September 21, 2004 ACTIVITY RATES WITH VERY HEAVY TAILS

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1 September 21, 24 ACTIVITY RATES WITH VERY HEAVY TAILS THOMAS MIKOSCH AND SIDNEY RESNICK Abstract. Consider a data networ model in which sources begin to transmit at renewal time points {S n}. Transmissions proceed for random durations of time {T n} and transmissions are assumed to proceed at fixed rate unity. We study Mt, the number of active sources at time t, a process we term the activity rate process, since Mt gives the overall input rate into the networ at time t. Under a variety of heavy-tailed assumptions on the inter-renewal times and the duration times, we can give results on asymptotic behavior of Mt and the cumulative input process At R t Ms ds. 1. The model, notation, preliminary results. Consider an ordinary renewal process {S n, n } such that S, S n n i, n 1, and { n, n 1} is a sequence of iid non-negative random variables with common distribution F. At time point S n, an event begins of duration T n, where we assume {T n, n } is a sequence of iid non-negative random variables with common distribution G and {T n } is independent of { n }. The event which was initiated at S n terminates at S n +T n. In a data networ context, S n would be the time a user initiates a file download and T n is the download time. In an insurance context, S n is the time of a disaster or accident and T n is the length of time during which all insurance claims from this incident are received so that S n + T n is the latest time a claim from the nth accident is received. A process of interest is 1.1 Mt 1 [Sn t<s n+t n], t >, n1 the number of active downloads at time t or the number of active claims at time t. If {S } are the points of a homogeneous Poisson process with intensity λ, the point process K 1 ɛ S,T is a Poisson random measure with state space [, 2 and mean measure λleb G; see Resnic 1992, Proposition on p Hence Resnic 1992, Proposition Mt K{x, y : x t < x + y} is a Poisson random variable and asymptotic analysis is relatively easy. It is the aim of this paper to deal with the case when the renewal process {S } is not a Poisson process. This creates many interesting problems many of which we have solved. We build a general theory about M which paralls and supplements the one for M Poisson. Sidney Resnic s research was partially supported by NSF grant DMS at Cornell University. Grateful acnowledgment is made to the University of Copenhagen for hospitality and support for a wee in summer 22 and also May 12-19, 24. Thomas Miosch s research is partially supported by MaPhySto, The Danish Research Networ for Mathematical Physics and Stochastics, DYNSTOCH, a research training networ under the program Improving Human Potential financed by the The 5th Framewor Program of the European Commission, and the Danish Research Council SNF Grant No

2 2 THOMAS MIKOSCH AND SIDNEY RESNICK In particular, we will consider the asymptotic behavior of Mt and obtain some novel approximations. We also see to understand the behavior of the cumulative process 1.2 At Mu du, t, which, in the data networs interpretation corresponds to cumulative wor inputted provided each transmission initiated at renewal epochs proceeds at unit rate. In particular, we consider the very heavy-tailed cases when F x 1 F x x α L F x, Ḡx 1 Gx x β L G x, x, and α, β 1 and some slowly varying functions L F, L G. Concerning the relationship of F and G, we assume one of the following: 1 Comparable tails: β α and F x c Ḡx, c >, as x so that the distribution tails of 1 and T 1 are essentially the same. To avoid having to eep writing annoying constants, we assume c 1. 2 G heavier-tailed: a < β < α < 1 or, if β α, then F x/ḡx as x so that the distribution tail of 1 is lighter than the distribution tail of T 1. b β < α < 1 so that the distribution tail of T 1 is slowly varying and thus again heavier than that of 1. 3 F heavier-tailed: β > α so that the distribution tail of 1 is heavier than the distribution tail of T 1. The process M has attracted attention in the data networ literature since, under the assumption of unit input rate, it corresponds to traffic per unit time which, in several data measurement studies, has been empirically identified as self-similar or possessing long range dependence; see Crovella and Bestavros 1996, Garrett and Willinger 1994, Leland et al. 1994, Par and Willinger 2. Some standard attempts to provide model based explanations of this empirically observed phenomenon use the infinite source Poisson model in which {S n } are homogeneous Poisson points and {T n } are iid with Ḡ regularly varying with index β > 1. This leads to M possessing long range dependence in the sense of covariances slowly decreasing with lag. See for example, the standard argument in Resnic 23 and Par and Willinger 2. However, the Poisson based model often does not fit collected data well Guerin et al. 23 and file sizes are sometimes modeled with heavier tails than β > 1 Arlitt and Williamson 1996, Resnic and Rootzén 2, and it is of interest to consider behavior of models with different assumptions. Hence the present study. In Resnic and Rootzén 2, queuing is allowed in the sense that inputs are processed by a server and the contents process is studied under the assumption that β < 1. We have not attempted to model the processing of offered load in this paper. Some of our composition arguments used later have the flavor of ones employed by Meerschaert and Scheffler 24, Becer-Kern et al. 24. However, ours are applied to random measures instead of càdlàg functions as in the latter reference. We finally mention that the methods and techniques of this paper are related to wor on Poisson shot noise processes with infinite variance stable limits see Klüppelberg et al. 23 and the references therein and to renewal reward processes with infinite variance stable limits; see Pipiras et al. 24. The novel approach of this paper is to avoid the Poisson assumption on the renewal process which leads to a variety of rather interesting technical difficulties which we could resolve in some cases. This paper is organized as follows. In Section 1.1 we give some of the notation used throughout the paper. We continue in Section 1.2 with a mean value analysis of Mt from which we gain preliminary information about the rate of growth of this process as t under different distributional assumption on F and G. In Section 1.3 we study the distributional limits of the renewal counting function of the points {S n } and of its inverse function. In Section 2 we study the case

