Extremes of Gaussian processes over an infinite horizon

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1 Extremes of Gassian processes over an inite horizon A B Dieer CWI O Box GB Amsterdam, the Netherlands and University of Twente Faclty of Mathematical Sciences O Box AE Enschede, the Netherlands Abstract Consider a centered separable Gassian process with a variance fnction that is reglarly varying at inity with index H 0, Let φ be a drift fnction that is strictly increasing, reglarly varying at inity with index β > H, and vanishing at the origin Motivated by qeeing and ris models, we investigate the asymptotics for of the probability t φt > as To obtain the asymptotics, we tailor the celebrated doble sm method to or general framewor Two different families of correlation strctres are stdied, leading to for qalitatively different types of asymptotic behavior A generalized icands constant appears in one of these cases Or reslts cover both processes with stationary increments inclding Gassian integrated processes and self-similar processes 1 Introdction Let be a centered separable Gassian process, and let φ be a strictly increasing drift fnction with φ0 = 0 Motivated by applications in telecommnications engineering and insrance mathematics, the probability t φt > 1 has been analyzed nder different levels of generality as In these applications, 0 is posed to be degenerate, ie, 0 = 0 Letting tend to inity is nown as investigating the large bffer regime, since can be interpreted as a bffer level of a qee Notice that 1 can be rewritten as µt 1 + t >, where µ is the inverse of φ Special attention has been paid to the case that has stationary increments eg, [5, 6, 9, 10, 11, 13, 16, 18, 0, 4, 5, 9, 30], and to the case that is self-similar or almost self-similar [3] The research was ported by the Netherlands Organization for Scientific Research NWO nder grant From a practical point of view, Gassian processes lead to parsimonios yet flexible models, since a broad range of correlation strctres can be described by few parameters The stdy of Gassian processes can also be jstified by an approximation argment; they can appear as stochastic process limits, often as a reslt of a second-order scaling as in the central limit theorem However, a warning is in place here: Wischi [40] arges that it is extremely important to chec the appropriateness of this scaling before resorting to Gassian models The main contribtion of the present paper is that we extend the nown reslts on the asymptotics of 1 For this, we introdce a wide class of local correlation strctres, covering both processes with stationary increments and almost self-similar processes A motivation for stdying the problem in this generality is to gain insight into the case that is the sm of a nmber of independent Gassian processes, eg, of a Gassian integrated process and a nmber of fractional Brownian motions with different Hrst parameters We stdy this case in somewhat more detail in forthcoming wor Some words for the technical aspects of this paper We se the doble sm method to find the asymptotics of, see iterbarg [34] or iterbarg and Fatalov [35] This method has been applied scessflly to find the asymptotics of t [0,T ] Xt >, where X is either a stationary Gassian process [33, 37] or a Gassian process with a niqe point of maximm variance [36] These reslts are also available for fields, see [34, Section 8] However, they cannot be applied to find the asymptotics of 1 In this paper, we approach the doble sm method differently The idea in [36] is to first establish the asymptotics of a certain stationary Gassian process on a sbinterval of [0, T ] Then a comparison ineqality is applied to see that the asymptotics of t [0,T ] Xt > eqal the asymptotics of this stationary field Here, we do not mae a comparison to stationary processes, bt we apply the ideas nderlying the doble sm method directly to the processes µt /1 + t Given or reslts, it can be seen immediately that the comparison approach cannot wor in the generality of this paper: a so-called generalized icands constant appears, which is not present in the stationary case It is also obtained in the analysis of rema of Gassian integrated processes, see Dȩbici [11] The appearance of this constant in the present stdy is not srprising, since or reslts also cover Gassian integrated processes Several related problems appear in the vast body of literatre on asymptotics for Gassian processes For instance, Dȩbici and Rolsi [17] stdy the asymptotics of 1 over a finite horizon, ie, the remm is taen over [0, T ] for some T > 0 We remar that the asymptotics fond in [17] differ qalitatively from the asymptotics established in the present paper Another problem closely related to the present setting is where has the form Z/ n for some Gassian process Z independent of n One then fixes and stdies the probability 1 as n The reslting asymptotics were stdied by Dȩbici and Mandjes [1]; these asymptotics are often called many sorces asymptotics, since convoltion of identical Gassian measres amonts to scaling a single measre It is worthwhile to compare or reslts with those of Berman [] on extremes of Gassian processes with stationary increments Berman stdies the probability t B t > for, where is constrcted from by standardization so that its variance is constant and B is some fixed compact interval The problem of finding the asymptotics of does not fit into Berman s framewor: or assmptions will imply that µt /1 + t has a point of maximm variance, which is asymptotically niqe Another difference is that this point depends asymptotically linearly on, so that it cannot belong to B for large The paper is organized as follows The main reslt and its assmptions are described in Section In Section 3, we wor ot two cases of special interest: processes with stationary increments and self-similar processes Frthermore, we relate or formlas with the literatre

