Tail Asymptotics of Supremum of Certain Gaussian Processes over Threshold Dependent Random Intervals
aa r X i v : . [ m a t h . P R ] N ov Tail Asymptotics of Supremum of Certain Gaussian Processes overThreshold Dependent Random Intervals
Krzysztof D¸ebicki ∗ , Enkelejd Hashorva † , Lanpeng Ji † University of Wroc law and University of LausanneApril 15, 2019
Abstract:
Let { X ( t ) , t ≥ } be a centered Gaussian process and let γ be a non-negative constant. In thispaper we study the asymptotics of P ( sup t ∈ [0 , T /u γ ] X ( t ) > u ) as u → ∞ , with T an independent of X non-negativerandom variable. As an application, we derive the asymptotics of finite-time ruin probability of time-changedfractional Brownian motion risk processes. Key Words:
Tail asymptotics; large deviations; Weibullian tails; supremum over random intervals; Gaussianprocess; fractional Brownian motion; fractional Laplace motion; Gamma process; ruin probability.
AMS Classification:
Primary 60G15; Secondary 60G70, 60K30, 91B30.
One of the seminal results in the extreme value theory of Gaussian processes concerns the asymptotic behaviorof P ( sup t ∈ [0 ,T ] X ( t ) > u ) , u → ∞ , (1)where { X ( t ) , t ≥ } is a centered Gaussian process with almost surely (a.s.) continuous sample paths, variancefunction that attains its unique maximum at exactly one point t ∈ [0 , T ] and T > T being an independentof X non-negative random variable, possibly depending on u , namely p ( X, T , γ, u ) := P ( sup t ∈ [0 , T /u γ ] X ( t ) > u ) , with γ ≥ u → ∞ .Let in the following B α denote a fractional Brownian motion (fBm) with Hurst index α/ ∈ (0 , { Y ( t ) , t ≥ } independent of B α be a non-negative, non-decreasing stochastic process. ∗ Mathematical Institute, University of Wroc law, pl. Grunwaldzki 2/4, 50-384 Wroc law, Poland † University of Lausanne, Faculty of Business and Economics (HEC Lausanne), 1015 Lausanne, Switzerland ψ T defined by ψ T ( u ) = P (cid:26) inf t ∈ [0 ,T ] ( u + cY ( t ) − B α ( Y ( t ))) < (cid:27) (3)for some c > , T > , u ≥ Y is known. In view of ourfindings, for all large u compact formulas for approximation of ψ T ( u ) can be given explicitly.In the context of theoretical actuarial models, u is the so-called initial reserve , c models the premium incomerate, and B α ( t ) represents the total amount up to time t > ψ T ( u ), with Y ( t ) = t is a good approximation of the finite-time ruinprobability for the classical risk process with claims possessing long range dependence property. The role ofthe random process Y is crucial in order to make such models adequate for applications. It is a substitute forthe real time, where Y ( t ) stands for the random business time, which is consistent with the insurance practicewhere both claims and premiums may not be received immediately at time t of the event, but at a later randomtime modeled by Y ( t ). Indeed, if { Y ( t ) , t ≥ } has additionally a.s. continuous sample paths, then re-writing(3) as ψ T ( u ) = P ( sup t ∈ [0 , T /u ] B α ( t )1 + ct > u − α/ ) , T = Y ( T ) (4)we see that the asymptotic analysis of ψ T ( u ) reduces to the analysis of the asymptotic tail behaviour of supremumof a specific Gaussian process over a random interval.Other branch of motivations to analyze (2) stems from recently studied problems in fluid queueing theory. Inparticular, the tail asymptotics of the stationary buffer content of a hybrid fluid queue , with input modelledby a superposition of integrated alternating on-off process and a Gaussian process with stationary increments,reduces (under some assumptions) to (2) with suitably chosen random T ; see e.g., Zwart et al. (2005).In the case that { Y ( t ) , t ≥ } is a Gamma process and c = 0, investigation of ψ T ( u ), as u → ∞ reduces to theanalysis of the tail asymptotics of the fractional Laplace motion; see Kozubowski et al. (2006), Arendarczykand D¸ebicki (2011). For the related literature on time-changed models we refer to Fotopoulos and Luo (2011),who considered the case of Brownian motion ( α = 1) and Wu and Wang (2012), where a model based on theCox risk process, which is a time-changed compound Poisson risk process, is considered. For more applicationsin finance and insurance, see e.g., Geman et al. (2001) and Palmowski and Zwart (2007), among many others. Contribution : a) In Theorem 3.1, under some canonical asymptotic assumptions of the Gaussian process { X ( t ) , t ≥ } , see Section 2, if T has a sufficiently heavy tail distribution (which is manifested by insensi-tivity property of the tail distribution of T ), we derive exact asymptotics of p ( X, T , γ, u ).b) Theorem 3.3 deals with a more general class of random variables T . Under a log-power tail asymptoticassumption (see (6)), we obtain the logarithmic asymptotics of p ( X, T , γ, u ). It appears that, depending on theinterplay between heaviness of T and local properties of the variance function of X at 0 we can distinguish fourscenarios, leading to four qualitatively different asymptotics.Our novel result complement recent findings of Arendarczyk and D¸ebicki (2011, 2012) and Tan and Hashorva(2013), where extremes over a random-time interval were analyzed for stationary centered Gaussian processesand centered Gaussian processes with stationary increments respectively. We refer to Zwart et al. (2005),D¸ebicki and van Uitert (2006), Palmowski and Zwart (2007) for other related results.The organization of the paper is as follows: In Section 2 we introduce some notation and formulate the mainassumptions imposed to the Gaussian process { X ( t ) , t ≥ } . Section 3 presents the main results. In Section 4we discuss the finite-time ruin probability of the time-changed fBm risk processes. The proofs of all the resultsare relegated to Section 5. Throughout this paper Ψ( · ) denotes the survival function of a standard Gaussian random variable N (0 , a ( · ) , a ( · ) on [0 , ∞ ) we write a ( u ) ∼ a ( u ) if lim u →∞ a ( u ) /a ( u ) = 1 and a ( u ) log ∼ a ( u )if lim u →∞ log( a ( u )) / log( a ( u )) = 1. In the sequel Γ( · ) denotes the Euler Gamma function.Following, e.g., Foss et al. (2013) we say that a non-negative random variable T is h − insensitive if P {T > u ± h ( u ) } ∼ P {T > u } as u → ∞ for some function h ( · ). Our first main result in this paper is the exact asymptotic behaviour of p ( X, T , γ, u ) defined in (2) for T being h − insensitive. Two large classes of distributions for T that satisfy theinsensitivity criteria are:i) T is regularly varying at ∞ with index λ >
0, which means that P {T > u } = L ( u ) u − λ , where L ( · ) is slowlyvarying at ∞ , i.e., for any x > u →∞ L ( xu ) / L ( u ) = 1 (see e.g., Resnick (1987));ii) T is asymptotically Weibullian , i.e., P {T > u } ∼ L ( u ) u δ exp( − Lu p ) (5)as u → ∞ , where L, p > δ ∈ IR and L ( · ) is slowly varying at ∞ ; we abbreviate (5) as T ∈ W ( p, L, L , δ ).A significant relaxation of the Weibullian-tail assumption (5) for T is that it has asymptotically a log-power tailwith coefficient L > p >
0, i.e.,lim u →∞ log P {T > u } u p = − L. (6)As above, hereafter B α stands for a fBm with Hurst index α/ ∈ (0 , σ B α ( t ) = t α . Following Piterbarg(1996), we introduce two key constants, namely Pickands constant H α := lim S →∞ S − E exp (cid:18) sup t ∈ [0 ,S ] (cid:16) √ B α ( t ) − t α (cid:17)(cid:19)! ∈ (0 , ∞ )and Piterbarg constant P Rα := lim S →∞ E exp (cid:18) sup t ∈ [ − S,S ] (cid:16) √ B α ( t ) − (1 + R ) t α (cid:17)(cid:19)! ∈ (0 , ∞ ) , R ∈ (0 , ∞ ) . In this paper we tacitly assume that { X ( t ) , t ≥ } is a centered Gaussian process with a.s. continuous andbounded sample paths and σ ( t ) := V ar ( X ( t )) that attains its maximum over [0 , ∞ ) at the unique point t = t > σ ( t ) = 1. Additionally we suppose that: A1 . There exist some positive constants a, β such that σ ( t ) = 1 − a | t − t | β + o ( | t − t | β ) , t → t . (7) A2 . There exist d > , α ∈ (0 ,
2] such that C ov (cid:18) X ( s ) σ ( s ) , X ( t ) σ ( t ) (cid:19) = 1 − d | t − s | α + o ( | t − s | α ) , s → t , t → t . A3 . There exist constants Q > H > t and r ∈ (0 ,
2] such that, for all s, t ∈ [0 , H ] with | s − t | < E (cid:0) ( X ( t ) − X ( s )) (cid:1) ≤ Q | t − s | r . (8) A4 . lim sup t →∞ σ ( t ) < Theorem 2.1
If the assumptions
A1-A4 are satisfied, then for any
T > t we have P ( sup t ∈ [0 , ∞ ) X ( t ) > u ) ∼ P ( sup t ∈ [0 ,T ] X ( t ) > u ) ∼ C α,β Λ α,β ( u )Ψ( u ) , u → ∞ , (9) where C α,β = H α Γ(1 /β + 1) d /α a − /β , if α < β, P a/dα , if α = β, , if α > β, and Λ α,β ( u ) = ( u /α − /β , if α < β, , if α ≥ β. Remark : The exact asymptotics for the infinite-time interval case can be obtained by a direct applicationof the Borell-TIS inequality (e.g., Adler (1990) or Adler and Taylor (2007)) using further the result for theasymptotics of the supremum of X over any finite-time interval [0 .T ]. In this section we present our main results. We begin with the derivation of the exact asymptotic behaviour of p ( X, T , γ, u ) as u → ∞ , which is presented in Theorem 3.1. Then, under milder assumptions on T , we providea complete study of the logarithmic asymptotics of p ( X, T , γ, u ) as u → ∞ , see Theorem 3.3. Theorem 3.1