3 ACTIVITY RATES 3 of very heavy-tailed F and G when α, β < 1. In Section 2.1 we start by studying the asymptotic behavior of Mt as t in the Case 1 of comparable tails. It turns out that Mt converges in distribution to a random variable which is conditionally Poisson distributed. Section 2.2 is devoted to Case 2 of heavier-tailed G. In this case [ F t/ḡt]mt converges in distribution to some random variable. In Section 3 we study the case of lighter-tailed F in the sense that α 1 or E 1 <. In Section 3.1 we study the case when E 1 < and β, 1. In this case, [t Ḡt] 1 Mt converges in probability to a deterministic limit. A similar result holds when α 1 and E 1 ; see Section 3.2. When both T 1 and 1 have finite mean it is natural to wor with a stationary version of M; see Section 3.3 for such a construction. Section 4 deals with the asymptotic behavior of the cumulative wor process A. We understand its limit behavior when E 1 < and β 1, 2 infinite variance stable limits; see Section 4.1, when both T 1 and 1 have finite variance Brownian motion limits,; see Section 4.2 and when < α, β < 1 the limit is an integral with respect to the inverse of an infinite variance stable subordinator; see Section 4.3. We conclude in Section 5 with some unresolved problems. We present in Table 1 a summary of some of the limiting behavior of Mt. Table 1. Limiting behavior of Mt as t. Conditions Limit behavior of Mt as t < α < 1 Mt random limit. F Ḡ β < α < 1 or < α β < 1 and F oḡ < β < 1 E 1 < < β α 1 E 1 E 1 < ET 1 < F t Ḡt Mt random limit. Mt tḡt constant Mt random centering t Ḡt Mt Gaussian rv constant tḡtµt µt truncated 1st moment Stationary version of M exists 1.1. Basic notation. In this section we introduce some of the basic notation used throughout the paper. E 1, µ T ET 1, σ 2 Var 1, σt 2 VarT 1, E [,, ], C + K S the space of continuous functions on S with compact support, equipped with the uniform topology D[, D[,, R 2 D [, ɛ x the Sorohod space of real-valued càdlàg functions on [, equipped with the J 1 -topology the Sorohod space of R 2 -valued càdlàg functions on [, equipped with the J 1 -topology subspace of D[, containing the non-decreasing functions f such that f and f lim x fx point mass as x

4 4 THOMAS MIKOSCH AND SIDNEY RESNICK f LEB M + S M p E the right-continuous inverse of a monotone function f f x inf{y : fy > x} Lebesgue measure the space of non-negative Radon measures on S the space of Radon point measures on E ν γ a measure on, ] given by ν γ x, ] x γ, γ >, x >. PRMµ Poisson random measure on E with mean measure µ. convergence in distribution For information on the space D[, we refer to Billingsley 1968, Resnic 1986, Whitt 22. For information on point processes, random measures and vague convergence, see Kallenberg 1983, Resnic There one can also find information about the spaces M +, M p Mean value analysis when α, β < 1. The mean value asymptotic behavior of Mt can be obtained essentially from Karamata s Tauberian theorem. Let Ux F n x, x >, n be the renewal function for the ordinary renewal sequence {S n }. Since < α < 1 we have Feller 1971, p. 471, 1.3 Ux Γ1 α Γ1 + α F x 1 cα x α /L F x, x. Therefore it follows that, as t, 1.4 EMt cα Udx Ḡt x Ḡt1 s Ḡt 1 s β αs α 1 ds Ḡt F t Utds ḠtUt Ut c αḡt F t. Thus, in Case 1 of comparable tails, EMt converges to a constant while in Case 2, where G is heavier-tailed, EMt. In Case 3, EMt and hence Mt L 1, so Case 3 may be of lesser interest. It corresponds to the case where renewals are so sparse relative to event durations that at any time there is not liely to be an event in progress. We will not consider this case Behavior of the renewal counting function when < α < 1. Define for x, Nx 1 [Sn x] inf{n : S n > x}. n Note that Nx S x, where S {S [t], t }. Next, let ɛ t,j be PRMLEB ν α on E. The process α t t t j, t, is α-stable Lévy motion with Lévy measure ν α ; see Samorodnitsy and Taqqu define the quantile function of F : bt 1/ F t, t. Finally, When α >, we can always choose b as continuous and strictly increasing function; see for example, Seneta 1976 and Bingham et al

5 ACTIVITY RATES 5 A standard result is that the renewal epochs are asymptotically stable. In fact, if s t S [st] bs, t, then in D[, we have as s, see, for example, Resnic s α. Furthermore, the inverse processes also converge in D[, : Unpacing this last result, we get 1.6 s α. Nbs s α in D[, or, equivalently, F sns α or, equivalently, 1 ɛ Sn α s, bs n in M + [,, where we have used α to indicate both the monotone function and the measure. The inverse α of the stable subordinator α, α, 1, is a well-studied process in the Lévy process literature; see, for example, Bertoin 1996, Section III.2, or Sato 1999, Chapter Activity rates when α, β < Case 1: Comparable tails. Consider Case 1, where the tails of F and G are asymptotically equivalent. We begin with a result which describes the behavior of the counting function of the points {S, T, }. Define the mapping T : D [, M + E M + E by 2.1 T x, m m, where m is defined by mf fxu, v mdu, dv, f C + K E. This means that T replaces the usual time scale of m by one determined by the function x. If m is a point measure with representation m ɛ τ,y, then T x, m ɛ xτ,y. Theorem 2.1. Suppose the Case 1 assumptions hold with F x Ḡx, as x, < α < 1, and let N ɛ t,j be PRMLEB ν α. Then in M p E we have as s, 2.2 N s ɛ S bs, T bs N T α, N ɛ αt,j. Remar 2.2. The distribution of N can be specified by giving its Laplace functional. For f : E [,, we have, E e N f E exp{ 1 e f αs,y ds ν α dy}. E