2 by giving some examples Sections 4 7 are devoted to proofs In Section 4, the classical icands lemma is generalized into an appropriate direction Section 5 distingishes for instances of this lemma The reslting observations are ey to the derivation of the pper bonds, which is the topic of Section 6 Lower bonds are given in Section 7, where we se a doble sm-type argment to see that the pper and lower bonds coincide asymptotically To slightly redce the length of the proofs and mae them more readable, details are often omitted when a similar argment has already been given, or when the argment is standard We then se crly bracets eg, {T1} to indicate which assmptions are needed to mae the claim precise We freqently apply standard reslts for reglarly varying fnctions, for which the main reference is Bingham, Goldie and Tegels [4] Recall that a positive fnction f is reglarly varying at inity with index ρ if for all t > 0, fαt lim α fα = tρ Implicitly, this convergence is niform on intervals of the form [a, b] with b > a > 0 by the Uniform Convergence Theorem Theorem 15 in [4] Often one can obtain niformity on a wider class of intervals, althogh additional conditions may be reqired see Theorem 15 and Theorem 153 in [4] The Uniform Convergence Theorem is sed extensively, and therefore abbreviated as UCT It is applied withot reference to the specific version that is sed Description of the reslts and assmptions This section presents or main theorem Since many yet natral and wea assmptions nderly or reslt, we defer a detailed description of these assmptions to Section 1 Main theorem The remm in is asymptotically most liely attained at a point where the variance is close to its maximm vale Let t denote a point that maximizes the variance σ µt/1+ t existence will be ensred by continity conditions Or main assmptions are that σ defined by σ t := Var t and µ defined as the inverse of φ in 1 are reglarly varying at inity with indices H 0, and 1/β < 1/H respectively Note that the UCT implies that t converges to t := H/β H In that sense, t is asymptotically niqe For an appropriately chosen δ with δ/ 0 and σµ/δ 0, 1 and are asymptotically eqivalent to µt t [t ±δ/] 1 + t >, see Lemma 7 Hence, in some sense, the variance σ µt of µt determines the length of the most probable hitting interval by the reqirement that σµ/δ 0 Not only the length of this interval plays a role in the asymptotics of There is one other important element: the local correlation strctre of the process on [t ± δ/] Traditionally, it was assmed that Var µs /σµs µt /σµt behaves locally lie s t α for some α 0, ] [3] It was soon realized that s t α can be replaced by a reglarly varying fnction at zero with minimal additional effort [37]; see also [3, 11, 3], to mention a few recent contribtions 3 However, by imposing sch a correlation strctre, it is impossible to find the asymptotics of 1 for a general Gassian process with stationary increments, for instance We solve this problem by introdcing two wide classes of correlation strctres, reslting in qalitatively different asymptotics in for cases These specific strctres mst be imposed to be able to perform explicit calclations The main novelty of this paper is that the local behavior may depend on Or framewor is specific enogh to derive generalities, yet general enogh to inclde many interesting processes as special cases to or best nowledge, all processes are covered for which the asymptotics of 1 appear in the literatre; see the examples in Section 33 Often there is a third element playing a role in the asymptotics: the local variance strctre of µt /1+t near t = t By the strctre of the problem and the differentiability assmptions that we will impose on σ and µ, this third element is only implicitly present in or analysis However, if one is interested in the asymptotics of some probability different from 1, it may play a role In that case, the reasoning of the present paper is readily adapted We now introdce the first family of correlation strctres, leading to three different types of asymptotics Sppose that the following holds: µs Var σµs µt σµt s,t [t ±δ/] Dτ νs νt /τ ν 1 0, 3 s t as, where D is some constant and τ and ν are sitable fnctions It is assmed that τ and ν are reglarly varying at inity with indices ι τ 0, 1 and ι ν > 0 respectively To gain some intition, pose that ν is the identity, and write τt = ltt ιτ for some slowly varying fnction at inity l The denominator in 3 then eqals D s t ιτ l s t /l From the analysis of the problem it follows that one mst consider s t /, where is some fnction satisfying = oδ As a reslt, the denominator is of the order [ /] ιτ l /l ; de to the term l, three cases can now be distingished: tends to inity, to a constant, or to zero Interestingly, the icands constant appearing in the asymptotics is determined by the behavior of τ at inity in the first case, and at zero in the last case one needs an additional assmption on the behavior of τ at zero The second intermediate case is special, reslting in the appearance of a so-called generalized icands constant The second family of correlation strctres, reslting in the forth type of asymptotics, is given by µs Var σµs µt σµt s,t [t ±δ/] τ νs νt /ν 1 0, 4 s t where ν is reglarly varying at inity with index ι ν > 0 and τ is reglarly varying at zero with index ι τ 0, 1 the tilde emphasizes that we consider reglar variation at zero A detailed description of the assmptions on each of the fnctions are given in Section Here, if ν is the identity, the denominator eqals l s t s t ιτ for some slowly varying fnction at zero l Therefore, it cannot be written in the form 3 nless l is constant Having introdced the for cases intitively, we now present them in somewhat more detail The cases are referred to as case A, B, C, and D We set assming the limit exists σµτν G := lim, 5 4

3 A Case A applies when 3 holds and G = B Case B applies when 3 holds and G 0, C Case C applies when 3 holds and G = 0 We then also pose that τ be reglarly varying at zero with index ι τ 0, 1 D Case D applies when 4 holds In order to state the main reslt, we first introdce some frther notation For a centered separable Gassian process η with stationary increments and variance fnction σ η, we define 1 H η := lim T T H 1 ηt := lim T T E exp t [0,T ] [ ] ηt σηt, 6 provided both the expectation and the limit exist Depending on the context, we also write H σ η for H η If η is a fractional Brownian motion with Hrst parameter H 0, 1, it is denoted as B H throghot this paper Recall that a fractional Brownian motion is defined by setting σηt = t H, and that these constants are strictly positive in particlar, they exist These constants appear in icands classical analysis of stationary Gassian processes [3, 33] In the present generality, they have been introdced by Dȩbici [11], and the field analoge shows p in the stdy of Gassian fields; see iterbarg [34] Given a stochastic process, we se both t and t for the vale of at time epoch t Moreover, we write Ψx = 1 e 1 w dw, π x and it is standard that, for x, πxψx e x /, 7 where