On the infimum attained by the reflected fractional Brownian motion

Let $\{B_H(t):t\ge 0\}$ be a fractional Brownian motion with Hurst parameter $H\in(\frac{1}{2},1)$. For the storage process $Q_{B_H}(t)=\sup_{-\infty\le s\le t} \left(B_H(t)-B_H(s)-c(t-s)\right)$ we show that, for any $T(u)>0$ such that $T(u)=o(u^\frac{2H-1}{H})$, \[\mathbb P (\inf_{s\in[0,T(u)]} Q_{B_H}(s)>u)\sim\mathbb P(Q_{B_H}(0)>u),\quad\text{as}\quad u\to\infty.\] This finding, known in the literature as the strong Piterbarg property, is in line with previously observed properties of storage processes with self-similar and infinitely divisible input without Gaussian component.


Introduction
The analysis of distributional properties of reflected stochastic processes is continuously motivated both by theory-and applied-oriented open problems in probability theory. In this paper we analyze the asymptotic properties of tail distribution of infimum of an important class of such processes, that naturally appear in models of storage (queueing) systems and, by duality to ruin problems, gained broad interest also in problems arising in finance and insurance risk; see, e.g., [4,5,14,18] or a novel work [10].
Consider a fluid queue with infinite buffer capacity, service rate c > 0 and the total inflow by time t modeled by a stochastic process with stationary increments X = {X(t) : t ∈ R}. Following Reich [20], the stationary storage process that describes the stationary buffer content process, has the following representation Q X (t) = sup There is a strong motivation for modeling the input process X by a fractional Brownian motion (fBm) B H = {B H (t) : t ∈ R} with H > 1/2, i.e., a centered Gaussian process with stationary increments, continuous sample paths a.s., and variance function σ 2 BH (t) = t 2H . On one hand, such structural properties of fBm as self-similarity and long range dependence, have been statistically confirmed in data analysis of many real traffic processes in modern data-transfer networks. On the other hand, in [13,22] it was proven that appropriately scaled aggregation of large number of (integrated) On-Off input processes with regularly varying tail distribution of successive On-times, converges to an fBm with H > 1/2.
The importance of fBm storage processes resulted in a vast interest of analysis of the process Q BH . In particular finding the properties of finite-dimensional (or at least 1-dimensional) distributions of Q BH has been a long standing goal; see [14,18]. The stationarity of increments of B H implies the stationarity of the process Q BH , so that, for any fixed t, the random variable Q BH (t) has the same distribution as Q BH (0). Nevertheless, apart from the Brownian case H = 1 2 , the exact distribution of Q BH (0) is not known. Therefore, one usually resorts to the exact asymptotics of P (Q BH (0) > u), as u → ∞. These have been found for the full range of parameter H ∈ (0, 1) in [11], leading to, This property is nowadays referred to as the generalized Piterbarg property; see [2]. As a corollary from (2) one easily gets that for any fixed n > 0 and t 1 , . . . , t n ∈ [0, T ], with u → ∞, This leads to the natural question, whether the minimum over finite number of points can be substituted with the infimum functional, which then leads to This property shall be referred to as the strong Piterbarg property.
The above terminology has been coined by Albin and Samorodnitsky [2], who, motivated by [18], considered the case when the input process X belongs to the class of self-similar infinitely divisible stochastic processes with no Gaussian component. They provide general conditions under which (2) and (3) hold with Q X instead of Q BH . The approach in [2] is based on the assumption that the Lévy measure associated with X has heavy tails, which combined with the absence of a Gaussian component allows for more direct and less delicate methods to be employed. It is the light-tailed nature of the Gaussian distribution that renders the problem of the asymptotics of suprema of Gaussian processes hard. Furthermore, infima of Gaussian processes (apart perhaps from the Brownian case) have not been considered systematically. On the high level, the problem stems from the fact that an infimum is, by definition, an intersection of events. If the number of events grows to infinity, then the intersection is much harder to handle than, for instance, the sum of events (which corresponds to the supremum).
In this paper we derive exact asymptotics of (4) P inf and prove that the strong Piterbarg property (3) holds for the same range of functions T (u) as in the generalized Piterbrag property (2), i.e., T (u) = o(u The idea of the proof is based on finding the exact asymptotics of for a broad class of functionals Φ : C(T ) → R acting on the space C(T ) of continuous functions on compacts T ⊂ R d + , d ≥ 1, and a broad class of Gaussian fields The connection between (4) and (5) can be seen by setting d = 1, Φ(f ) = inf t∈[0,1] f (t) and X u (t) = Q BH (T (u)t), although the relation is far from straight forward since Q BH is not Gaussian.
Structure of the paper: The exact asymptotics of (5) are given in Lemma 1 (see Section 3), which is the first contribution of this paper. Interestingly, the asymptotics of (5) involve a new type of constants of the form , where η is a Gaussian random field with variance function σ 2 η . These new constants extend the notion of the classical Pickands' constants H sup ), S > 0, dating back to Pickands [16]. Recall that H sup (1). In Theorem 1 (Section 4) we give the strong Piterbarg property, which is the second contribution of this paper. More precisely, we show that (3) holds for H > 1 2 and T (u) = o(u 2H−1 H ), i.e., the same order of functions for which (2) holds. In Section 5 and Section 6 we give the proofs of our main results.