6 6 THOMAS MIKOSCH AND SIDNEY RESNICK Proof. Begin with the statement Resnic 1986, 1987 that in M p E we have as s, ɛ s, T N. bs Since {S } is independent of {T }, we then get the joint convergence in D[, M p E, using 1.5, S[s ] bs, ɛ s, T bs α, N. The function T is a.s. continuous at α, N. Hence S[s ] T bs, ɛ s, T bs ɛ S [s/s] ɛ, T bs bs S bs, T bs T α, N. From this result, we get the desired result about M, the number of active sources or events. Corollary 2.3. The finite-dimensional distributions of the counting function Mt defined in 1.1 satisfy as s, Ms t 1 S [ s t< S +T ] M t 1 [αt t< αt +j ]. s Conditionally on α, the limit M t is Poisson with mean Λt t u α dα u and hence the generating function of M t is E τ M t E exp{τ 1 Λt}, τ, 1. Mt Mt t t Figure 1. A path of the process M for α β.9 left and α β.6 right. Proof. Fix t >. An important point to note is that Λt < a.s. To prove this claim we first note that Eα u uα E α α 1 d α u α. This results from the self-similar scaling of the Lévy process α : Eα u P[α u > x] dx P[u > α x] dx

7 ACTIVITY RATES 7 P[u > x 1/α α 1] dx u α E α α 1 d α u α. The quantity d α is finite; see Zolotarev We prove Λt < a.s. for t 1 as an example of the method. Writing fu 1 u α, < u < 1, and observing that f 1, we have fu dα u α 1 1 fu f d α u α u dα α1 u s α 1 ds s α 1 α s 1 s α 1 ds. Taing expectations, we have E fu dα u d α + α d α 1 s α 1 s α 1 ds. f s ds d α u Now, apart from constants, the second term is 1 1 s α s α 1 ds. The problem for integrability is near. But as s, the integrand is asymptotic αs α which, for < α < 1, is integrable. This verifies Λ1 < a.s. Next we prove Mbst M t for fixed t >. As before we choose t 1 in order to demonstrate the method. For positive ɛ, let B ɛ {u, v : u 1 < u + v, v > ɛ}, which is relatively compact in E. By virtue of Theorem 2.1, N s B ɛ N B ɛ. Also, by monotone convergence and using Λ1 <, with probability 1, N B ɛ N B M 1 < From the Converging Together Theorem Billingsley 1968, Theorem 4.2, p. 25, it suffices to show, for any δ >, that 2.3 Observe that lim ɛ lim sup P[ Ns B ɛ Ns B > δ]. n N s B N s B ɛ 1 [S bs<s +T,T ɛbs], By Chebyshev s inequality, it suffices to show that the expectation of this last quantity has a double limit which is zero. We have P[S bs < S + T, T ɛ bs] 1 ɛ F bs dx P[1 x < T /bs ɛ] 1 ɛ Ubs dx [Ḡbs 1 x Ḡbs ɛ] 1 ɛ Ḡbs1 x Ḡbs ɛ Ḡbs Ubs dx Ubs Ubs Ḡbs

8 8 THOMAS MIKOSCH AND SIDNEY RESNICK cα 1 ɛ as ɛ. [1 x α ɛ α ] dx α as s Thus we proved Mbs t M t for fixed t >. The convergence of the finite-dimensional distributions follows analogously by an application of Theorem 2.1. Since b can be chosen continuous and strictly increasing, we may rephrase the latter limit relation as Ms t M t. Remar 2.4. The above proof rests on a continuous mapping argument applied to the wea convergence relation 2.2. A similar argument ensures the joint convergence In particular, F s Ns, Ms α 1, M 1. Ms Ns d F s M 1 α o P1. Thus Ms/Ns is essentially of the order F s Ḡs. Compare this with the case when Ḡ is heavier-tailed than F Remar 2.8. Then Ms/Ns Ḡs Case 2: G is heavier-tailed. In this section we assume the Case 2 conditions β α < 1 and if < α β, then F t/ḡt, as t. Recall the definition of the measure ν γ given by ν γ x, ] x γ, for x >, some γ >. For γ, we interpret this as ν ɛ, i.e., the unit mass at. As in the previous section we first prove a limit result for the point process generated by the scaled points bs 1 S, T. Later we use this result in order to derive a distributional limit for Ms as s. Theorem 2.5. Assume the Case 2 conditions. Then in M + E we have F bs 2.4 ɛ S Ḡbs bs, T bs T α, LEB ν β, where T was defined in 2.1. Remar 2.6. Note that the normalization in 2.4 for both S and T is by the quantile function bs 1/ F s for the lighter-tailed distribution. Since this is inappropriate for T, it should not be too surprising that the pre-multification by the ratio of the tails which goes to is necessary. Proof. Begin by observing that s F bs Ḡbs Ḡbs v ν β, in M +, ], where v denotes vague convergence in the Borel σ-field of, ]. Hence from Resnic 1987, Example 3.5.7, see also a proof in Resnic 1986, we get F bs Ḡbs [s] ɛ T bs ν β. This may be extended as in the proof of Resnic 1987, Proposition 3.21, to show in M + E, F bs ɛ Ḡbs s, T LEB ν bs β.