asymptotic eqivalence f g as x X [, ] means fx = gx1 + o1 as x X rovided it exists, we denote an asymptotic inverse of τ by τ ; recall that it is asymptotically niqely defined by τ τt τ τ t t 8 It depends on the context whether τ is an asymptotic inverse near zero or inity, ie, whether 8 holds for t 0 or t respectively Unless stated otherwise, reglar variation shold always be nderstood as reglar variation at inity, and measrability of sch fnctions is implicit it is often ensred by continity assmptions It is convenient to introdce the notation and, for case B, C H,β,ιν,ιτ β 1/ιτ H ιν+ H := β H ιτ β 1 1/ιτ πι ν H β H M := β G H H/β β H H/β, where G 0, is defined as in 5 Here is or main reslt The assmptions are detailed in Section 5 Theorem 1 Let µ and σ satisfy assmptions M1 M4 and S1 S4 below for some β > H In case A, ie, when A1, A, T1, T, N1, N below hold, we have σµν 1 + t µt t > H Bιτ C H,β,ιν,ιτ D 1/ιτ Ψ τ σµτν σµt In case B, ie, when B1, B, T1, T, N1, N below hold, then H DMτ exists and we have H ιν+ H β 1 σµν 1 + t µt t > H DMτ πιν Ψ β H σµt In case C, ie, when C1 C3, T1, N1, N below hold, we have σµν µt t > H B ιτ C H,β,ιν, ιτ D 1/ ιτ Ψ τ σµτν In case D, ie, when D1, D, N1, N below hold, we have σµ µt t > H B ιτ C H,β,ιν, ιτ Ψ τ σµ 1 + t σµt 1 + t σµt Observe that τ is an asymptotic inverse of τ at inity in case A, and at zero in case C and D Hence, the factors preceding the fnction Ψ are reglarly varying with index H/β + ι ν ι τ 11 1/ι τ + 1 ι τ ι ν in case A, with index H/β + ι ν 1 in case B, with index H/β + ι ν 1 H/β + ι τ ι ν 1/ ι τ in case C, and with index H/β 11 1/ ι τ in case D Note that case B is special in a nmber of ways: a non-classical icands constant is present and no inverse appears in the formla We now formally state the nderlying assmptions Assmptions Two types of assmptions are distingished: general assmptions and case-specific assmptions The general assmptions involve the variance σ of, the time change µ, and the fnctions ν and τ appearing in 3 and 4 The case-specific assmptions formalize the for regimes introdced in the previos sbsection 1 General assmptions We start by stating the assmptions on µ M1 µ is reglarly varying at inity with index 1/β, M µ is strictly increasing, µ0 = 0, M3 µ is ltimately continosly differentiable and its derivative µ is ltimately monotone M4 µ is twice continosly differentiable and its second derivative µ is ltimately monotone Assmption M is needed to ensre that the probabilities 1 and be eqal The remaining conditions imply that β µ µ and β µ 1 βµ, see Exercise of [4] In particlar, µ and µ are reglarly varying with index 1/β 1 and 1/β respectively Now we formlate the assmptions on σ and one assmption on both µ and σ 6

4 S1 σ is continos and reglarly varying at inity with index H for some H 0, 1, S σ is ltimately continosly differentiable and its first derivative σ is ltimately monotone, S3 σ is ltimately twice continosly differentiable and its second derivative σ is ltimately monotone, S4 There exist some T, ɛ > 0, γ 0, ] sch that 1 lim s,t 0,1+ɛT 1/β ] Var s t σ s t γ <, and σ lim µ log µt t T 1+t > < 1 1+t t H/β We emphasize that σ denotes the derivative of σ, and not the sqare derivative of σ As earlier, conditions S1 S3 imply that σ Hσ and σ HH 1σ The first point of S4, which is Kolmogorov s wea convergence criterion, ensres the existence of a modification with continos sample paths; we always assme to wor with this modification The second point of S4 ensres that the probability t T µt t > cannot dominate the asymptotics We choose to formlate this as an assmption, althogh it is possible to give sharp conditions for S4 to hold However, these conditions loo relatively complicated, while the second point is in general easier to verify on a case by case basis In the next section, we show that it holds for processes with stationary increments and self-similar processes Note that if M1 M4 and S1 S4 hold, the first and second derivative of σ µ are also reglarly varying, with indices H/β 1 and H/β respectively It is this fact that garantees the existence of the limits that are implicitly present in the notation in Theorem 1 The fnction ν appearing in 3 and 4 also has to satisfy certain assmptions, which are similar to the assmptions imposed on µ: N1 ν is reglarly varying at inity with index ι ν > 0, N ν is ltimately continosly differentiable and its derivative ν is ltimately monotone Finally, we formlate the assmptions on τ in 3 or 4 T1 τ is continos and reglarly varying at inity with index ι τ for some ι τ 0, 1, T τt Ct γ on a neighborhood of zero for some C, γ > 0 Assmption T is essential to prove niform tightness at some point in the proof, which yields the existence of the icands constants Case-specific assmptions We now formlate the case-specific assmptions in each of the cases A, B, C, and D These assmptions are also mentioned in the Introdction, bt it is convenient to label them for reference prposes If we write that the correlation strctre is determined by 3 or 4, the fnction δ is posed to satisfy δ = o and σµ = oδ as After recalling the definition of G in 5, we start with case A A1 the correlation strctre is determined by 3, 7 A G = Similar conditions are imposed in case B B1 the correlation strctre is determined by 3, B G 0, In case C, we need an additional condition C3 Note that the index of variation in C3 appears at several places in the asymptotics, cf Theorem 1 It also implies the existence of an asymptotic inverse τ at zero, cf Theorem 151 of [4] C1 the correlation strctre is determined by 3, C G = 0, C3 τ is reglarly varying at zero with index ι τ 0, 1 Case D is slightly different from the previos three cases, althogh here the reglar variation of τ at zero also plays a role In fact, the index of variation appears in exactly the same way in the asymptotics as in case C D1 the correlation strctre is determined by 4, D τ is reglarly varying at zero with index ι τ 0, 1 3 Special cases: stationary increments and self-similarity In this section, we apply Theorem 1 to calclate the asymptotics of for two specific cases: i has stationary increments and ii is self-similar In both examples, the imposed assmptions imply that σ 0 = 0, so that 0 = 0 almost srely In case has stationary increments, the finite-dimensional distribtions are completely determined by the variance fnction σ For self-similar processes, has been stdied by Hüsler and iterbarg [3] We show that their reslts are reprodced and even slightly generalized by Theorem 1 We conclde this section with some examples that have been stdied in the literatre 31 Stationary increments Since σ determines the finite-dimensional distribtions of, it also fixes the local correlation strctre; we record this in the next proposition To get some feeling for the reslt, observe that for s, t [t ± δ/], µs Var σµs µt Var µs µt σµt σ µt = σ µs µt σ µt This intitive reasoning is now made precise Note the proposition also entails that case D does not occr in this setting roposition 1 Let S1 S, M1 M3 hold for some β > H Let δ be reglarly varying with index ι δ 1 1/β, 1 Then 3 holds with τ = σ, ν = µ and D = t H/β 8