Notation
Before we begin, let us set the notation that will be used throughout the paper. By B H = {B H (t) : t ∈ R} we denote the fBm with Hurst parameter H ∈ (0, 1), that is, a Gaussian process with zero mean and covariance function given by Let Ψ be the right tail of the standard normal distribution. Recall that For we denote a centered Gaussian field, with almost surely continuous sample paths, η(0) = 0 and variance function σ 2 η (t) = Var(η(t)). Let us introduce the following condition: Condition E1 is a standard regularity requirement; see, e.g., [17]. Now let Φ : C(T ) → R be a functional acting on C(T ), the space of continuous functions on compacts T ⊂ R d + , d ≥ 1. Assume that: Note that the dependence on T is implicit via Φ : Now since η is continuous, then it has bounded sample paths a.s. and σ 2 η = sup t∈T σ 2 η (t) < ∞. Let m = E sup t∈T η(t). Borell's inequality; see, e.g., [1], implies that for x > m, P (sup t∈T η(t) > x) ≤ 2 exp −(x − m) 2 /(2σ 2 η ) and, as a consequence,

Generalized Pickands' lemma
In this section we present a lemma that shall play a crucial role in proving the strong Piterbarg property in the remaining part of the paper.
Lemma 1 (Generalized Pickands' lemma). For any u > 0, let X u = {X u (t) : t ∈ R d + } be a centered Gaussian field with a constant variance equal to one. Let the correlation function r u (t 1 ; t 2 ) = Corr(X u (t 1 ), X u (t 2 )) satisfy for some compact set T ⊂ R d + , some function f (u) → ∞, as u → ∞, and η satisfying E1. Let Φ : C(T ) → R be a functional satisfying F1-F2. Then, for any function n(u) such that n(u) ∼ f (u), P (Φ(X u ) > n(u)) ∼ H Φ η (T )Ψ(n(u)), as u → ∞. Remark 1. Conditions similar to assumption (8) have been introduced in, among others, [6,7,8,12] as a standard way of capturing nonstationarity. The shape of Lemma 1 is tailored to the needs of the next section, where asymptotics of tail distribution of inf sup functionals of Gaussian processes are analyzed. Various further extensions of Lemma 1 can be thought of along the lines of already existing extensions of the classical Pickands' lemma, especially in the direction allowing nonconstant variance function of the family (X u ), as in Piterbarg and Prisyazhnyuk [19] or Hashorva et al. [10].
is a centered Gaussian field with unit variance and correlation function satisfying Hi constitute independent fBm's with Hurst parameters H i . Hence the conclusion of Lemma 1 holds for any functional Φ on C(T ) satisfying F1-F2. In the following section we shall encounter this example in the setting of d = 2, H 1 = H 2 , a 1 = a 2 and Φ(f ) = inf t1∈[0,λ1] sup t2∈[0,λ2] f (t), for some λ 1 , λ 2 > 0. In this case, with H = H 1 and a = a 1 , for any function n(u) ∼ u,

Strong Piterbarg property
In this section we present the main result of this paper. Let us first recall the definition of the storage process Q BH with service rate c > 0 and input B H , In particular, Let us recall that Q B 1 2 (0) has 1 2 -exponential distribution. On the other hand, [18,Theorem 6], gives (note that the original formula in [18] has a misprint) Therefore, we see that the strong Piterbarg property does not hold in the case of H = 1 2 . Remark 4. One can envision that the strong Piterbarg property can be applied to functionals Φ : C([0, T ]) → R of Q BH that can be majorized, up to the same magnitude, by the infimum and supremum functionals. A simple example is the integral functional. Theorem 1 yields, for every H > 1 2 , , as u → ∞, . The problem of the area under the graph of the storage process fed by the Brownian motion, i.e., the case when H = 1 2 , has been considered in [3].

