w + t) ), (6.11) 26 (1) where F0 = σ(N0). For the second summand of Hn we have for large n by Potter’s inequality nMn (1,2,M) 2 Hn (w) bn 2 1 N0 (Mnw, Mn(w + t)] {K>N0(0,Mn(w+t)]} ≤ E 1 −1 −1 F (M )−1 P(K > F (M )−1) {N0(0,Mnw]6=N0(0,Mn(w+t)] (1) By Potter’s inequality the last term of Hn has the upper bound nMn (1,3,M) 2 Hn (w) bn nM ≤ n E N (0, M (w + t)]2 1 P(K > N (0, M (w + t)] |F ) 2 0 n {F (Mn)N0(0,Mn(w+t)]≥M} 0 n 0 bn 2−α1 N0 (0, Mn(w + t)] 1 ≤ C2E −1 F (M )−1 {N0(0,Mn(w+t)]≥F (Mn) M} n ! ∞ N (0, M (w + t)] 2−α1 = C P 0 n >y dy 2 −1 2−α1 F (M ) ZM n ! 2−α1 2−α1 N0 (0, Mn(w + t)] 2−α1 +C2M P > M . F (M )−1 n ! The first term on the right hand side above can be bounded as follows. For some constant C3 > 0 we obtain as in (6.3) that ∞ N (0, M (w + t)] 2−α1 P 0 n >y dy −1 2−α1 F (M ) ZM n ! ∞ 1−α1 −1 = (2 − α1) z P(N0 (0, Mn(w + t)] > zF (Mn) ) dz M Z ∞ 1−α1 ≤ (2 − α ) z P(T −1 ≤ M (w + t)) dz 1 ⌊zF (Mn) ⌋+1 n M Z ∞ 1−α1 −1 ≤ (2 − α1) z exp(−zF (Mn) F (Mn(w + t))) dz ZM ∞ − 1 β −1 1−α1 −C3(ω+t) 1 z ≤ C3 z e dz −→ 0 as M →∞. ZM Similarly, N (0, M (w + t)] 2−α1 M 2−α1 P 0 n > M 2−α1 −→ 0 as M →∞. F (M )−1 n ! Hence, the result follows. 27 (2) Lemma 6.10 Let Assumption A hold and let Hn (w) be as in (5.9). Then nM α 2α lim n H(2)(w) = E I(w + t)2−α + I(w)I(w + t)1−α n→∞ b2 n 2 − α α − 1 n 2 −E I(w)2I(w + t)−α + I(w)2−α . (2 − α)(α − 1) Proof. We define −1 An,M = {M ≤ F (Mn)N0 (0, Mnw] ≤ F (Mn)N0 (0, Mn(w + t)] ≤ M, N0 (Mnw, Mn(w + t)] ≤ ǫbn} and −1 AM = {M ≤ I(w) ≤ I(w + t) ≤ M}. By Karamata’s theorem and the uniform convergence of regularly varying functions on compact sets we have 1 1 2 {K≤N0(0,Mn(w+t)]} − {K≤N0(0,Mnw]} E K 1A F (M )−2P(K > F (M )−1) n,M n n α n−→→∞ E (I(w + t)2−α − I(w)2−α) 1 , (6.13) 2 − α AM and 1 1 N0 (0, Mnw] {K>N0(0,Mnw]} − {K>N0(0,Mn(w+t)]} E K 1A F (M )−1 F (M )−1P(K > F (M )−1) n,M n n n n→∞ α −→ E I(w)(I(w)1−α − I(w + t)1−α) 1 . (6.14) α − 1 AM Further, 2 1 − 1 E N0 (0, Mnw] {K>N0(0,Mnw]} {K>N0(0,Mn(w+t)]} 1 −2 P −1 An,M F (Mn) (K > F (Mn) ) ! n→∞ E 2 −α −α 1 −→ I(w) (I(w) − I(w + t) ) AM . (6.15) Thus, (6.13)-(6.15) give us nMn E 2 1 1 lim lim 2 Nc (Mnw, Mn(w + t)] {N0(0,Mnw] The following theorem is needed to apply dominated convergence to establish the convergence − of ξn in Lemma 5.3. 