9 ACTIVITY RATES 9 From independence we get the joint convergence in D[, M + E, S[s ] bs, F bs ɛ Ḡbs s, T bs α, LEB ν β. Now apply the a.s. continuous map T see 2.1 to get 2.4. From this result, we get the desired result about M, the number of active sources or events. Corollary 2.7. The finite-dimensional distributions of the counting function M defined in 1.1 satisfy as s, F s t 2.5 Ms t t u β dα Ḡs u. For any fixed t, t u β dα u d t β α 1 u β dα u. Mt Mt t t Figure 2. A path of the process M for α.9, β.2 left and α.9, β.4 right. Remar 2.8. In particular, for β < α < 1, we get F s Ḡs Ms t α t. Coupled with 1.6 we conclude as s, Ms Ns Ḡs P. Proof. We again consider the case of a fixed t > ; the convergence of the finite-dimensional distributions is analogous. We evaluate the convergence in 2.4 on the set {u, v : u t < u + v}. After a truncation and Slutsy style argument outlined in 2.3, we get F bs 2.6 Ḡbs Mbs t T α, LEB ν β f, where T is the mapping defined in 2.1 and fu, v 1 [u t<u+v]. Evaluating the right side, we find α t T α, LEB ν β f f α v, x dv dν β x t α v β dv

10 1 THOMAS MIKOSCH AND SIDNEY RESNICK t v β d α v, which is the convolution of the measure ν β and the non-decreasing function α. The integral also equals t β 1 v β dα tv d t βα 1 v β dα v. Since b can be chosen continuous and strictly increasing, the Mbst in 2.6 may be replaced by Ms t. This concludes the proof. 3. Activity rates when α 1 or < In this section we collect some results about the activity rates when either is finite or and F is regularly varying with index The case when F has finite mean and < β < 1. For mean value analysis of Mt, we have from 1.4, Hence EMt EMs s Ḡs Ḡt x Udx Ḡt1 x Ut dx. 1 x β µ 1 dx µ 1 1 β 1. This suggests what the correct normalization for Mt should be. Proposition 3.1. Under the assumptions < < and β, 1, the finite-dimensional distributions of M satisfy 1 t µ 1 s ḠsMs 1 β 1 t 1 β, s. Mt Mt t t Figure 3. A path of the process M for α 2, β.2 left and α 2, β.5 right. Proof. Since β, 1, we have sḡs as s. Therefore as s 3.1 s s Ḡs 1 Gs v ν β

11 ACTIVITY RATES 11 in, ]. This is equivalent to see Resnic 1987, Example 3.5.7, see also proof in Resnic sḡs in M + [,, and this can be extended to sḡs ɛ T s ν β, ɛ s, T s LEB ν β, in M + E. The law of large numbers for {S } together with 3.2 yields as s, S[s ] s, 1 sḡs ɛ s, T µ s, LEB ν β. Therefore, as in earlier sections, for any fixed t, as s, Mst s Ḡs T, LEB ν β f, where fu, v 1 [u t<u+v]. Evaluating the right side, one obtains /µ 1 [µ v t< v+x] dv ν β dx t v β dv µ 1 1 β 1 t 1 β. v Since the limit is deterministic, this implies the convergence of the finite-dimensional distributions in D[,. This result generalizes equation 2.7 in Resnic and Rootzén 2. Since the limit is deterministic, Proposition 3.1 should be regarded as the first order behavior of M and suggests there may be second order behavior involving a Gaussian limit as in Theorem 1, p. 76 in Resnic and Rootzén 2. We have the following result. Proposition 3.2. Suppose that β < 1 and <, and define for s > W s t : Mst Nst 1 Ḡst S, t. s Ḡs Then as s, the finite dimensional distributions of W s converge to those of W µ 1 β where W, is a mean-zero Gaussian process with covariance function Ct 1, t 2 : t 1 β 2 t 2 t 1 1 β, t 1 t 2. Remar 3.3. The limiting process is self-similar with index 1 β. Except for the case β, W does not have stationary increments and then it is Brownian motion. It would be desirable to replace the random centering Nst 1 Ḡst S by st Ḡst u du but it is not clear this is in general possible since Ns s/ is of order s while sḡs is of order s 1 β/2 and 1 β/2 < 1/2. In the case when S are the points of a homogeneous Poisson process the hypothesis of replacing the random centering by the expectation EMs can be made to wor by following the lines of the proof in Klüppelberg and Miosch There it is assumed that the shot noise has non-decreasing sample paths which is inessential for the proof in our situation. In this case, the convergence can be strengthened to a functional CLT in D[,, J 1 with the limiting process described above.