Let { X ( t ) , t ≥ } be a centered Gaussian process such that assumptions A1-A4 are satisfied,and let T be a non-negative random variable independent of the Gaussian process X .i) If γ = 0 and P {T ≥ t } > , then p ( X, T , γ, u ) ∼ C α,β Λ α,β ( u )Ψ( u ) P {T ≥ t } . (10) ii) If γ > and T is u − / ( γ (1+ β )) -insensitive, then p ( X, T , γ, u ) ∼ C α,β Λ α,β ( u )Ψ( u ) P {T ≥ t u γ } . (11)The proof of Theorem 3.1 is given in Section 5.1.Theorem 3.1 complements recent results of Arendarczyk and D¸ebicki (2011, 2012) and Tan and Hashorva (2013),where the class of centered stationary Gaussian processes and Gaussian processes with stationary incrementswas analyzed.As a straightforward consequence of Theorem 3.1 we obtain the following results. Corollary 3.2
Suppose that the assumptions of Theorem 3.1 are satisfied and that γ > .i) If T is regularly varying at ∞ with index λ > , then p ( X, T , γ, u ) ∼ C α,β Λ α,β ( u )Ψ( u ) P {T ≥ t u γ } . (12) ii) If T ∈ W ( p, L, L , δ ) with p ∈ (cid:16) , γ (1+ β ) (cid:17) , then (12) holds. Complementary to the above exact asymptotics, in the next theorem we derive the logarithmic asymptoticbehaviour of (2) for a class of log-power tailed random variables T .Let us introduce the following notation b σ ( s ) := sup t ∈ [0 ,s ] σ ( t ) , e σ L,γ ( s ) := 12 b σ ( s ) + Ls /γ , s ≥ A = inf (cid:26) A : A = arg inf t ≤ t ( e σ L,γ ( t )) (cid:27) . (14) Theorem 3.3
Let { X ( t ) , t ≥ } be a centered Gaussian process such that assumptions A1-A4 are satisfied andlet T be a non-negative random variable independent of the Gaussian process X with asymptotically log-powertail with coefficient L > and power p > .i) If γp < , then lim u →∞ log p ( X, T , γ, u ) u = − . (15) ii) If γp = 2 , A > , and on any compact subset of (0 , ∞ ) b σ ( s ) is differentiable and | b σ ′ ( s ) | is uniformlybounded, then lim u →∞ log p ( X, T , γ, u ) u = − e σ L,γ ( A ) . (16) iii) If γp > and σ (0) > , then lim u →∞ log p ( X, T , γ, u ) u = − σ (0) . (17) iv) If γp > and σ ( t ) = Dt η (1 + o (1)) as t ↓ for some positive constants D and η , then lim u →∞ log p ( X, T , γ, u ) u p ( ηγ +1) / (2 η + p ) = − A , (18) where A = D − p η + p ( Lp/η ) η η + p + L η η + p ( η/ ( pD )) p η + p . A complete proof of Theorem 3.3 is given in Section 5.2.
Remark 3.4
It follows straightforwardly from the proof of Theorem 3.3 that statements i ) and iii ) also hold if − log P {T > u } = L ( u ) u p with some slowly varying function L ( · ) at infinity.When pγ = 2 we imposed the assumption that A > . The special case A = 0 , which is also possible, is muchmore involved, and will therefore be considered elsewhere. Consider an extension of the time-changed fBm risk process defined in the Introduction, by allowing a powertrend-function; i.e., let Z ( t ) := u + c ( Y ( t )) θ − B α ( Y ( t )) , t ≥ , (19)where u ≥ , c > , θ > α/ { Y ( t ) , t ≥ } is a non-negative non-decreasing stochastic process beingindependent of { B α ( t ) , t ≥ } . Clearly, θ = 1 is a choice leading to our risk model in the Introduction. Relatedrisk models were studied for instance in D¸ebicki and Rolski (2002), where the finite-time ruin probability withthe choice Y ( t ) ≡ t was analyzed, whereas the infinite-time ruin counterpart was considered in H¨usler andPiterbarg (1999), see also the recent contribution Hashorva et al. (2013).As in the Introduction the finite-time ruin probability for the risk process (19) is given by ψ T ( u ) = P (cid:26) inf t ∈ [0 ,T ] (cid:16) u + c ( Y ( t )) θ − B α ( Y ( t )) (cid:17) < (cid:27) = P ( sup t ∈ [0 ,T ] (cid:16) B α ( Y ( t )) − c ( Y ( t )) θ (cid:17) > u ) , with T > u ≥ Y In this subsection, we apply the results of Theorems 3.1 and 3.3 to the analysis of the asymptotics of thefinite-time ruin probability of the time-changed fBm risk process (19) as u → ∞ , where the time process Y hasa.s. continuous sample paths.Before stating our results for this risk model we need to introduce the following notation Q := 2 + α √ πc − α θ α α − − θ θ θ − αα (2 θ − α ) θα − θ +2 α − α θα ,s := (cid:18) αc (2 θ − α ) (cid:19) /θ , V := 2 θ − α θ s α/ = 2 θ − α θ (cid:18) αc (2 θ − α ) (cid:19) α θ . The main results are presented in Propositions 4.1 and 4.2; their proofs are relegated to Section 5.3.