28 Theorem 6.11 Let Assumption A and B hold. Then there exists a non-negative measurable R R ∞ N function g : + → + such that 0 g(w) dw < ∞ and for every n ∈ R nMn E 2 2 (Nc (Mnw, Mn(w + 1)] ) ≤ g(w) ∀ w> 0. bn Proof. The existence of the required function on the interval (0, M] for an arbitrary M > 0 follows from Lemma 6.16 below, so we only need to construct a required function on the interval (M, ∞). We define An,w = {N0 (0, Mnw] 6= N0 (0, Mn(w + 1)]}, Bn,w = {N0 (0, Mn]= N0 (0, Mnw]}∩ An,w, Cn,w = {N0 (0, Mn] 6= N0 (0, Mnw]}∩ An,w. We have for w > M nMn E 2 nMn E 2 1 2 (Nc (Mnw, Mn(w + 1)] ) ≤ 2 (Nc (Mnw, Mn(w + 1)] Bn,w ) bn bn nMn E 2 1 1 + 2 (N0 (Mnw, Mn(w + 1)] {K>N0(0,Mnw]>0} Cn,w ) bn =: J2,1(n, w)+ J2,2(n, w). (6.16) Potter’s inequality and Lemma 6.2 result in J2,2(n, w) 2 −α1 −α2 N0 (Mnw, Mn(w + 1)] N0 (0, Mnw] N0 (0, Mnw] ≤ E C1 + C2 1C F (M )−2 F (M )−1 F (M )−1 n,w n " n n # ! 2 −α1 −α2 (N0 (0, Mn] + 1) N0 (0, Mnw] N0 (0, Mnw] ≤ E E C1 + C2 1C . F (M )−2 F (M )−1 F (M )−1 n,w n " n n # ! By (6.3) we have for large n 2 ∞ N0 (0, Mn] 1 2 E = P(N0 (0, Mn] >x) dx F (M )−2 F (M )−2 n ! n Z0 2 ∞ ≤ yP(T⌊y⌋+1 ≤ Mn) dy F (M )−2 n Z0 2 ∞ ≤ ye−yF (Mn) dy −2 F (Mn) 0 ∞ Z = 2 ze−z dz < ∞. (6.17) Z0 Hence, (6.16), (6.17) and Proposition 6.13 below show that J2,2(n, w) is uniformly in n bounded from above by an integrable on [M, ∞) function. The fact that the same is true for J2,1(n, w) follows from Lemma 6.17 below. The last piece needed to prove Lemma 5.3 is the next lemma. 29 Lemma 6.12 Let Assumption A and B hold and let I3,1(n) and I3,2(n), respectively, be as in (5.10). Then (a) limn→∞ I3,1(n) = 0. (b) limn→∞ I3,2(n) = 0. Proof. (a) We assume, once again for the ease of notation, that ǫ = 1. Let θ > 0. We have ∞ −θ nMn E 2+θ 1 I3,1(n) ≤ bn 2 Nc (Mnw, Mn(w + t)] {Nc(Mnw,Mn(w+t)]>bn} dw bn M Z ∞ 1 N (M w, M (w + t)] 2+θ ≤ b−θF (M )−θ E c n n dw. n n −1 M P(K > F (Mn) ) F (Mn) Z ∞ −r As in the proof of Theorem 6.11 we have that the integral is bounded above by C1 M w dw for some C1 > 0,r > 1. Then by (6.1) we conclude that I3,1(n) → 0 as n →∞. R (b) For I3,2, notice that by Lemma 6.3 E N0 (0, Mn(M + t)] 1 −1 {N0(0,Mn(M+t)]>bn} F (Mn) ∞ = bnF (Mn)P(N0 (0, Mn(M + t)] > bn)+ F (Mn) P(N0 (0, Mn(M + t)] >x) dx bn ∞ Z −bnF (Mn(M+t)) −xF (Mn(M+t)) ≤ bnF (Mn)e + F (Mn) e dx Zbn −bnF (Mn(M+t)) −1 −bnF (Mn(M+t)) = bnF (Mn)e + F (Mn(M + t)) F (Mn)e . Therefore, nMn E 1 1 I3,2(n) ≤ M (N0 (0, Mn(M + t)] {N0(0,Mn(M+t)]>bn} {K>bn}) bn P E N0 (0, Mn(M + t)] 1 (K > bn) ≤ MF (Mn)bn {N (0,M (M+t)]>b } F (M )−1 0 n n P(K > F (M )−1) n n 2 −1 −bnF (Mn(M+t)) −α1 ≤ C1 (F (Mn)bn) + F (Mn(M + t)) F (Mn)F (Mn)bn e (bnF (Mn)) n→∞ −→ 0 by (6.1). 