12 12 THOMAS MIKOSCH AND SIDNEY RESNICK Proof. We begin by showing one-dimensional convergence and then give the covariance calculation. Define S σs, 1, for the σ-field generated by the renewal times. Conditionally on S, Ms is a sum of independent, non-identically distributed Bernoulli random variables Thus Ns Ms Ns 1 1 [T >s S ]. 1 1 [T >s S ] Ns 1 Ḡs S Ns N, 1, Var 1 1 [T >s S ] S provided the denominator converges to as s. To see this note that Var Ns 1 Now almost surely, as s, locally uniformly, and therefore also locally uniformly. Thus it follows that Ns Var 1 1 [T >s S ] S sḡs Ns 1 [T >s S ] S Ḡs S Gt S 1 s Ḡs ugs undu Ḡs1 ugs1 unsdu. S [s ] /s, Ns /s 1, if β < 1. Note in this case, that sḡs. Thus we conclude that Ḡs1 u Gs1 u Nsdu Ḡs s β du 1 u 1 1 β, P[W s 1 x S] P[W 1/ 1 β x], and taing expectations, we get the same result unconditionally. For the covariance calculation we again proceed conditionally on S. Suppose t 1 t 2. Then CovW s t 1, W s t 2 S Nst 1 1 sḡs Cov 1 [T >st 1 S ], 1 [T >st 2 S ] S 1 since the sums for W s t 2 involving terms with Nst 1 < Nst 2 are conditionally independent of terms appearing for W s t 1 1 sḡs Nst 1 1 Ḡst2 S Ḡst 1 S Ḡst 2 S 1 Ḡst 2 u Ḡs Nsdu s 1 Ḡst 1 u Ḡst 2 u Nsdu Ḡs s

13 ACTIVITY RATES 13 1 β du t 2 u t1 β 2 t 2 t 1 1 β 1 β The case < β < α 1 with E 1. Then x F u du is slowly varying which is the necessary and sufficient condition for relative stability in probability to hold Feller 1971, p. 236; that is where S n nµn P 1, n, µn E 1 1 [1 bn]. As in 3.1, since sḡsµs, this leads to 1 s sḡsµs Gsµs v ν β and therefore we have as s, S[s ] sµs, 1 Ḡsµs Applying composition yields Finally, we get for t >, 1 Ḡsµs ɛ s, T, LEB ν s µs β. ɛ S sµs, T T, LEB ν β. s µs Msµs t t1 β sḡsµs 1 β. Since the limit is deterministic the convergence of the finite-dimensional distributions is immediate. This implies the following result which is analogous to Proposition 3.1. Proposition 3.4. Under the assumptions and < β < α 1, the finite-dimensional distributions of M satisfy Ms t t1 β sḡs/µs 1 β, s The case when F and G have finite mean. Then we have from the Key Renewal Theorem EMt Ḡt xudx µ T. This suggests that there exists a stationary version of the process M. We mae this precise in what follows. As s, in M p [,, where {S ɛ S s ɛ S,, } is the stationary renewal sequence, so that P[S > x] 1 x F udu,

14 14 THOMAS MIKOSCH AND SIDNEY RESNICK Resnic Since {T } is independent of {S } we get ɛ S s,t ɛ S,T, in M p [, 2, where {T } is independent of {S }. We therefore conclude that as s, for any t > 3.3 ɛ S s,t {u, v : u t < u + v} ɛ S,T {u, v : u t < u + v} 1 [S t<s +T ]. Note that the left side of 3.3 is not all of Mt + s, since from 3.3 we only have 1 [s S t+s S +T ]. 1 The difference between this and Mt + s has expectation s UduḠt + s u 1 µ T Ḡu + tdu 1 µ T t Ḡudu. However, the way to construct a stationary version of M is clear: start with {S } a stationary renewal sequence on all of R and define for t > M t 1 [S t<s +T ]. We observe, additionally, that even when the renewal process is a Poisson process, M is only stationary if one defines the Poisson process on all of R. Mt t Figure 4. A path of the process M for α 2, β 2.

15 ACTIVITY RATES The cumulative wor process In the Introduction we mentioned that the worload process At is of major interest in the networ context. The following decomposition of At will be useful: At Nt I 1 + I 2 Nt Ms ds Nt Nt T i 1 [Si +T i t] + Nt T i I 11 I 12 + I 2. mint i, t S i t S i 1 [Si +T i >t] Nt T i 1 [Si +T i >t] + t S i 1 [Si +T i >t] 4.1. The case <, β 1, 2. Define the quantile function of G: σt 1/Ḡ t, t. We always choose σt continuous and strictly increasing. Theorem 4.1. Assume β 1, 2. Moreover, assume that the renewal process N is non-arithmetic and that either F is regularly varying with index α [ 2, 1 or σ 2 < Suppose F is regularly varying and either a α > β or b α β and F x oḡx or c σ 2 <. Set A s u σs 1 Asu suµ T /, u. Then as s, A s µ 1/β β, where β is a β-stable spectrally positive Lévy motion on [,. 2 If F is regularly varying α β and F x c Ḡx, then 4.3 holds, where β is β-stable Lévy motion with sewness parameter If F is regularly varying and α < β or α β and Ḡx o F x, then, as s bs 1 [A s s µ T / ] µ 1/α α, where α is spectrally negative β-stable Lévy motion. Here refers to convergence of the finite-dimensional distributions; it cannot be strengthened to wea convergence in the Sorohod space D[,, J 1 since A has continuous sample paths and the limiting process has jumps. Proof. We have for γ, 1 σt 1 EI 2 σt 1 t x Ḡt x Udx σt γ t x Ḡt x σt x 1+γ Udx