5 roof Since s, t [t ± δ/], we have by the UCT {S1, M1}, lim σ µ s,t [t Dσµsσµt 1 = 0 ±δ/] s t Moreover, the stationarity of the increments implies that [ σµsσµt Cov µs, µt ] = σ µs µt [σµs σµt] Hence, it sffices to prove that lim s,t [t ±δ/] s t [σµs σµt] σ µs µt = 0 9 For this, observe that the left hand side of 9 is majorized by t 1 t, where t 1 := s,t [t ±δ/] s t [σµs σµt] [µs µt] ; t := s,t [t ±δ/] s t [µs µt] σ µs µt As for t 1, by the Mean Vale Theorem {S, M3} there exist t, s, t, t, s, t sch that, for large enogh, [ σ µ t, s, t] t [t t 1 = s,t [t [ µt ±δ/], s, t] ±δ/] σ µ t, t [t ±δ/] µt s t where σ µ denotes the derivative of σµ As a conseqence of the UCT {M1, M3, S1, S}, t 1 can therefore be pper bonded by C σ µ/µ for some constant C < We now trn to t A sbstittion {M} shows that s t t = s,t [µt δ,µt σ +δ] s t = t 0<t µt +δ µt σ δ t s>t Observe that, again by the Mean Vale Theorem and the UCT {M1, M3}, µt + δ µt δ µtδ t [t β t 1/β 1 µδ/, ±δ/] which tends to inity by the assmption on ι δ Sppose for the moment that the map x x /σ x is bonded on sets of the form 0, ] Since it is reglarly varying with index H > 0 {S1}, we have by the UCT and the assmption that ι δ > 1 1/β, for large enogh, [µδ/]t 3 H t 0<t 3/βt 1/β 1 σ µδ/t β t 1/β 1 [µδ/] σ µδ/ In conclsion, there exists a constant K < sch that [σµs σµt] s,t [t σ ±δ/] K σ µδ / µs µt σ µδ/, s t 9 which is reglarly varying with index 1 Hι δ 1 < 0, so that 9 follows It remains to show that x x /σ x is locally bonded To see this, we se an argment introdced by Dȩbici [11, Lemma 1] By S, one can select some large s 0 sch that σ is continosly differentiable at s Then, for some small x > 0, σ s σ s x σ s + σ x σ s x = Cov s, x σsσx, and by the Mean Vale Theorem there exists some ρ x [s x, s] sch that σ s σ s x = σ ρ x x By continity of σ at s, lim x 0 x σx lim σs x 0 σ ρ x = σs σ s < The claim follows pon combining this observation with S1 Lemma 1 Let have stationary increments, and pose that S1 and M1 hold If σ t Ct γ on a neighborhood of zero for some C, γ > 0, then S4 holds roof By the stationarity of the increments, the first point of S4 follows immediately from the UCT for t σ tt γ this map is locally bonded by the condition in the lemma In fact, it holds for all T, ɛ > 0 To chec the second reqirement of S4, select some ω sch that H/β < ω < 1 By the UCT {M1} lim lim T t T t ω µ µt 1/β = lim T ω 1 = 0 T Hence, we may pose withot loss of generality that T is sch that µ µt 1/β / t ω for every t T and large This implies that µt t T 1 + t > µt1/β t [µt /µ] β 1 + t ω > µt 1/β t T/ 1 + t ω > We now apply some reslts from earlier wor [18] By Corollary 3 and the argments in the proof of roposition 1 of [18], we have lim σ µ log t T µt 1 + t > 1 t T/ 1 + t ω t H/β Note that we have sed the continity of the fnctional x t T/ 1/β xt/1 + t ωβ in a certain topology, cf Lemma of [18] The claim is obtained by choosing T large enogh, which is possible since t ω /t H/β as t With roposition 1 and Lemma 1 at or disposal, we readily find the asymptotics of 1 when has stationary increments roposition Let have stationary increments Sppose that S1 S3 hold, and that σ t Ct γ on a neighborhood of zero for some C, γ > 0 Moreover, pose that M1 M4 hold for some β > H If σ µ/, then β H µt t > H BH C H,β,1/β,H H 10 1/β σµµ Ψ σ σ µ 1 + t σµt

6 If σ µ/ G 0,, then µt t > H /G σ π/ H σµµ Ψ 1 + t σµt If σ µ/ 0 and σ is reglarly varying at zero with index λ 0, 1, then β H µt t > H Bλ C H,β,1/β,λ H H/βλ σµµ Ψ σ σ µ 1 + t σµt roof Directly from Theorem 1 For the case σ µ/ G 0,, observe that necessarily H = β 3 Self-similar processes We now pose that is a self-similar process with Hrst parameter H, ie, Var t = t H and for any α > 0 and s, t 0, Cov αt, αs = α H Cov t, s 10 The self-similarity property has been observed statistically in several types of data traffic, see, eg, [31] Two examples of self-similar Gassian processes are the fractional Brownian motion and the Riemann-Lioville process Another ndobtedly related reason why self-similar processes are interesting is that the wea limit obtained by scaling a process both in time and space mst be self-similar if it exists; see Lamperti [7] In the setting of Gassian processes with stationary increments, a strong type of wea convergence is stdied in [18] We also mention the interesting fact that self-similar processes are closely related to stationary processes by the so-called Lampertitransformation; see [1] for more details We mae the following assmption abot the behavior of the standardized variance of near t = t : for some fnction τ which is reglarly varying at zero with index ι τ 0, 1, s 1/β Var s lim H/β t 1/β t H/β s,t t τ = 1 11 s t By the self-similarity, one may eqivalently reqire that a similar condition holds for s, t tending to an arbitrary strictly positive nmber; see [3] In the proof of roposition 3 below we show that 11 implies that self-similar processes are covered by case D We also need the following assmption on the variance strctre of : for some γ > 0, Var s t s t γ < 1 s,t 0,1] This Kolmogorov criterion ensres that there exists a continos modification of Notice that withot loss of generality it sffices to tae the remm over any interval 0, ] by the self-similarity The following proposition generalizes Theorem 1 of Hüsler and iterbarg [3]; it is left to the reader to chec that the formlas indeed coincide when φt = ct β for some c > 0 Althogh no condition of the type 1 appears in [3], it is implicitly present; the process Z in [3] is claimed to satisfy condition E3 on page 19 of [34] 11 roposition 3 Let be self-similar with Hrst parameter H, and let µ satisfy M1 M4 for some β > H If 11 and 1 hold, then, µ H 1 + t µt t > H B ιτ C H,β,1, ιτ Ψ τ µ H µt H roof Note that by 11, for δ with δ = o, µs/µ Var µt/µ µs/µ lim H µt/µ H s,t [t ±δ/] τ µs β µt β /µ β 1 = 0 The self-similarity