Proposition 4.1
Assume that θ > α/ , c > and { Y ( t ) , t ≥ } has a.s. continuous sample paths.i) If Y ( T ) is regularly varying at ∞ with index λ > , i.e., P { Y ( T ) > u } = L ( u ) u − λ , then ψ T ( u ) ∼ Q H α s − λ u (2 θ − α )(2 − α ) − λα θα L ( s u θ )Ψ (cid:16) V − u θ − α θ (cid:17) . (20) ii) If Y ( T ) ∈ W ( p, L, L , δ ) with p ∈ (cid:0) , θ − α (cid:1) , L > , and δ ∈ IR , then ψ T ( u ) ∼ Q H α L ( s u /θ ) s δ u (2 θ − α )(2 − α )+2 δα θα exp (cid:16) − Ls p u pθ (cid:17) Ψ (cid:16) V − u θ − α θ (cid:17) . (21) Proposition 4.2
Under the setup of Proposition 4.1, suppose further that Y ( T ) has asymptotically log-powertail with coefficient L > and power p > .i) If θ − α > p , then lim u →∞ log ψ T ( u ) u θ − αθ = − V . (22) ii) If θ − α < p , then lim u →∞ log ψ T ( u ) u pα + p = − (cid:18) pα (cid:19) αα + p + (cid:18) α p (cid:19) pα + p ! L αα + p . (23) iii) If θ − α = p , then lim u →∞ log ψ T ( u ) u θ − αθ = − (cid:18) (1 + cA θ ) A α + LA θ − α (cid:19) , (24) where A = c ( α − θ ) + p c ( θ − α ) + 2 α ( c ( θ − α/
2) + L (2 θ − α )) c (2 θ − α ) + 2 L (2 θ − α ) ! /θ . Y In several models the time-process Y has discontinuous sample paths. Therefore, in this section we investigateadditional cases relaxing the assumption on continuity of sample paths of Y . Proposition 4.3
Assume that θ > α/ , c > and the random variable Y ( T ) possesses an absolutely continuousdistribution with probability density function which is regularly varying at ∞ with index λ + 1 > .Then lim u →∞ log ψ T ( u ) u θ − αθ = − V . (25) Proposition 4.4
Assume that θ > α/ , c > and Y ( T ) possesses absolutely continuous distribution withprobability density function ρ Y ( T ) ( · ) such that lim u →∞ log( ρ Y ( T ) ( u )) /u p = − L for some p, L > .i) If θ − α > p , then lim u →∞ log ψ T ( u ) u θ − αθ = − V . (26) ii) If θ − α < p , then lim u →∞ log ψ T ( u ) u pα + p = − (cid:18) pα (cid:19) αα + p + (cid:18) α p (cid:19) pα + p ! L αα + p . (27) iii) If θ − α = p , then lim u →∞ log ψ T ( u ) u θ − αθ = − (cid:18) (1 + cA θ ) A α + LA θ − α (cid:19) , (28) where A = c ( α − θ ) + p c ( θ − α ) + 2 α ( c ( θ − α/
2) + L (2 θ − α )) c (2 θ − α ) + 2 L (2 θ − α ) ! /θ . Proofs of Propositions 4.3 and 4.4 are given in Section 5.4.