Proof of Lemma 1
The general idea behind the proof follows the one in Piterbarg [17,Lemma D.2]. For any u > 0, where we have used the change of variable v = n(u) − w n(u) . Let ζ u = {ζ u (t) : t ∈ T } be a Gaussian field defined via ζ u (t) = n(u)(X u (t) − n(u)) + w. Then, using F2, the last integral can be written as where χ u = {χ u (t) : t ∈ T } is a Gaussian field defined as χ u (t) d = ζ u (t)|ζ u (0) = 0. For the family of Gaussian distributions that appear inside the integral, for every t ∈ T , Eχ u (0) = Eχ 2 u (0) = 0. Furthermore, for any t 1 , t 2 ∈ T , Hence from (8) it follows that, as u → ∞, uniformly on T , Thus the finite dimensional distributions of χ u converge to the finite dimensional distributions ofη = { √ 2η(t) − σ 2 η (t) : t ∈ T }. Therefore χ u d →η in C(T ), as u → ∞, provided that the family χ = {χ u : u > 0} is tight. For this let χ • u = {χ • u (t) : t ∈ T } be a centered Gaussian field defined by χ • u (t) = χ u (t) − Eχ u (t). In order to prove tightness of the family χ = {χ u : u > 0} it suffices to show tightness of the centered family χ • = {χ • u : u > 0}. Since χ • u (0) = 0 for all u > 0, then a straightforward consequence of Straf's criterion for tightness of Gaussian fields, [21], implies that it suffices to show that for any µ, ρ > 0, there exists δ ∈ (0, 1) and u 0 > 0 such that, for each t 1 ∈ T and u > u 0 , where t = max{|t 1 |, . . . , |t d |}. Note that, for sufficiently large u, for all t 1 , t 2 ∈ T and some constant C > 0. Thus, the assumption E1 implies, which combined with the application of Borell's inequality gives (13). Then, the continuous mapping theorem implies , provided we can interchange the limit with the integral in (14). From (8) it follows that (1−r u (t; 0)) → 0 uniformly in t ∈ T , therefore (10)- (11) imply that for any ε > 0 and sufficiently large u, Using (12) combined with Sudakov-Fernique's inequality yields, for sufficiently large u and some constant C > 0, Furthermore, (12) combined with E1 implies, for sufficiently large u, Now, by F1, Borell's inequality yields, for |w|(1 − ε) ≥ m, Hence the interchange of the limit with the integral in (14) follows by the dominated convergence theorem and the limit is finite, that is H Φ η (T ) < ∞. This completes the proof of Lemma 1.

Proof of Theorem 1
We divide the proof on a number of steps. Before we proceed, let us make the following observation. The time-reversibility property of fBm implies that (on the process level) which is the form of Q BH that we shall use in this section. The relations of Section 6.1 and Section 6.2 were derived in [18].
6.1. Reduction to a Gaussian field. Using new variables τ = (σ − t)/u and s = t/u, for any T > 0, where ν(τ ) = τ −H + cτ 1−H and Z u = {Z u (s, τ ) : s, τ ≥ 0} is a Gaussian field given by The distribution of Z u does not depend on u, hence we deal with Z = Z 1 . Note that Z(s, τ ) is stationary in s, but not in τ .
The correlation function r(s 1 , τ 1 ; s 2 , τ 2 ) of Z equals r(s 1 , τ 1 ; s 2 , τ 2 ) = EZ(s 1 , τ 1 )Z(s 2 , τ 2 )ν(τ 1 )ν(τ 2 ) as s 1 − s 2 → 0, τ 1 → τ 0 , τ 2 → τ 0 . 6.3. Asymptotic properties of Z. In this step we will be concerned with the asymptotic properties of (17) P inf AZ(s, τ ) > u as u grows to infinity. Note that we normalized Z such that now the variance of AZ(s, τ ) equals one at τ = τ 0 (Z is stationary in s). It follows from [18, Lemma 1] that there exists a constant C such that, for any T > 0 and sufficiently large u, If we restrict ourselves to the neighborhood {τ : |τ − τ 0 | ≤ log u/u} of τ 0 , then the following step shows that the probability in (17), with Z restricted to the neighborhood of τ 0 , on the logarithmic scale decays as − u 2 2 when u grows large. Therefore, the neighborhood of τ 0 has the largest contribution to the asymptotic behavior of (17). In the following step we present its asymptotic contribution.