implies µs/µ Var µs/µ H µt/µ µs µt/µ H = Var µs H µt µt H, so that 4 holds for νt = µt β and the τ of 11; then we have N1 and N as a conseqence of the assmption that M1 M3 hold Moreover, it is trivial that σ t = t H satisfies S1 S3 We now show that S4 holds By the self-similarity, for any T > 0, Var s t s,t 0,T ] H s t γ = T H γ Var s t s,t 0,1] s t γ, so that the first condition of S4 is satisfied de to 1 As for the second point, by the self-similarity and the reasoning in the proof of Lemma 1, it sffices to show that for large T µ H lim log t 1/β t T/ 1 + t ω > µ H < t t H/β, for some ω satisfying H/β < ω < 1 This follows from Borell s ineqality eg, Theorem D1 of [34] once it has been shown that t /t ωβ 0 as t We se a reasoning as in Lemma 3 of [18] to see that this is the case First, one can exploit the fact that ωβ > H to establish lim / ωβ = 0 by the Borel-Cantelli lemma It then sffices to show that also Z / ωβ 0, where Z := s [, +1 ] s Note that Z has the same distribtion as H Z 0 by the self-similarity of The almost sre convergence follows again from the Borel-Cantelli lemma: for α, ɛ > 0, Z / ωβ > ɛ Z 0 > ɛ ωβ H exp αɛ ωβ H E exp αz0 If one chooses α > 0 appropriately small, E exp αz0 is finite as a conseqence of Borell s ineqality which can be applied since is continos In conclsion, case D applies and the asymptotics are given by Theorem 1 Hüsler and iterbarg [3, Section 3] also consider a class of Gassian processes that behave somewhat lie a self-similar processes Althogh we do not wor this ot, this class is also covered by case D of Theorem 1; note that their condition 18 is a special case of 4, for νt = t 33 Examples We now wor ot some examples that appear in the literatre In all examples, we obtain modest extensions of what is nown already For Gassian integrated processes Section 33, we also remove some technical conditions 1

7 331 Fractional Brownian motion In some sense, fractional Brownian motion fbm is the easiest instance of a process that fits into the framewor of roposition Indeed, the variance fnction σ of fbm is the canonical reglarly varying fnction, σ t = t H for some H 0, 1 A fractional Brownian motion B H is self-similar in the sense of 10 Therefore, it can appear as a wea limit of a time- and space-scaled process; for examples, see [18, 39] The increments of a fractional Brownian motion are long-range dependent if and only if H > 1/, ie, the covariance fnction of the increments on an eqispaced grid is then nonsmmable For more details on long-range dependence and an extensive list of references, see Dohan et al [19] As fbm is both self-similar and has stationary increments, the asymptotics can be obtained by applying either roposition or roposition 3 Interestingly, this implies that it shold be possible to write the formlas in the three cases of roposition as a single formla for fbm The proof given below is based on roposition, bt the reader easily verifies that roposition 3 yields the same formla; one then ses β H β 1/β C H,β,1/β,H = β H βh C H,β,1,H Note that fbm is the only process for which both roposition and 3 can be applied: it is the only Gassian self-similar process with stationary increments Corollary 1 Let B H be a fractional Brownian motion with Hrst parameter H 0, 1 If µ satisfies conditions M1 M4 for some β > H, then B H µt t > β H 1/β 1/H 1 H BH C H,β,1/β,H H µ 1 H Ψ 1 + t µt H roof First note that µ H / has a limit in [0, ] as a conseqence of M If µ H / tends to either zero or inity, the formla follows readily from roposition by setting σ t = t H so that λ = H in case C In case µ H / G 0,, the generalized icands constant can be expressed in a classical one by exploiting the self-similarity of B H ; one easily checs that H /GBH = /G 1/H H BH The above formla is then fond by noting that β = H and µ H+1 1/H 1/H 1 G µ 1 H For a standard Brownian motion H = 1/, icands constant eqals H B1/ = 1, so that the formla redces to B µt t > πββ 1 1 1/β t Ψ 13 µ µt This probability has been extensively stdied in the literatre; the whole distribtion of B µt t is nown in a nmber of cases We refer to some recent contribtions [7, 1, ] for bacgrond and references The tail asymptotics of B µt t are stdied in Dȩbici [10], bt we believe that formla 13 does not appear elsewhere in the literatre Gassian integrated process A Gassian integrated process has the form t t = Zsds, 14 0 where Z is a centered stationary Gassian process with covariance fnction R We pose that R be ltimately continos and that R0 > 0 It is easy to see that t s σ t = Rvdvds 0 0 In the literatre, µ is assmed to be of the form µt = t/c for some c > 0, so that M1 M4 obviosly hold For an easy comparison, we also adopt this particlar choice for µ here simple scaling argments show that we may have assmed c = 1 withot loss of generality Evidently, the reslts of this paper allow for mch more general drift fnctions, and the reader has no difficlties to write ot the corresponding formla The strctre of the problem ensres that S and S3 hold, and that σt Ct γ for some C, γ > 0 since σ t/t = 1 s 0 0 Rtvdvds tends to R0 as t 0 Short-range dependent case A nmber of important Gassian integrated processes have short-range dependent characteristics erhaps the most well-nown example is an Ornstein-Uhlenbec process, for which Rt = exp αt, where α > 0 is a constant Dȩbici and Rolsi [16] stdy the more general case where Z = r X for some -vector r and X is the stationary soltion of the stochastic differential eqation dxt = AXtdt + σdw t, for matrices A, σ satisfying certain conditions and a standard -dimensional Brownian motion W Then Rt = r Σe ta r for some covariance Σ By stating that a Gassian integrated process is short-range dependent, we mean that t R := lim t 0 Rsds exists as a strictly positive real nmber and that R is integrable, ie, 0 Rs ds < We can now specialize roposition to this case Corollary Let be a Gassian integrated process with short-range dependence Then R 1 + ct t ct > H c π Ψ R c 3/ t 15 s 0 0 Rvdvds roof By the existence of R, continity of t t 0 Rsds, and bonded convergence, we have σ t/c lim = 1 st t t c lim Rvdvds = R t 0 0 c <, so that S1 holds with H = 1/ and we are in the second case roposition with G = R/c Notice that Corollary is a modest generalization of the reslts of Dȩbici [11] To see this, note that 15 is asymptotically eqivalent with R H c R c exp ct t s 0 0 Rvdvds, 14