Example 4.5
Assume that Y ( t ) = P N ( t ) i =1 Z i is a compound Poisson process with Z i , i ≥ non-negative inde-pendent random variables with common probability density function h ( · ) which is regularly varying at ∞ withindex λ + 1 > . If h ( · ) is monotone and { N ( t ) , t ≥ } is a homogeneous Poisson process with intensity µ > ,then by Proposition 4.3 lim u →∞ log ψ T ( u ) u θ − αθ = − V . Example 4.6
Let { Γ t , t ≥ } be a Gamma process with parameter ν > , i.e., a L´evy process such that theincrements Γ t + s − Γ t have Gamma distribution with probability density function f ( x ) = 1Γ( sν ) x sν − exp( − x ) , x > . By fractional Laplace motion { L α ( t ) , t ≥ } we denote a random process defined as { L α ( t ) , t ≥ } d = { σB α (Γ t ) , t ≥ } , σ > . A standard fractional Laplace motion corresponds to σ = ν = 1 ; see, e.g., Kozubowski et al. (2004), (2006).Choosing Y ( t ) = Γ t , we consider below finite-time ruin probability of risk process modelled by fractional Laplacemotion ψ T ( u ) = P (cid:26) inf t ∈ [0 ,T ] (cid:16) u + c Γ t − B α (Γ t ) (cid:17) < (cid:27) = P ( sup t ∈ [0 ,T ] (cid:16) B α (Γ t ) − c Γ t (cid:17) > u ) . For this model Proposition 4.4 implies:i) If α < , then lim u →∞ log ψ T ( u ) u − α = − − α ) (cid:18) c (2 − α ) α (cid:19) α . ii) If < α < , then lim u →∞ log ψ T ( u ) u α +1 = − (cid:18) α (cid:19) αα +1 − (cid:16) α (cid:17) α +1 . iii) If α = 1 , then lim u →∞ log ψ T ( u ) u = − (1 + cA ) A − A , where A = √ c +2 . This section is dedicated to proofs of our results. In what follows, the positive constant Q may be different fromline to line. We begin with a lemma which is of some interests on its own. Lemma 5.1
Let X be a non-negative random variable which is u − p -insensitive, with p > . Then, for anypositive constant B lim u →∞ exp( − Bu p ) P { X > u } = 0 . Proof of Lemma u there exists some θ u ∈ [0 ,
1] such that (setnext Y = X p ) P { Y > u } ≤ P { Y > u − } = P n X > ( u − /p o = P n X > u /p − (1 /p )( u − θ u ) /p − o ≤ P n X > u /p − (2 /p )( u ) /p − o . By insensitivity of X , we immediately get that P n X > u /p − (2 /p )( u ) /p − o ∼ P n X > u /p o = P { Y > u } as u → ∞ . Hence P { Y > u − } ∼ P { Y > u } , u → ∞ . Consequently, for any ε ∈ (0 ,
1) there exists
A > v > A we have P { Y > v } ≥ (1 − ε ) P { Y > v − } . Thus, for sufficiently large u P { Y > u } ≥ (1 − ε ) P { Y > u − } ≥ (1 − ε ) P { Y > u − } ≥ (1 − ε ) u − A P { Y > A } implying that for each B > u →∞ exp( − Bu ) P { Y > u } = 0and thus the claim follows. (cid:3) i) We first give the upper bound. For any ε ∈ (0 , t ) , u >
0, we derive that P ( sup t ∈ [0 , T ] X ( t ) > u ) ≤ P ( sup t ∈ [0 , T ] X ( t ) > u, T < t − ε ) + P ( sup t ∈ [0 , T ] X ( t ) > u, T ≥ t − ε ) ≤ P ( sup t ∈ [0 ,t − ε ] X ( t ) > u ) + P ( sup t ∈ [0 , ∞ ) X ( t ) > u ) P {T ≥ t − ε } , Similarly, for any u > P ( sup t ∈ [0 , T ] X ( t ) > u ) ≥ P ( sup t ∈ [0 ,t + ε ] X ( t ) > u ) P {T ≥ t + ε }≥ P ( sup t ∈ [0 , ∞ ) X ( t ) > u ) − P ( sup t ∈ [ t + ε, ∞ ) X ( t ) > u )! P {T ≥ t + ε } . Choosing ε small enough such that lim sup t →∞ σ ( t ) < σ ( t ± ε ) < P ( sup t ∈ [0 ,t − ε ] X ( t ) > u ) = o P ( sup t ∈ [0 , ∞ ) X ( t ) > u )! and P ( sup t ∈ [ t + ε, ∞ ) X ( t ) > u ) = o P ( sup t ∈ [0 , ∞ ) X ( t ) > u )! as u → ∞ . Thus the claim of the first statement follows by letting ε → T is u − / ( γ (1+ β )) -insensitive and let ε u = u − / (1+ β ) . With similar arguments as in the proofof statement i ) we obtain p ( X, T , γ, u ) = P ( sup t ∈ [0 , T /u γ ] X ( t ) > u ) ≤ P ( sup t ∈ [0 ,t − ε u ] X ( t ) > u ) + P ( sup t ∈ [0 , ∞ ) X ( t ) > u ) P {T > ( t − ε u ) u γ } (29)and p ( X, T , γ, u ) ≥ P ( sup t ∈ [0 ,t + ε u ] X ( t ) > u ) P {T > ( t + ε u ) u γ }≥ P ( sup t ∈ [0 , ∞ ) X ( t ) > u ) − P ( sup t ∈ [ t + ε u , ∞ ) X ( t ) > u )! P {T > ( t + ε u ) u γ } . Further, by insensitivity of T as u → ∞ P {T > ( t ± ε u ) u γ } ∼ P {T > t u γ } . (30)Thanks to Piterbarg inequality (cf. Theorem 8.1 of Piterbarg (1996)), combined with A3 , for all u large wehave P ( sup t ∈ [0 ,t − ε u ] X ( t ) > u ) ≤ Q u /r − exp (cid:18) − u (cid:19) exp (cid:18) − au ( ε u ) β (cid:19) Q not depending on u . Similarly, using additionally Borell-TIS inequality, for u largeand some G > t P ( sup t ∈ [ t + ε u , ∞ ) X ( t ) > u ) ≤ P ( sup t ∈ [ t + ε u ,G ] X ( t ) > u ) + P ( sup t ∈ [ G, ∞ ) X ( t ) > u ) ≤ Q u /r − exp (cid:18) − u (cid:19) exp (cid:18) − au ( ε u ) β (cid:19) + exp − u − E sup t ∈ [ G, ∞ ) X ( t ) !! t ∈ [ G, ∞ ) σ ( t ) . Hence, using that ε u = u − / (1+ β ) and the results of Lemma 5.1lim u →∞ exp (cid:16) − a u β (cid:17) P {T > t u γ } = 0and thus P ( sup t ∈ [ t + ε u , ∞ ) X ( t ) > u ) = o P ( sup t ∈ [0 , ∞ ) X ( t ) > u )! , (31) P ( sup t ∈ [0 ,t − ε u ] X ( t ) > u ) = o P ( sup t ∈ [0 , ∞ ) X ( t ) > u ) P {T > ( t − ε u ) u γ } ! . (32)Consequently, combining (29)-(32) we obtain p ( X, T , γ, u ) ∼ P ( sup t ∈ [0 , ∞ ) X ( t ) > u ) P {T > t u γ } as u → ∞ , which completes the proof. (cid:3) For the proof we need to distinguish between three different cases depending on the value of γp . For notationalsimplicity, let T u := T /u γ .Case γp <
2. For some ε > u large p ( X, T , γ, u ) = P ( sup t ∈ [0 , T u ] X ( t ) > u, T u > u − γpp − ε ) + P ( sup t ∈ [0 , T u ] X ( t ) > u, T u ≤ u − γpp − ε ) ≥ P n T > u γ + − γpp − ε o P sup t ∈ (cid:20) ,u − γpp − ε (cid:21) X ( t ) > u ≥ P n T > u γ + − γpp − ε o P { X ( t ) > u } log ∼ P { X ( t ) > u } , which follows from the fact that, by the assumption on T , P n T > u γ + − γpp − ε o log ∼ exp( − Lu − εp ), while P { X ( t ) > u } log ∼ exp( − u / p ( X, T , γ, u ) ≤ P ( sup t ∈ [0 , ∞ ) X ( t ) > u ) log ∼ P { X ( t ) > u } , then the claim (15) follows.1Case γp = 2. Let b σ ( s ) , e σ L,γ ( s ) , s ≥ A be defined as in (13) and (14) respectively. The lower boundfollows from the fact that p ( X, T , γ, u ) = P ( sup t ∈ [0 , T u ] X ( t ) > u, T u > A ) + P ( sup t ∈ [0 , T u ] X ( t ) > u, T u ≤ A ) ≥ P {T > A u γ } P { σ ( A ) N > u } log ∼ exp (cid:16) − LA /γ u (cid:17) exp (cid:18) − u σ ( A ) (cid:19) = exp (cid:0) − e σ L,γ ( A ) u (cid:1) , (33)where N is a standard Gaussian (i.e., an N (0 , < m < < M (to be determined later) we have p ( X, T , γ, u ) = − Z ∞ P ( sup t ∈ [0 ,s ] X ( t ) > u ) d P {T u > s } = − Z mA P ( sup t ∈ [0 ,s ] X ( t ) > u ) d P {T u > s } − Z MA mA P ( sup t ∈ [0 ,s ] X ( t ) > u ) d P {T u > s }− Z ∞ MA P ( sup t ∈ [0 ,s ] X ( t ) > u ) d P {T u > s } := I ( u ) + I ( u ) + I ( u ) . We analyze I i ( u ) , i = 1 , , I ( u ). Using Piterbarg inequality and integration by parts, we have that I ( u ) ≤ − Z MA mA (cid:18) Q M A u /r Ψ (cid:18) u b σ ( s ) (cid:19)(cid:19) d P {T u > s } = − Q M A u /r Ψ (cid:18) u b σ ( M A ) (cid:19) P {T > M A u γ } − Ψ (cid:18) u b σ ( mA ) (cid:19) P {T > mA u γ } + Z MA mA u √ π (cid:18) b σ ( s ) (cid:19) ′ exp − u b σ ( s ) ! P {T u > s } ds ! := I , ( u ) + I , ( u ) + I , ( u ) ≤ I , ( u ) + I , ( u ) . Next we find bounds for I ,i ( u ) , i = 2 , I , ( u ) log ∼ exp − (cid:18) b σ ( mA ) + L ( mA ) /γ (cid:19) u ! (34)and, for any ε > u large I , ( u ) ≤ Q M A u /r +1 √ π max s ∈ [ mA ,MA ] (cid:18) − b σ ( s ) (cid:19) ′ Z MA mA exp (cid:18) − (cid:18) b σ ( s ) + L (1 − ε ) s /γ (cid:19) u (cid:19) ds ≤ Q ( M A ) u /r +1 √ π max s ∈ [ mA ,MA ] (cid:18) − b σ ( s ) (cid:19) ′ exp − inf s ∈ [ mA ,MA ] (cid:18) b σ ( s ) + L (1 − ε ) s /γ (cid:19) u ! . Consequently, letting ε → u →∞ log I , ( u ) u ≤ − e σ L,γ ( A ) . (35)Ad. I ( u ). By the Piterbarg inequality we have that I ( u ) ≤ P ( sup t ∈ [0 ,mA ] X ( t ) > u ) ≤ Q u /r +1 exp (cid:18) − u b σ ( mA ) (cid:19) , (36)2where r ∈ (0 ,
2] is as in A3 .Ad. I ( u ). We straightforwardly have that I ( u ) ≤ P {T u > M A } log ∼ exp( − L ( M A ) /γ u ) . (37)Now we are ready to determine both constants m and M . First, choose m such that12 b σ ( mA ) > e σ L,γ ( A ) , and then choose M such that L ( M A ) /γ > e σ L,γ ( A ) . We conclude from (34)-(37) that lim sup u →∞ log p ( X, T , γ, u ) u ≤ − e σ L,γ ( A ) . (38)Consequently, combination of (33) with (38) leads tolim u →∞ log p ( X, T , γ, u ) u = − e σ L,γ ( A ) . Case γp > σ (0) >
0. Let ε ∈ (0 , γp − p ). Then we have (set u ε := u − γpp + ε ) p ( X, T , γ, u ) = P ( sup t ∈ [0 , T u ] X ( t ) > u, T u > u ε ) + P ( sup t ∈ [0 , T u ] X ( t ) > u, T u ≤ u ε ) ≤ P n T > u p + ε o + P ( sup t ∈ [0 ,u ε ] X ( t ) > u ) . Since − log P {T > u } is regularly varying at infinity with index p , thenlim u →∞ log P n T > u p + ε o u = −∞ and, by Borell-TIS inequality, for sufficiently large u P ( sup t ∈ [0 ,u ε ] X ( t ) > u ) ≤ − u − E sup t ∈ [0 ,u ε ] X ( t ) !! t ∈ [0 ,u ε ] σ ( t ) . Combining the above with p ( X, T , γ, u ) ≥ P { X (0) > u } we obtain that lim u →∞ log p ( X, T , γ, u ) u = − σ (0) , which proves (17).Case γp > σ ( t ) = Dt η (1 + o (1)) as t ↓
0. Let g ( t ) = D t η + Lt p , t ≥
0, which has a unique minimumpoint at t ∗ = (cid:16) ηpLD (cid:17) / (2 η + p ) with A := g ( t ∗ ) = 12 D − p η + p ( Lp/η ) η η + p + L η η + p ( η/ ( pD )) p η + p . µ = pγ − η + p > p ( X, T , γ, u ) = P ( sup t ∈ [0 , T u ] X ( t ) > u, T u > t ∗ u − µ ) + P ( sup t ∈ [0 , T u ] X ( t ) > u, T u ≤ t ∗ u − µ ) ≥ P ( sup t ∈ [0 ,t ∗ u − µ ] X ( t ) > u ) P (cid:8) T > t ∗ u γ − µ (cid:9) ≥ P (cid:8) X ( t ∗ u − µ ) > u (cid:9) P (cid:8) T > t ∗ u γ − µ (cid:9) log ∼ exp (cid:18) − u ηµ D ( t ∗ ) η − L ( t ∗ ) p u p ( γ − µ ) (cid:19) = exp (cid:16) − A u ηγp +2 p η + p (cid:17) . In order to derive an upper bound, we replace A with t ∗ u − µ in the upper estimate of the case γp = 2. Followingstep-by-step the same argument as in the upper bound of the case γp = 2, we conclude thatlim sup u →∞ log p ( X, T , γ, u ) u ηγp +2 p η + p ≤ − A and thus the proof is complete. (cid:3) By the self-similar property of the fBm we see that P ( sup t ∈ [0 ,T ] (cid:16) B α ( Y ( t )) − c ( Y ( t )) θ (cid:17) > u ) = P ( sup t ∈ [0 ,Y ( T ) /u /θ ] B α ( t )1 + ct θ > u − α θ ) . (39)The function V ( t ) = t α/ ct θ attains its maximum at the unique point t = (cid:18) α/ c ( θ − α/ (cid:19) /θ and V = V ( t ) = 2 θ − α θ (cid:18) αc (2 θ − α ) (cid:19) α θ . Re-writing (39), we have P ( sup t ∈ [0 ,T ] (cid:16) B α ( Y ( t )) − c ( Y ( t )) θ (cid:17) > u ) = P ( sup t ∈ [0 ,Y ( T ) /u /θ ] Z ( t ) > V − u − α θ ) = P sup t ∈ " , Y ( T )( V − θ − α ( v ( u )) 22 θ − α Z ( t ) > v ( u ) , (40)where Z ( t ) = B α ( t ) t α/ V ( t ) V , t ≥ v ( u ) = V − u − α θ . It is straightforward to check that p E ( Z ( t )) = 1 − c θ α − θ (2 θ − α ) θ ( t − t ) + o (cid:0) ( t − t ) (cid:1) as t → t and C ov Z ( t ) p