16 16 THOMAS MIKOSCH AND SIDNEY RESNICK µ 1 σt γ x Ḡx σx 1+γ dx. The right hand integral is finite for small γ. We conclude that EI 2 oσt. We have EI 12 By Karamata s Theorem see Bingham et al. 1987, E[T 1 1 [T1 >t x]] Udx. E[T 1 1 [T1 >t]] β 1 1 t P[T 1 > t]. Mohan 1976 proved for a non-arithmetic renewal process N that Ut µ 1 t Ũt is regularly varying with index 2 α if F is regularly varying with index α, α 1, 2], and Ũt c for some positive c if σ 2 < cf. Resnic 1992, p Hence, for F regularly varying with index α 2, 1, EI 12 µ 1 E[T 1 1 [T1 >x]] dx + cβ t 2 P[T 1 > t] + cα, β oσt. E[T 1 1 [T1 >t1 x]] Ũt dx Ũt E[T1 1 E[T 1 1 [T1 >t]] [T1 >t]] Ũt 1 x 1 β x 1 α dx Ũt t P[T1 > t] Now consider the case when σ 2 < or σ2 and F is regularly varying with index 2. Then, as above, We integrate by parts: EI 12 cβ t 2 P[T 1 > t] + E[T 1 1 [T1 >t x]] Ũdx. E[T 1 1 [T1 >t x]] Ũdx E[T 11 >]]Ũt E[T [T1 11 >t]]ũ [T1 Ũx de[t 1 1 [T1 >t x]] E[T 1 1 >]]Ũt [T1 Ũx P[T 1 > t x] dx. Since Ũ is slowly varying and µ T < it also follows in this case that EI 12 oσt. Notice that I 11 µ T t Nt T i µ T + µ T Nt t/ Nt T i µ T + µ T Nt S Nt + O Nt+1 Nt From the above decomposition we conclude that T i µ T i + O µ Nt+1. Nt σt 1 [At t µ T / ] σt 1 T i µ T i + o P 1,

17 which equation holds for the finite-dimensional distributions. Since Ḡ is regularly varying we have 4.4 ACTIVITY RATES 17 P[T i µ T i > x] Ḡx, cf. Resnic 1986, Lemma 4.2. If F is regularly varying with positive index we also have If σ 2 < 4.5 P[T i µ T i x] F x/µ T µ T / α F x. P[T i µ T i x] P[ µ T i x] oḡx. Regular variation of Ḡ and the conditions 4.4 and 4.5 imply that in D[,, J 1 see Gihman and Sorohod 1969, Chapter I.6 σt 1 [t ] T i µ T i [t ] [t ] σt 1 T i µ T + σt 1 µ T µ T i [t ] σt 1 T i µ T + o P 1 β, where β is a spectrally positive β-stable Lévy motion. Notice that 4.5 also holds when α > β or α β and F x oḡx. Hence the same result applies. If α β and F x c Ḡx, the corresponding limit theory yields that σt 1 [t ] T i µ T i β, where β is a β-stable Lévy motion with sewness parameter 4.6 If α < β or if α β and Ḡx o F x, then c 1 /µ T α 1 [ 1, 1]. bt 1 [t ] T i µ T i α for a spectrally negative α-stable Lévy motion. Therefore Nt [t ], bt 1 T i µ T µ 1 t, α in D[,, R 2. By a continuous mapping argument we conclude that bt 1 At µ T t µ 1/α α, where refers to the convergence of the finite-dimensional distributions. The cases when β appears in the limit is completely analogous and therefore omitted.

18 18 THOMAS MIKOSCH AND SIDNEY RESNICK 4.2. The case when 1 and T 1 have finite variance. Under the assumptions σt 2 < and <, the Key Renewal Theorem yields σ 2 EI 2 EI 12 t x Ḡt x Udx µ 1 E[T 1 1 [T1 >t x]] Udx µ 1 On the other hand, similar arguments as in Section 4.1 show that t 1/2 I 11 µ T t t 1/2 Nt x Ḡx dx <, E[T 1 1 [T1 >x]] dx <. T i µ T i + o P 1. Following the ideas of the proof on p. 18 in Embrechts et al. 1997, it is now easy to derive the following result: Proposition 4.2. Assume σt 2 < and σ2 <. Then t 1/2 At µ T t [σt 2 + µ T σ / 2 ] µ 1 1/2 B, where B is standard Brownian motion and refers to convergence of the finite-dimensional distributions The case < β, α < 1. Observe that EI 2 Ḡt Ut EI 1 E[T 1 1 [T1 t]] Ut t cα t, 1 x Ḡt1 x Ḡt E[T 1 1 [T1 t1 x]] E[T 1 1 [T1 t]] Ut dx Ut Ut dx Ut cα, β. This means that EI 2 and EI 1 are of the same order tḡtut t1 β+α Lt. The term I 11 is of order t α/β see Proposition 4.3 below and hence is either of larger order than EI 1 when α > β, or of smaller order when α < β. The analysis of At cannot be based just on I 11 in this case; one has to understand the interplay between I 1 and I 2. Since N and T i are independent, 4.7 [t ] t 1 Nbt, σt 1 T i α, β in D[,, R 2. Then by a continuous mapping argument Nbt σt 1 T i β α. Since bt and σt can be chosen as continuous functions, we can change time: Nt σb t 1 in D[,. Now observe that σb t σ1/ F t. T i β α,