8 since t = H/β H = 1 and σ R roposition 61 of [11] shows that this expression is in agreement with the findings of [11] Long-range dependent case Consider a Gassian integrated process as in 14, bt now with a covariance fnction R that is reglarly varying at inity with index H for some H 1/, 1 in addition to the reglarity assmptions above Since there is so mch long term correlation that 0 Rt dt =, the process is long-range dependent The motivation for stdying this long-range dependent case stems from the fact that it arises as a limit in heavy traffic of on-off flid models [15] By the direct half of Karamata s theorem Theorem 1511 of [4], we have for t, t s σ t = 0 0 Rvdvds t t 0 Rvdv H t Rt HH 1 Therefore, since H > 1/, σ t/t and we are in the first case of roposition Corollary 3 Let be a Gassian integrated process with long-range dependence Then t ct > is asymptotically eqivalent to H BH C H,1,1,H c 1 H 1 H 1 [HH 1] H 1 H R Ψ τ R where τ denotes an asymptotic inverse of t t Rt at inity t ct s 0 Rvdvds The case of a Gassian integrated process with long-range dependence is also stdied by Hüsler and iterbarg [4] The reasoning following Eqation 7 of [4] shows that the formlas are the same p to the constants; we leave it to the reader to chec that these coincide 4 A variant of icands lemma In this section, we present a generalization of a classical lemma by J icands III As we need a field version of this lemma, we let time be indexed by R n for some n 1, and we write t = t 1,, t n Given an even fnctional ξ η : R n R ie, ξ η t = ξ η t for t R n, we define the centered Gassian field η by its covariance, Covη s, η t = ξ η s + ξ η t ξ η s t, 16 provided it is a proper covariance in the sense that the field η exists A central role in the lemma is played by fnctions g, ξ η, and θ These fnctions are in principle arbitrary, bt they are assmed to satisfy certain conditions, which we now formlate To get some feeling for these conditions, the reader may want to loo in the proof of Lemma 3, for instance, to see how the fnctions are chosen in a particlar sitation Throghot, {K } denotes a nondecreasing family of contable sets say K Z, and {X, t : t [0, T ] n }, N, K denotes a collection of centered continos separable Gassian fields on [0, T ] n for some fixed T > 0 We pose that X, has nit variance It is important to notice that we do not assme stationarity of the X, 1 K g as, 15 for some even fnctional ξ η, K θ, s, t ξ η s t 0 for any s, t [0, T ] n, 3 for some γ 1,, γ n > 0, θ, s, t lim n K s,t [0,T ] n i=1 s <, γi i t i 4 t g Cov X, t, X, 0 is niformly continos in the sense that lim lim g Cov X s, X, ε 0 t, X, 0 = 0 K s t <ε s,t [0,T ] n We se the following lemma in Section 6 for n = 1 to establish the pper bond, and in Section 7 for n = to establish the lower bond The main assmption of the lemma is that Cov X s,, X, t tends niformly to nity at rate θ, s, t/g as Lemma Sppose there exist fnctions g, ξ η, and θ satisfying 1 4 If lim Var X, K s,t [0,T ] n g s X, t 1 θ, s, t = 0, 17 s t then for any K, as, where Moreover, we have X, t > g t [0,T ] n H η [0, T ] n = E exp lim K H η [0, T ] n Ψg, 18 ηt ξ η t t [0,T ] n t [0,T ] n X, t Ψg > g < 19 roof The proof is based on a standard approach in the theory of Gassian processes; see for instance the proof of Lemma D1 of iterbarg [34] First note that = X, t > g t [0,T ] n 1 πg exp 1 g expw exp 1 w R g X, t > g X, 0 = g w dw 0 t [0,T ] n g For fixed w, we set χ, t := g [X, t g ] + w, so that the conditional probability that appears in the integrand eqals t [0,T ] n χ, t > w χ, 0 = 0 16

9 We first stdy the field χ, χ, 0 = 0 as, starting with the finite-dimensional cylinder distribtions These converge niformly in K to the corresponding distribtions of η ξ η To see this, we set v, s, t := VarX s, X, t, so that by 1,, and 17, niformly in K, E[χ, t χ, 0 = 0] = 1 g v,0, t + 1 wv,0, t and similarly, also niformly in K, = 1 θ, 0, t1 + o1 + o1 ξ η t, Varχ, s χ, t χ, 0 = 0 = g v,s, t 1 4 g [v,0, t v, 0, s] = θ, s, t1 + o1 + o1 ξ η s t Denoting the law of a field X by LX, we next show that the family {Lχ, χ, 0 = 0 : N, K } is niformly tight Since t Eχ, t χ, 0 = 0 is niformly continos in the sense that 4 holds, it sffices to show that the family of centered distribtions is tight We denote the centered χ, by χ,, ie, χ, t := χ, t E[χ, t χ, 0 = 0] It is important to notice that L χ, χ, 0 = 0 does not depend on w To see that {L χ, χ, 0 = 0 : N, K } is tight, observe that for large enogh, niformly in s, t [0, T ] n and K, we have Var χ, s χ, t χ, 0 = 0 g v,s, t θ, s, t By 3, there exist constants γ 1,, γ n, C > 0 sch that, niformly for s, t [0, T ] n and K, n Var χ, s χ, t χ, 0 = 0 C s i t i γi, provided is large enogh As a corollary of Theorem 147 in Knita [6], we have the claimed tightness Since the fnctional x C[0, T ] n t [0,T ] n xt is continos in the topology of niform convergence, the Continos Mapping Theorem yields for w R, lim χ, t > w t [0,T ] n χ,0 = 0 = i=1 [ η t ξ η t] > w t [0,T ] n Using R ew t [0,T ] n[ η t ξ η t] > wdw = H η [0, T ] n and 7, this proves 18 once it has been shown that the integral and limit can be interchanged The dominated convergence theorem and Borell s ineqality are sed to see that this can indeed be done For arbitrary δ > 0 and large enogh, E[χ, t χ, 0 = 0] δ w, K t [0,T ] n Var[χ, t χ, 0 = 0] θ, t, 0, K t [0,T ] n K t [0,T ] n and the latter qantity remains bonded as as a conseqence of 3; let ξ η denote an pper bond Observe that for a R, again by the Continos Mapping Theorem, we have lim χ, t > a K t [0,T ] n χ,0 = 0 = ηt > a t [0,T ] n 17 Since η is continos as remared below, one can select an a independent of w,, sch that the conditions for applying Borell s ineqality eg, Theorem D1 of [34] are flfilled Hence, for every,, w, w δ w a χ, t > w t [0,T ] n χ,0 = 0 Ψ 3ξ η When mltiplied by expw exp 1 w /g, this pper bond is integrable with respect to w for large This not only shows that the dominated convergence theorem can be applied, it also implies 19 Indeed, sing 1, we have by standard bonds on Ψ lim e 1 g g K Ψg = π One observation in the proof deserves to be emphasized, namely the existence and continity of η If θ satisfies 17 and converges niformly in to some ξ η as in, the analysis of the finite-dimensional distribtions shows that there atomatically exists a field η with covariance 16 Moreover, η has continos sample paths as a conseqence of 3 and 4 ie, the tightness A nmber of special cases of Lemma appear elsewhere in the literatre erhaps the best nown example is the case where X is a stationary process with covariance fnction