6.3 Auxiliary Results for the Proof of Theorem 6.11 The following proposition is the first ingredient in the proof of Theorem 6.11. Proposition 6.13 Let η > 1 and M > 1, and suppose that Assumption A and B hold. Then R R ∞ there exists a non-negative measurable function g : + → + such that M g(w) dw < ∞ and N for every n ∈ R −η E N0 (0, Mnw] 1 {N (0,M ]6=N (0,M w]6=N (0,M (w+1)]} ≤ g(w) ∀ w ≥ M. F (M )−1 0 n 0 n 0 n n ! 30 The statement follows from Lemma 6.14 and Lemma 6.15 below. Lemma 6.14 Let η > 1 and M > 1, and suppose that Assumption A and B hold. Then there R R ∞ exists a non-negative measurable function g : + → + such that M g(w) dw < ∞ and for N every n ∈ R −η N0 (0, Mnw] 1 E {N (0,M (w−1)]6=N (0,M w]} ≤ g(w) ∀ w ≥ M. F (M )−1 0 n 0 n n ! −1 −1 Proof. Let w ≥ M and n so large such that Mn ≤ 2 . We have −η E N0 (0, Mnw] 1 J1(n, w) := {N (0,M (w−1)]6=N (0,M w]} F (M )−1 0 n 0 n n ! ∞ Mnw j −η = F (M w − y) P(T ∈ dy) n −1 j Mn(w−1) F (Mn) Z Xj=1 ∞ Mnw−2 Mnw j −η = + F (M w − y) P(T ∈ dy) n −1 j " Mn(w−1) Mnw−2# F (Mn) Z Z Xj=1 =: J1,1(n,ω)+ J1,2(n,ω). Now, ⌈Mnw−2⌉−1 ∞ k + 1 j −η J (n,ω) ≤ F M w − [P(T ≤ k + 1) − P(T ≤ k)] . 1,1 n −1 j j Mn F (Mn) k=⌊MnX(w−1)⌋−1 Xj=1 Since F is regularly varying of index −1/β, by Potter’s inequality there exists a constant 0 ≤ C1 < ∞ such that ⌈Mnw−2⌉−1 1 ∞ − β 1−η k + 1 1 F (Mn) 1−η −η J1,1(n,ω) ≤ C1 w − kF (k) j [P(Tj ≤ k + 1) − P(Tj ≤ k)] . Mn kF (k)1−η k=⌊MXn(w−1)⌋ Xj=1 Using Lemma 4.3, we obtain 1 ⌈Mnw−2⌉−1 − 1−η k + 1 β1 F (Mn) J1,1(n,ω) ≤ C2 w − . Mn kF (k)1−η k=⌊MXn(w−1)⌋ Taking again the regular variation of F and Potter’s Theorem into account yields ⌈Mnw−2⌉−1 − 1 1−η k + 1 β1 1 k β1 J1,1(n,ω) ≤ C3 w − Mn k Mn k=⌊MXn(w−1)⌋ ⌈Mnw−2⌉−1 − 1 1−η −1 1 k + 1 β1 k β1 = C3 w − Mn Mn Mn k=⌊MXn(w−1)⌋ w 1 1−η − β β −1 ≤ C4 (w − z) 1 z 1 dz w−1 Z 1−η β −1 ≤ C5w 1 , (6.18) 31 which is an integrable function on [M, ∞) since η> 1. Finally, using, once again, Lemma 4.3, we obtain ∞ −η −η J1,2(n,ω) ≤ F (Mn) j [P(Tj ≤ Mnw) − P(Tj ≤ Mnw − 2)] Xj=1 −η 1 η−1 ≤ C6F (Mn) F (Mnw) Mnw 1 1−η β −1 ≤ C7 w 1 , MnF (Mn) which is uniformly bounded by an integrable function. Lemma 6.15 Let η > 1 and M > 1, and suppose that Assumption A and B hold. Then there R R ∞ exists a non-negative measurable function g : + → + such that M g(w) dw < ∞ and for N every n ∈ R −η E N0 (0, Mnw] 1 {N (0,M ]6=N (0,M (w−1)]=N (0,M w]6=N (0,M (w+1)]} ≤ g(w) ∀ w ≥ M. F (M )−1 0 n 0 n 0 n 0 n n ! Proof. As in the previous lemma, −η N0 (0, Mnw] J1(n, w) := E 1 F (M )−1 {N0(0,Mn]6=N0(0,Mn(w−1)]=N0(0,Mnw]6=N0(0,Mn(w+1)]} n ! ∞ Mn(w−1) j −η = [F (M w − y) − F (M (w + 1) − y)] P(T ∈ dy) n n −1 j Mn F (Mn) Z Xj=1 ⌈Mn(w−1)⌉−1 ≤ F (Mnw − k − 1) − F (Mnw + Mn − k) k=X⌊Mn⌋ ∞ j −η × [P(T ≤ k + 1) − P(T ≤ k)] . −1 j j F (Mn) Xj=1 By Lemma 4.3 we have for n large ⌈M (w−1)⌉−1 n F (M w − k − 1) − F (M w + M − k) F (M )1−η J (n, w) ≤ C n n n n . 1 1 1−η F (Mn) kF (k) k=X⌊Mn⌋ Note that for every k in the above sum by Assumption B ⌈Mn⌉−1 F (Mnw − k − 1) − F (Mnw + Mn − k) ≤ F (Mnw − k + j) − F (Mnw − k + j + 1) j=−1 X ⌈Mn⌉−1 F (Mnw − k + j) F (Mnw − k − 1) ≤ C2 ≤ C3Mn . Mnw − k + j Mnw − k − 1 jX=−1 We conclude by Potter’s Theorem that for large n and all k as above − 1 −1 F (Mnw − k − 1) − F (Mnw + Mn − k) k + 1 β2 ≤ C4 w − . F (M ) Mn n 32 Hence, we obtain 1 ⌈Mn(w−1)⌉−1 − −1 1−η k + 1 β2 F (Mn) J1(n, w) ≤ C4 w − . Mn kF (k)1−η k=X⌊Mn⌋ Similar calculations as in (6.18) result in w−1 1 1−η η−1 − β −1 β −1 − β −1 J1(n, w) ≤ C5 (w − z) 2 z 2 dz ≤ C6w 2 , Z1 as an easy computation shows. This is an integrable on [M, ∞) function. The final two lemmas needed for the proof of Theorem 6.11 follow. Lemma 6.16 Let M > 0, and suppose that Assumption A hold. Then there exists a positive constant C < ∞ such that for every n ∈ N nMn E 2 2 (Nc (Mnw, Mn(w + 1)] ) ≤ C ∀ w ≤ M. bn Proof. It is clearly enough to establish the required bound for n large enough. By Potter’s inequality and Karamata’s theorem, we obtain for all n large enough and 0 < w ≤ M nMn E 2 2 (Nc (Mnw, Mn(w + 1)] ) bn nMn E 2 ≤ 2 (Nc (0, Mn(M + 1)] ) bn nMn E 2 1 nMn E 2 1 = 2 N0 (0, Mn(M + 1)] {K>N0(0,Mn(M+1)]>0} + 2 K {K≤N0(0,Mn(M+1)]} bn bn 2−α1 2−α2 N0 (0, Mn(M + 1)] N0 (0, Mn(M + 1)] ≤ C1E + C2E . F (M )−1 F (M )−1 n ! n ! The right hand side is bounded for n large enough by computations similar to (6.17). Lemma 6.17 Let M > 1 and suppose that Assumption A and B hold. Then there exists a R R ∞ N non-negative measurable function g : + → + such that M g(w) dw < ∞ and for every n ∈ nMn E 2 R 2 Nc (Mnw, Mn(w + 1)] 1{N0(0,Mn]=N0(0,Mnw]6=N0(0,Mn(w+1)]} ≤ g(w) ∀ w ≥ M. bn Proof. We define Bn,w := {N0 (0, Mn] = N0 (0, Mnw] 6= N0 (0, Mn(w + 1)]}. Notice that by Lemma 6.2, nMn E 2 1 nMn E 2 P 2 (Nc (Mnw, Mn(w + 1)] Bn,w ) ≤ 2 min N0 (0, Mn] , K (Bn,w). (6.19) bn bn Further, ∞ 2 E min N0 (0, Mn] , K = 2 t P N0 (0, Mn] >t P(K>t) dt 0 Z −1 F (Mn ) ∞ = 2 + t P N0 (0, Mn] >t P(K>t) dt. −1 Z0 ZF (Mn) h i 33 Since by Karamata’s theorem, as n →∞, −1 −1 F (Mn) F (Mn) t P N0 (0, Mn] >t P(K>t) dt ≤ tP(K>t) dt Z0 Z0 −1 2 −1 ∼ C1 F (Mn) P K > F (Mn) and by (6.17) ∞ −1 2 t P N0 (0, Mn] >t P(K>t) dt ≤ C2P K > F (Mn) E N0 (0, Mn] F (M )−1 Z n −1 2 −1 ≤ C3 F (Mn) P K > F (Mn) , we have the bound 2 −1 2 −1 E min N0 (0, Mn] , K ≤ C4 F (Mn) P K > F (Mn) . 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