E ( Z ( t )) , Z ( s ) p E ( Z ( s )) ! = 1 − t α | t − s | α + o ( | t − s | α )4as s, t → t . Moreover, for any given positive constant H , it is derived that, for t, s ∈ [0 , H ] with | t − s | < E ( Z ( t ) − Z ( s )) = V − E (cid:18)(cid:18) B α ( t )1 + ct θ − B α ( s )1 + ct θ (cid:19) + (cid:18) B α ( s )1 + ct θ − B α ( s )1 + cs θ (cid:19)(cid:19) ≤ V ) − E (cid:18) B α ( t )1 + ct θ − B α ( s )1 + ct θ (cid:19) + E (cid:18) B α ( s )1 + ct θ − B α ( s )1 + cs θ (cid:19) ! = 2( V ) − | t − s | α (1 + ct θ ) + s α (cid:18)
11 + ct θ −
11 + cs θ (cid:19) ! ≤ Q | t − s | α . Thus assumptions
A1-A4 are satisfied. Additionally, by Theorem 1 in H¨usler and Piterbarg (1999) P ( sup t ∈ [0 , ∞ ) Z ( t ) > v ( u ) ) ∼ Q H α u (2 θ − α )(2 − α )2 θα Ψ θ θ − α (cid:18) αc (2 θ − α ) (cid:19) − α θ u θ − α θ ! , u → ∞ , where Q := 2 + α √ πc − α θ α α − − θ θ θ − αα (2 θ − α ) θα − θ +2 α − α θα .The rest of the proof follows from an application of Theorem 3.1 statement ii) and Theorem 3.3. (cid:3) Note that, for each u >
0, using notation introduced in Section 5.3 P ( sup t ∈ [0 ,T ] B α ( Y ( t )) − c ( Y ( t )) θ > u ) ≤ P ( sup t ∈ [0 ,Y ( T ) /u /θ ] Z ( t ) > V − u − α θ ) . Thus it suffices to find logarithmically tight lower bounds for each subclass of densities of Y ( T ). The idea ofthe proof is the same both for the density of Y ( T ) being regularly varying and log-power tailed and heavily usesthe idea of getting the lower bound in the proof of Theorem 3.3. Hence we give only the argument for Y ( T )having regularly varying density function with index λ + 1. Under this scenario P ( sup t ∈ [0 ,T ] (cid:16) B α ( Y ( t )) − c ( Y ( t )) θ (cid:17) > u ) ≥ P (cid:26) B α ( Y ( T )) u + c ( Y ( T )) θ > (cid:27) ≥ min t ∈ [ t u /θ − u / (2 θ ) ,t u /θ + u / (2 θ ) ] P (cid:26) B α ( t ) u + ct θ > (cid:27) P n Y ( T ) ∈ [ t u /θ − u / (2 θ ) , t u /θ + u / (2 θ ) ] o . Using that P (cid:8) Y ( T ) ∈ [ t u /θ − u / (2 θ ) , t u /θ + u / (2 θ ) ] (cid:9) is regularly varying at ∞ andlim u →∞ u αθ − log (cid:18) min t ∈ [ t u /θ − u / (2 θ ) ,t u /θ + u / (2 θ ) ] P (cid:26) B α ( t ) u + ct θ > (cid:27)(cid:19) = − V we obtain a logarithmically tight lower bound, and thus the proof is complete. (cid:3) Acknowledgement : We are thankful to the referees for several comments and suggestions. K. D¸ebicki waspartially supported by NCN Grant No 2011/01/B/ST1/01521 (2011-2013). K. D¸ebicki, E. Hashorva and L. Jikindly acknowledge partial support by the Swiss National Science Foundation Grant 200021-140633/1 and bythe project RARE -318984, a Marie Curie FP7 IRSES Fellowship.
References [1] Adler, R.J., 1990.
An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes ,Inst. Math. Statist. Lecture Notes Monogr. Ser. 12, Inst. Math. Statist., Hayward, CA. [2] Adler, R.J., Taylor, J.E., 2007. Random Fields and Geometry . Springer.[3] Arendarczyk, M., D¸ebicki, K., 2011. Asymptotics of supremum distribution of a Gaussian process over a Weibulliantime. Bernoulli 17, 194-210.[4] Arendarczyk, M., D¸ebicki, K., 2012. Exact asymptotics of supremum of a stationary Gaussian process over arandom interval. Stat. Prob. Lett. 82, 645-652.[5] D¸ebicki, K., Rolski, T., 2002. A note on transient Gaussian fluid models. Queueing Systems 41, 321-342.[6] D¸ebicki, K., van Uitert, M., 2006. Large buffer asymptotics for generalized processor sharing queues with Gaussianinputs. Queueing Systems 54, 111-120.[7] Foss, S., Korshunov, D., Zachary, S., 2013.
An introduction to Heavy-tailed and Subexponential Distributions.