satisfying rt = 1 t α +o t α for some α 0, ] as t 0, see Lemma D1 of iterbarg [34] This lemma is obtained by letting K consist of only a single element for every, and by setting g =, X t = X /α t, η = B α/ and ξ η t = t α A generalization of Lemma D1 in [34] to a stationary field {Xt : t R n } is given in Lemma 61 of iterbarg [34], and we now compare this generalization to Lemma We se the notation of [34] Lemma deals with the case A = 0 and T in the notation of [34] eqal to [0, T ] n in or notation Again, let K consist of only a single element for every, and set g =, X t = X g 1 t, and ξ ηt = t E,α As the ideas of the proof are the same, Lemma can readily be extended to also generalize Lemma 61 of [34] However, we do not need this to derive the reslts of the present paper Theorem 1 of Dȩbici [11] can also be considered to be a special case of Lemma There, again, K consists of a single element, and X t1,,tn = 1 n n i=1 X i t i for independent processes X i satisfying a condition of the type 17, bt where θ does not depend on Lemma has some interesting conseqences for the properties of icands constant For instance, icands constant is readily seen to be sbadditive, ie, for T 1, T > 0 and n = 1, H η [0, T 1 + T ] H η [0, T 1 ] + H η [0, T ], with appropriate generalizations to the mltidimensional case This property garantees that the limit in 6 exists The vale of icands constant is only nown in two cases: H B1/ = 1 Brownian motion and H B1 = 1/ π degenerate case Frther properties of icands constants are explored both theoretically and nmerically by Shao [38], Dȩbici [8], and Dȩbici et al [14] 5 For cases We now specialize Lemma according to the for types of correlation strctres introdced in Section Throghot this section, we pose that S1 and M1 hold Let T > 0 be fixed, and write I T for the intervals [t +T /, t ++1T /], where is some fnction that depends on the correlation strctre, and = oδ 18

10 51 Case A We say that case A applies if A1, A, T1, T, N1, and N hold and is given by := 1 τν σµt τ νt D 1 + t, 1 where τ denotes an asymptotic inverse of τ at inity this exists when T1 holds, see Theorem 151 of [4] Note that the argment of τ tends to inity as a conseqence of A, and that therefore ν / It is easy to chec that is reglarly varying with index H/β 1/ι τ + 1 < 1 The next lemma shows that this particlar choice of balances the correlation strctre on the intervals I T note that the interval IT depends on Lemma 3 Let S1 and M1 hold and pose that case A applies Let δ be sch that δ = o and = oδ For any and δ T δ T, pic some t IT Then we have for, µt t I T σµt > 1 + t 1 + t σµt H Bιτ T Ψ σµt, where H Bιτ T is defined as in 6 Moreover, µt t I T lim σµt > 1+t σµt δ δ 1+t < Ψ T T σµt roof The main argment in the proof is, of corse, Lemma Set D 1 + t κ := τ νt σµt τν and note that by the UCT and 1, κ 1 0 δ δ s,t I T T T Eqation 3 implies that {A1} κ τ µs ν Var σµs µt σµt s,t [t ±δ/] Dτ νt τ νs νt /τ νt s t T / The preceding display sggests certain choices for the fnctions g and θ of Lemma, cf 17; we now show that 1 4 are indeed satisfied As for 1, one readily checs that κ τν g := D τ νt = 1 + t σµt tends to inity niformly in We set for s, t [0, T ] and δ T δ T 3 θ, s, t := τ νt + T + s νt + T + t τ νt 5 19 To chec that θ, s, t converges niformly in as, we note that by the Mean Vale Theorem {N} there exists some t, s, t [0, T ] sch that νt + T + s νt + T + t = νt + [T + t, s, t]s t Now note that we have for s, t [0, T ], + θ, s, t s t ιτ νt θ, s, t + [T + t, s, t] ιτ νt s t ιτ νt + [T + t, s, t] ιτ νt 1 s t ιτ =: I + II As a conseqence of the UCT {N1, N}, we have lim s,t [0,T ] δ δ T T νt + [T + t, s, t] νt s t = T 6 Since ν tends to inity {A}, this shows that I is majorized by τ νt t t [0,T ] τ νt tιτ 0 II also tends to zero by the UCT Hence, holds with ξ η t = t ιτ, so that η is a fractional Brownian motion with Hrst parameter ι τ A similar reasoning is sed to chec 3 Notice that τ tt γ is bonded on intervals of the form 0, ] {T}, and that we may pose that γ < ι τ withot loss of generality Again sing 6 and the UCT, we observe that for large, = s,t 0,T ] s>t s,t 0,T ] s>t θ, s, ts t γ τ νt + [T + t, s, t]s t τ νt s t γ τ νt t t»0, 3 τ νt t γ 1/ιτ γ T 4T ιτ γ, which is clearly finite the factor 4 trns p again in the proof of Lemma 9 below It remains to chec 4 For this, observe that it sffices to show that g µs Cov µt lim ε 0 lim s,s,t [t ±δ/] s t <εt / s t <εt / σµs µs σµs, σµt = 0, and hence that lim lim g µs ε 0 s,t [t ±δ/] Var s t <ε / σµs µt = 0 7 σµt 0

11 Observe that for large, by 4 and the Mean Vale Theorem, niformly for s, t [t ± δ/], g µs Var s t <ε / σµs µt τ νs νt 4 σµt s t <ε / τ νt τ νt t 4 t<ε / τ νt 8ε ιτ 0, as ε 0 Having checed that Lemma can be applied, we se the definition of to see that µt t I T σµt > 1 + t µt σµt = t I T σµt > κ τν D τ νt κ τν H Bιτ T Ψ D τ νt 1 + t = H Bιτ T Ψ σµt, as claimed 5 Case B Case B is different from the other cases in the sense that no asymptotic inverse is involved in the definition of As a conseqence, a non-classical icands constant appears in the asymptotics We say that case B applies when B1, B, T1, T, N1, and N hold and is given by Moreover, we set := 1 νt 8 F := D1 + t G t H/β Under these assmptions, lim ν / exists in 0, Lemma 4 Let S1 and M1 hold and pose that case B applies Let δ be sch that δ = o and = oδ For any and δ T δ T, pic some t IT For T large enogh, we have for, µt t I T σµt > 1 + t 1 + t σµt H Fτ T Ψ σµt, where H Fτ T is defined as in 6 Moreover, holds roof Define D 1 + t κ := F τνσµt which converges niformly in to nity as a conseqence of the fact that by B, Fτ ν D = 1 + t G t H/β τ ν 1 + t σ µt 1 Therefore, as in Lemma 3, we have by 3, Fκ τ µs ν Var σµs µt σµt δ δ s,t I T T T D Fτ νs νt Z 1 0 Again, this shold be compared to 17 Set g := F/Dκ τν, and θ, s, t := Fτ νt + T + s νt + T + t 9 Obviosly, we have 1 We now chec that holds with ξ η t = Fτ t Let s, t [0, T ], and observe that by the Mean Vale Theorem there exist t, s, t [0, T ] sch that for every ɛ > 0, θ, s, t Fτ s t = Fτ νt + [T + t, s, t] s t Fτ s t F τ st τ t s [1 ɛ,1+ɛ] t [0,T ] F τ s τ t, s,t [0,T ] s t ɛt where we sed the definition of and the UCT By continity of τ {T1}, this pper bond which is a modls of continity tends to zero as ɛ 0 As for 3, the same argments show that for large T by the UCT {T1, T} θ, s, t s,t [0,T ]»0, 3 1/ιτ γ T s t γ F t τ t t γ 4FT ιτ γ It remains to verify 4 As in the proof of Lemma 3, it sffices to show that 7 holds By again applying the UCT, one can chec that for s, t [t ± δ/], g µs Var s t <ε / σµs µt F τ t, σµt t [0,ε] showing 4 since τ is continos at zero In conclsion, Lemma can be applied and ths µt t I T σµt > 1 + t µt F σµt = t I T σµt > D κ τν F H Fτ T Ψ D κ τν 1 + t = H Fτ T Ψ σµt, as claimed

12 53 Case C We say that case C applies when C1 C3, T1, N1, and N hold and is given by := 1 τν σµt τ νt D 1 + t, 30 where τ denotes an asymptotic inverse of τ at zero which exists de to T1, see Theorem 151 of [4] Here, the argment of τ tends to zero as a conseqence of C, and therefore ν / 0 Note that we do not impose T, since i is atomatically satisfied once C3 holds The following lemma is the analog of Lemma 3 and Lemma 4 for case C Lemma 5 Let S1 and M1 hold and pose that case C applies Let δ be sch that δ = o and = oδ For any and δ T δ T, pic some t IT Then we have for, µt t I T σµt > 1 + t 1 + t σµt H B ιτ T Ψ σµt, where H B ιτ T is defined as in 6 Moreover, holds roof The proof is exactly the same as the proof of Lemma 3, except that now ι τ is replaced by ι τ 54 Case D We say that case D applies when D1, D, N1, N hold and is given by := σµt τ ι ν t ιν t 31 The local behavior is described by the following lemma Lemma 6 Let S1 and M1 hold and pose that case D applies Let δ be sch that δ = o and = oδ For any and δ T δ T, pic some t IT Then we have for, µt t I T σµt > 1 + t 1 + t σµt H B ιτ T Ψ σµt, where H B ιτ T is defined as in 6 Moreover, holds roof The argments are similar to those in the proof of Lemma 3 Therefore, we only show how the fnctions in Lemma shold be chosen in order to match 4 with 17 Define for δ T δ T κ := τι νt ιν 1 /1 + t κ, g σµt := τι ν t ιν 1 /, and θ, s, t := τ ν t + T + s νt + T + t /ν τ ι ν t ιν 1 / 3 It follows from Lemma with η = B ιτ that µt t I T σµt > 1 + t µt σµt = t I T σµt > κ τι ν t ιν 1 / κ H B ιτ T Ψ τι ν t ιν 1 / 1 + t = H B ιτ T Ψ σµt, as claimed 6 Upper bonds In this section, we prove the pper bond part of Theorem 1 in each of the for cases Since the proof is almost exactly the same for each of the regimes, we only give it once by sing the following notation in both the present and the next section We denote the icands constants H Bιτ T, H DMτ T, and H B ιτ T by HT The abbreviation H := lim T HT /T is sed for the corresponding limits The definition of also depends on the regime; it is defined in 1, 8, 30, and 31 for the cases A, B, C, and D, respectively Notice that the dependence on is pressed in the notation I T = [t + T /, t + + 1T /] It is convenient to define t T and tt as the left and right end of I T respectively In the proofs of the pper and lower bonds, we write C := 1 d 1 + t dt t H/β = t H/β 1 3 t=t We start with an axiliary lemma, which shows that it sffices to focs on local behavior near t This observation is important since the lemmas of the previos section only yield local niformity note that I T [t ± δ/] and δ = o Lemma 7 Sppose that S1 S4, and M1 M4 hold for some β > H Let δ be sch that δ = o and σµ = oδ Then we have µt t [t ±δ/] 1 + t > σµ = o Ψ 1 + t σµt 33 roof The proof consists of three parts: we show that the intervals [0, ω], [ω, T ]\[t ±δ/] and [T, ] play no role in the asymptotics, where ω, T > 0 are chosen appropriately We start with the interval [T, If T is chosen as in S4, this interval is asymptotically negligible by assmption As for the remaining intervals, by S4 we can find some ɛ, C 0,, γ 0, ] sch that for each s, t [0, 1 + ɛt 1/β ] Var s t Cσ s t γ, 34 where is large Starting with [0, ω], we select ω so that for large, σµt 1 σµt t [0,ω] 1 + t 1 + t 35 4

13 The main argment is Borell s ineqality, bt we first have to mae sre that it can be applied For a > 0, there exists constants c γ, C independent of and a sch that for large, {M} µt t [0,ω] σµ1 + t > a µt 1/β» µω β σµ > a t 0, µ µt 1/β t [0,ω] σµ > a 4 exp c γa, C where the last ineqality follows from 34 and Ferniqe s lemma [8, p 19] as γ 0, ] By choosing a sfficiently large, we have by Borell s ineqality eg, Theorem D1 of [34] µt t [0,ω] 1 + t > Ψ 1 aσµ/ σµt t [0,ω] 1+t Since 35 holds, there exist constants K 1, K < sch that µt t [0,ω] 1 + t > K 1 exp 1 + t σ µt + K 1 + t σµt This shows that the interval [0, ω] is asymptotically negligible in the sense of 33 We next consider the contribtion of the set [ω, T ]\[t ± δ/] to the asymptotics Define σµt σµt σ = = max δ t [ω,t ]\[t ±δ/] 1 + t 1 + t δ/, σµt + δ 1 + t, + δ/ where the last eqality holds for large Now observe that by the UCT {M1}, for large, µt t [ω,t ]\[t 1 + t > µt ±δ/] t [ω,t ]\[t σµt > σ ±δ/] µt t [ω 1/β /,T 1/β ] σµt > σ In order to bond this qantity frther, we se 34 and the ineqality ab a + b : for s, t [ ω 1/β /, T 1/β], {M} µs Var σµs µt σµt Var µs µt σµsσµt Var µs µt v [ω 1/β /,T 1/β ] σ µv 1+H ω H/β σ Var µ µs µt K s t γ, 5 where K < is some constant depending on ω and T Hence, by Theorem D4 of iterbarg [34] there exists a constant K depending only on K and γ sch that µt /γ t [ω,t ]\[t 1 + t > T K Ψ σ σ ±δ/] Consider the expression 1 + t + δ/ σ µt + δ 1 + t / [ ] δ σ µt C, 36 σµ where C is given by 3 By Taylor s Mean Vale Theorem {S3, M4}, there exists some t # = t # [t, t + δ/] sch that this expression eqals 1 δ d 1 + t / [ ] δ dt σ µt C t=t# σµ Recall that σ µ is reglarly varying with index H/β > 0, and that nder the present conditions both its first and second derivative are reglarly varying with respective indices H/β 1 and H/β The UCT now yields lim σ µ d 1 + t dt σ µt = C t=t# Since σµ = oδ, the expression in 36 converges to one as Hence, we have Ψ σ = exp 1 Ψ 1+t C δ σ 1 + o1 1 + o1, µ σµt showing that the interval [ω, T ]\[t ± δ/] plays no role in the asymptotics We can now prove the pper bonds In the proof, it is essential that σµ/ in all for cases To see that this holds, note that this fnction is reglarly varying with index 1 H/β1/ι τ 1 > 0 in case A and B se ι ν = 1 H/β/ι τ in the latter case In case C, the index of variation is H β + ι ν ι τ ι ν H/β ι τ > 1 ι τ ι ν H 1 1 > 0 β ι τ Finally, it is reglarly varying with index 1 H/β1/ ι τ 1 > 0 in case D The pper bonds are formlated in the following proposition roposition 4 Let µ and σ satisfy assmptions M1 M4 and S1 S4 for some β > H Moreover, let case A, B, C, or D apply We then have µt t > π lim H σµ Ψ 1+t C σµt roof Select some δ sch that δ = o, σµ = oδ, = oδ, and = oδν While the specific choice is irrelevant, it is left to the reader that sch δ exists in each of the for cases In view of Lemma 7, we need to show that µt t [t ±δ/] 1+t > π lim H σµ Ψ 1+t C 37 σµt 6