19 Proposition 4.3. Assume < β, α < 1. Then Nt σ1/ F t 1 ACTIVITY RATES 19 T i β α, where the convergence is in D[,, J 1, β is β-stable spectrally positive Lévy motion on [, and α is the inverse process to the α-stable Lévy motion defined in Section 1.3, and both processes are independent. Despite this result, it turns out that At needs a different normalization and we must proceed by relying on Theorem 2.5. Theorem 4.4. Suppose the Case 2 assumptions hold: β α < 1 and if α β, then F s/ḡs, as s. Then A satisfies the relation F s t t u 1 β 4.8 Ast dα sḡs 1 β u, t, in D[,, J 1. Remar 4.5. The convergence in 4.8 is the result one expects by integrating to the limit in 2.5. It suggests that Corollary 2.7 may hold in the M 1 -topology Whitt 22 since integration is continuous in that topology. However, we have not been able to verify this. Proof. We start by verifiying the convergence of the finite-dimensional distributions and focus on the case of a fixed t. We again decompose At I 1 + I 2 as defined in 4.1. The idea is to express both I 1 and I 2 as functions of the random measure in 2.4. Fix δ >. The map 4.9 m u t,δ<v u+v t v mdu, dv, u t,δ<v u+v>t t u mdu, dv from M + E [, 2 is continuous at measures in { Λ : m M + E : m{ [, } m{u, v : u + v t, v δ} } m[, {δ} m{t} [δ,. To see this, write, for instance v mdu, dv 1 u t,δ<δ [,t] u v 1 {u t,δ<v,u+v t} u, v mdu, dv [, u+v t 2 fu, v mdu, dv, and proceed as in the proof of the Helly-Bray lemma. An almost identical argument applies to the continuity of the second integral. Referring to Theorem 2.5, note that, Therefore by continuous mapping, as s, I 1,δ t, I 2,δ t P[T α, LEB ν β Λ c ]. F Nbst bs T Ḡbs bs 1 [ T bs t S bs, T δ], bs 1 Nbst 1 t S bs 1[ T bs >t S bs, T bs δ] I 1,δ t, I 2,δ t

20 2 THOMAS MIKOSCH AND SIDNEY RESNICK As δ, u t,v δ u+v t I 1,δ t, I 2,δ t vt α, LEB ν β du, dv, u t,v δ u+v>t β 1 β t u1 β d α u, Note the sum of the last two terms is 1 1 β t u1 β dα u, as claimed in the statement 4.8. So it remains to show for any η >, 4.1 lim lim sup P[ I j,δ t I j > η], j 1, 2. δ For j 1 the probability is s [ Nbst F bs T ] P Ḡbs bs 1 [ T bs t S bs, T bs δ] > η 1 η 1 F bs Ḡbs E Nbst 1 bst T1 t ut α, LEB ν β du, dv T bs 1 [ T bs t S bs, T bs δ] t u 1 β d α u. Udu η F 1 bs E Ḡbs bs 1 [T 1 bst u,t 1 bsδ] E T 1 1 η 1 [T1 t y δbs] bsḡbs F bsubsdy E T 1 1 [T1 t ybs] t δ bsḡbs F bsubsdy δ E T η 1 [T1 δbs] A + B. bsḡbs F bsubsdy Now for A we have the bound apart from the factor η 1, E T 1 1 [T1 δbs] A F bsubst bsḡbs F bsubst δ and as s. This is asymptotic to For B we have c 1 δ 1 β c 2 t α c 2 t δ α, δ. B cδ 1 β F bsubs1 δ cδ 1 β 1 δ α, δ. For j 2 in 4.1, we have for the probability [ F bs P Ḡbs Nbst 1 S ] t 1[ T bs bs >t S bs, T > η bs δ]. Chebyshev

21 ACTIVITY RATES 21 η F 1 bs t Ḡbs E S 1[ T bs bs >t S bs, T bs δ] > η, Letting S σs, 1, we get by iterating expectations η 1 F bs Ḡbs EE η 1 F bs Ḡbs E η 1 F bs Ḡbs E η 1 F bs Ḡbs Nbst 1 Nbst 1 Nbst 1 S t 1[ T bs bs >t S bs, T bs δ] > η S t S bs t S bs P [bst S < T bsδ S] Ḡbst S Ḡbsδ t uḡbst u Ḡbsδ Ubsdu + η 1 u t uḡbst F bsubsdu t δ Ḡbs c t u 1 β du α s t δ δ. This proves convergence of the one-dimensional distributions in Theorem 4.4. The convergence of the finite dimensional distributions is straightforward: The multivariate analog of the map in 4.9 is also almost surely continuous and once this is noted, it is clear how to proceed. The tightness of the converging processes in D[,, J 1 follows from the convergence of the finite-dimensional distributions together with the observation that the sample paths of A and of the limiting process are monotone and continuous; see Jacod and Shiryaev 1987, Theorem VI Several questions remain unanswered. 5. Unresolved problems 5.1. The case < and β, 1. An analysis similar to what was performed at the beginning of Subsection 4.3, shows that EI 1 and EI 2 are of the same order and of lower order than I 11 ; see below. Hence I 11 does not help here. By the independence of N and T i, [t ] t 1 Nt, σt 1 T i µ 1, β, in D[,, R 2, where β is spectrally positive β-stable Lévy motion. By a continuous mapping argument, Nt σt 1 T i µ 1/β β, in D[,, J 1. A similar argument as for Proposition 4.3 finally gives the following result:

22 22 THOMAS MIKOSCH AND SIDNEY RESNICK Proposition 5.1. Assume β, 1 and <. Then Nt σt 1 T i µ 1/β β in D[,, J 1, where β is spectrally positive β-stable Lévy motion on [,. Referring to Proposition 3.2, we would expect a Gaussian limit for At in this case Other problems. Here is a list of problems whose resolution is unsatisfactory: 1 The Gaussian limit in Proposition 3.2 is only obtained after a random centering. It can be replaced by the expected value if {S } constitutes a Poisson process. When can the random centering be replaced by a non-random centering? 2 The Gaussian approximation in Proposition 3.2 is only in the sense of convergence of finitedimensional distributions. We suspect that the convergence can be considerably strengthened allowing integration to the limit which would resolve the asymptotic behavior of At. 3 We expect that the mode of convergence in Corollary 2.7 can be strengthened. If so, this would provide a convenient way to obtain Theorem Connections to data networs rarely occur according to a Poisson process, and it is unliely they occur according to a renewal process Guerin et al. 23. What more general class of connection models would be tractable? 5 Transmissions do not occur at unit rate as assumed here and more general models are needed. References M. Arlitt and C. Williamson. Web server worload characterization: The search for invariants extended version. In Proceedings of the ACM Sigmetrics Conference, Philadelphia, Pa, Available from {mfa126,carey}@cs.usas.ca. P. Becer-Kern, M. Meerschaert, and H.-P. Scheffler. Limit theorems for coupled continuous time random wals. Annals of Probability, 32:73 756, J. Bertoin. Lévy Processes. Cambridge University Press, P. Billingsley. Convergence of Probability Measures. Wiley, New Yor, N. Bingham, C. Goldie, and J. Teugels. Regular Variation. Cambridge University Press, M. Crovella and A. Bestavros. Self-similarity in world wide web traffic: evidence and possible causes. In Proceedings of the ACM SIGMETRICS 96 International Conference on Measurement and Modeling of Computer Systems, volume 24, pages , P. Embrechts, C. Klüppelberg and T. Miosch Modelling Extremal Events for Insurance and Finance. Springer, Berlin, W. Feller. An Introduction to Probability Theory and Its Applications, volume 2. Wiley, New Yor, 2nd edition, M. Garrett and W. Willinger. Analysis, modeling and generation of self similar vbr video traffic. In Proceedings of the ACM SigComm, London, I.I. Gihman and A.V. Sorohod. Introduction to the Theory of Random Processes, Saunders, Philadelphia, PA. C. Guerin, H. Nyberg, O. Perrin, S. Resnic, H. Rootzen, and C. Stărică. Empirical testing of the infinite source poisson data traffic model. Stochastic Models, 192:151 2, 23. J. Jacod and A.N. Shiryaev. Limit Theorems for Stochastic Processes. Springer, Berlin, O. Kallenberg. Random Measures. Aademieverlag, Berlin, C. Klüppelberg and T. Miosch. Explosive Poisson shot noise processes with applications to ris reserves. Bernoulli, 1: , 1995.

23 ACTIVITY RATES 23 C. Klüppelberg, T. Miosch and A. Schärf. Regular variation in the mean and stable limits for Poisson shot noise. Bernoulli, 93: , 23. W. Leland, M. Taqqu, W. Willinger, and D. Wilson. On the self-similar nature of Ethernet traffic extended version. IEEE/ACM Transactions on Networing, 2:1 15, M. Meerschaert and H. Scheffler. Limit theorems for continuous time random wals with infinite mean waiting times. J. Applied Probability, 41: , 24. N.R. Mohan. Teugels renewal theorem and stable laws. Annals of Probability, 4: K. Par and W. Willinger. Self-similar networ traffic: An overview. In K. Par and W. Willinger, editors, Self-Similar Networ Traffic and Performance Evaluation. Wiley-Interscience, New Yor, 2. V. Pipiras, M. S. Taqqu, J. B. Levy. Slow, fast and arbitrary growth conditions for renewal-reward processes when both the renewals and the rewards are heavy-tailed. Bernoulli, 11: , 24. S. Resnic. Point processes, regular variation and wea convergence. Adv. Applied Probability, 18: , S. Resnic. Extreme Values, Regular Variation and Point Processes. Springer-Verlag, New Yor, S. Resnic, Adventures in Stochastic Processes. Birhäuser, Boston, S. Resnic. Modeling data networs. In B. Finenstadt and H. Rootzen, editors, SemStat: Seminaire Europeen de Statistique, Exteme Values in Finance, Telecommunications, and the Environment, pages Chapman-Hall, London, 23. S. Resnic and H. Rootzén. Self-similar communication models and very heavy tails. Ann. Applied Probability, 1: , 2. G. Samorodnitsy and M. S. Taqqu. Stable Non-Gaussian Random Processes. Chapman & Hall, New Yor, K.-i. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, E. Seneta. Regularly Varying Functions. Springer-Verlag, New Yor, Lecture Notes in Mathematics, 58. W. Whitt. Stochastic-Process Limits. Springer-Verlag, New Yor, 22. V. Zolotarev. One-Dimensional Stable Distributions, volume 65 of Translations of mathematical monographs. American Mathematical Society, Translation from the original 1983 Russian edition. Thomas Miosch, Laboratory of Actuarial Mathematics, University of Copenhagen, Universitetsparen 5, DK-21 Copenhagen, Denmar, and, MaPhySto, The Danish Research Networ for Mathematical Physics and Stochastics miosch address: Sidney Resnic, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY sid address: