An Erd\"os-R\'ev\'esz type law of the iterated logarithm for reflected fractional Brownian motion

Let $B_H=\{B_H(t):t\in\mathbb R\}$ be a fractional Brownian motion with Hurst parameter $H\in(0,1)$. For the stationary storage process $Q_{B_H}(t)=\sup_{-\infty<s\le t}(B_H(t)-B_H(s)-(t-s))$, $t\ge0$, we provide a tractable criterion for assessing whether, for any positive, non-decreasing function $f$, $\mathbb P(Q_{B_H}(t)>f(t)\, \text{ i.o.})$ equals 0 or 1. Using this criterion we find that, for a family of functions $f_p(t)$, such that $z_p(t)=\mathbb P(\sup_{s\in[0,f_p(t)]}Q_{B_H}(s)>f_p(t))/f_p(t)=\mathscr C(t\log^{1-p} t)^{-1}$, for some $\mathscr C>0$, $\mathbb P(Q_{B_H}(t)>f_p(t)\, \text{ i.o.})= 1_{\{p\ge 0\}}$. Consequently, with $\xi_p (t) = \sup\{s:0\le s\le t, Q_{B_H}(s)\ge f_p(s)\}$, for $p\ge 0$, $\lim_{t\to\infty}\xi_p(t)=\infty$ and $\limsup_{t\to\infty}(\xi_p(t)-t)=0$ a.s. Complementary, we prove an Erd\"os--R\'ev\'esz type law of the iterated logarithm lower bound on $\xi_p(t)$, i.e., $\liminf_{t\to\infty}(\xi_p(t)-t)/h_p(t) = -1$ a.s., $p>1$; $\liminf_{t\to\infty}\log(\xi_p(t)/t)/(h_p(t)/t) = -1$ a.s., $p\in(0,1]$, where $h_p(t)=(1/z_p(t))p\log\log t$.


Introduction and Main Results
The analysis of properties of reflected stochastic processes, being developed in the context of classical Skorokhod problems and their applications to queueing theory, risk theory and financial mathematics, is an actively investigated field of applied probability. In this paper we analyze 0-1 properties of a class of such processes, that due to its importance in queueing theory (and dual risk theory) gained substantial interest; see, e.g., [1,2,13,14] or novel works on γ-reflected Gaussian processes [7,12].
Consider a reflected (at 0) fractional Brownian motion with drift Q BH = {Q BH (t) : t ≥ 0}, given by the following formula With no loss of generality in the reminder of this paper we assume that the drift parameter c ≡ 1. An important stimulus to analyze the distributional properties of Q BH and its functionals stems from the Gaussian fluid queueing theory, where the stationary buffer content process in a queue which is fed by B H and emptied with constant rate c = 1 is described by (2); see e.g. [13]. In particular, in the seminal paper by Hüsler and Piterbarg [8] the exact asymptotics of one dimensional marginal distributions of Q BH was derived; see also [3,4,6] for results on more general Gaussian input processes. The purpose of this paper is to investigate the asymptotic 0-1 behavior of the processes Q BH . Our first contribution is an analog of the classical finding of Watanabe [18], where an asymptotic 0-1 type of behavior for centered stationary Gaussian processes was analyzed. is finite or infinite.
The exact asymptotics, as u grows large, of the probability in I f was found by Piterbarg [14,Theorem 7]. Namely, for any T > 0, , Φ is the distribution function of the unit normal law and the constants a, b, A, H BH are given explicitly in Section 2. Since relation (3) also holds when Theorem 1 provides a tractable criterion for settling the dichotomy of , p ∈ R, H ∈ (0, 1).
One can check that, as u → ∞, Hence, for any p ∈ R, This result extends findings of Zeevi and Glynn [19,Theorem 1], where it was proven that the above convergence holds weakly as well as in L p for all p ∈ [1, ∞). Now consider the process ξ p = {ξ p (t) : t ≥ 0} defined as ξ p (t) = sup{s : 0 ≤ s ≤ t, Q BH (s) ≥ f p (s)}.
Let, cf. (5), The second contribution of this paper is an Erdös-Révész type of law of the iterated logarithm for the process ξ p . We refer to Shao [16] for more background and references on Erdös-Révész type law of the iterated logarithm and a related result for centered stationary Gaussian processes; see also De ֒ bicki and Kosiński [5] for extensions to order statistics.
If p ∈ (0, 1], then then it follows that Theorem 2 shows that for t big enough, there exists an s in [t − h p (t), t] (as well as in [t, t + h p (t)] by (6)) such that Q BH (s) ≥ f p (s) and that the length h p (t) of the interval is the smallest possible. This shines new light on results, which are intrinsically connected with Gumbel limit theorems; see, e.g., [11], where the function h p (t) plays crucial role. We shall pursue this elsewhere.
The paper is organized as follows. In Section 2 we introduce some useful properties of storage processes fed by fractional Brownian motion. In Section 3 we provide a collection of basic results on how to interpret extremes of the storage process Q BH as extremes of a Gaussian field related to the fractional Brownian motion B H . Furthermore, in Section 4 we prove lemmas, which constitute building blocks of the proofs of the main results.

Properties of the storage process
In this section we introduce some notation and state some properties of the supremum of the process Q BH as derived in [10,14]. We begin with the relation is a Gaussian field. Note that the self-similarity property of B H implies that the field Z u has the same distribution for any u. Thus, we do not use u as an additional parameter in the following notation whenever it is not needed; let Z(s, τ ) := Z 1 (s, τ ). Furthermore, the field Z(s, τ ) is stationary in s, but not in τ . The variance σ 2 Z (τ ) of the field Z(s, τ ) equals ν −2 (τ ) and σ Z (τ ) has a single maximum point at Taylor expansion leads to Let us define the correlation function of the process Z u as follows By series expansion we find for any fixed τ 1 < τ 0 < τ 2 and τ, τ ′ with 0 < τ 1 < τ, τ ′ < τ 2 < ∞, provided that | u us−u ′ s ′ | and | u ′ us−u ′ s ′ | are sufficiently small. For 2H = 1, we have r u,u ′ (s, τ, s ′ , τ ′ ) = 0 since the increments of Brownian motion on disjoint intervals are independent. Therefore, (9) r * (t) := sup for λ = 2 − 2H > 0, t sufficiently large and some positive constant K depending only on H, τ 1 and τ 2 .
Similarly, from (8) it follows that for any fixed M there exists δ ∈ (0, 1) such that for sufficiently small m.

2.1.
Asymptotics. Due to the following lemma, while analyzing tail asymptotics of the supremum of Z, we can restrict the considered domain of (s, τ ) to a strip with |τ − τ 0 | ≤ log v/v.
For a fixed T, θ > 0 and some v > 0, let us define a discretization of the set [0, T ] × J(v) as follows Along the same lines as in [10, Lemma 6] we get the following lemma.
Finally, it is possible to approximate tail asymptotics of the supremum of Z on the strip [0, T ] × J(v) by maximum taken over discrete time points. The proof of the following lemma follows line-by-line the same as the proof of [14, Lemma 4] and thus we omit it. Similar result can be found in, e.g., [10,Lemma 7].
It follows easily that H θ BH → H BH as θ → 0, so that the above asymptotics is the same as in Lemma 1 when the discretization parameter θ decreases to zero so that the number of discretization points grows to infinity.

Auxiliary Lemmas
We begin with some auxiliary lemmas that are later needed in the proofs. The first lemma is a slightly modified version of [11,Theorem 4 Lemma 4 (Berman's inequality). Suppose that ξ 1 , . . . , ξ n are normal random variables with correlation matrix Λ 1 = (Λ 1 i,j ) and η 1 , . . . , η n similarly with correlation matrix The following lemma is a general form of the Borel-Cantelli lemma; cf. [17].
Lemma 6. For any ε ∈ (0, 1), there exist positive constants K and ρ depending only on ε, H, p and λ such that Proof. Let ε ∈ (0, 1) be some positive constant. For the reminder of the proof let K and ρ be two positive constants depending only on ε, H, p and λ that may differ from line to line. For any k ≥ 0 put s 0 = S, From this construction, it is easy to see that the intervals I k are disjoint. Furthermore, δ(I k , I k+1 ) = εx k , and 1 − ε ≤ y k /x k ≤ 1, for any k ≥ 0 and sufficiently large S. Note that, for any k ≥ 0, |I k | ∼ f p (S) as S grows large, therefore if T (S, ε) is the smallest number of intervals {I k } needed to cover [S, T ], then T (S, ε) ≤ [(T − S)/(f p (S)(1 + ε))]. Moreover, since f p (T )/f p (S) is bounded by the constant C > 0 not depending on S and ε, it follows that, x k /x t ≤ C for any 0 ≤ t < k ≤ T (S, ε). Now let us introduce a discretization of the setĨ k × J(v k ) as in Section 2.2. That is, for some θ > 0, define grid points Since f p is an increasing function, it easily follows that, where the last inequality follows from Berman's inequality with Estimate of P 1 .
Note that we can use the fact that Z x k has the same distribution as Z 1 ≡ Z for any x k . Since the process Z is stationary with respect to the first variable, from Lemma 3, for any ε ∈ (0, 1), sufficiently large S and small θ, Then, by (7) combined with (3), Estimate of P 2 . 6 where the last inequality holds provided that k − t ≥ s 0 with s 0 sufficiently large. Therefore, c.f. (9), r * k,t := sup 0≤l≤L k ,0≤p≤Lt |n|≤N k ,|m|≤Nt |r x k ,xt (s k,l , τ k,n , s t,p , τ t,m )| ≤ r * ((k − t)ε) ≤ K(k − t) −λ ≤ min(1, λ)/4.
Let S > 0 be any fixed number, a 0 = S, y 0 = f p (a 0 ) and b 0 = a 0 + y 0 . For i > 0, define From this construction it is easy to see that the intervals M i are disjoint, ∪ i j=0 M j = (S, b i ] and |M i | = 1. Now let us introduce a discretization of the setM i × J(v i ) as in Section 2.2. That is, for some θ > 0, define grid points With the above notation, we have the following lemma.
Lemma 7. For any ε ∈ (0, 1) there exist positive constants K and ρ depending only on ε, H, p and λ such that, with for any T − f p (S) ≥ S ≥ K, with f p (T )/f p (S) ≤ C and C being some universal positive constant.
Similarly as in the proof of Lemma 6 we find that Berman's inequality implies withr yi,yj (s i,l , τ i,n , s j,p , τ j,m ) = −r yi,yj (s i,l , τ i,n , s j,p , τ j,m ). Estimate of P ′ 1 .
By Lemma 1 the correction term θ H 2 i /v i does not change the order of the asymptotics of the tail of Z. Furthermore, the tail asymptotics of the supremum on the strip (s, τ ) ∈M i × J(v i ) are of the same order if τ ≥ 0. Hence, for every ε > 0, following the same lines of reasoning as in the estimation of P 1 in Lemma 6, provided that S is sufficiently large.
Completely similarly to the estimation of P 2 in the proof of Lemma 6, we can get that there exist positive constants K and ρ such that, for sufficiently large S, The next lemma is a straightforward modification of [ 2 , it is true without the additional condition.

Proof of the main results
Proof of Theorem 1. Note that the case I f < ∞ is straightforward and does not need any additional knowledge on the process Q BH apart from the stationarity property. Indeed, consider the sequence of intervals M i as in Lemma 7. Then, for any ε > 0 and sufficiently large T , and the Borel-Cantelli lemma completes this part of the proof since f is an increasing function. Now let f be an increasing function such that I f ≡ ∞. Using the same notation as in Lemma 6 with f instead of f p , we find that, for any S, ε, θ > 0, For sufficiently large S and θ; c.f. estimation of P 1 , we get Note that The first limit equals to zero as a consequence of (19). The second limit equals to zero because of the asymptotic independence of the events E k . Indeed, there exist positive constants K and ρ, depending only on H, ε, λ, such that for any n > m, by the same calculations as in the estimate of P 2 in Lemma 6 after realizing that, by Lemma 8, we might restrict ourselves to the case when (18) holds. Therefore P (E c i i.o.) = 1, which completes the proof. Proof of Theorem 2 In order to make the proof more transparent we divide it on several steps.
Step 1. Let p > 1. Then, for every ε ∈ (0, 1 4 ), Since h p (t) = O(t log 1−p t log 2 t), then, for p > 1, S k ∼ T k , as k → ∞, and from Lemma 6 it follows that Moreover, as k → ∞, Now take T k = exp(k 1/p ). Then, Hence, by the Borel-Cantelli lemma, we have Since ξ p (t) is a non-decreasing random function of t, for every T k ≤ t ≤ T k+1 , we have For p > 1 elementary calculus implies which completes the proof of this step.
Step 2. Let p > 1. Then, for every ε ∈ (0, 1), Proof. As in the proof of the lower bound (Step 1), we put It suffices to show P (B n i.o.) = 1, that is Define J k to be the biggest number such that b k Since f p is an increasing function, Analogously to (14), define a discretization of the setM k i × J(v k i ) as follows Observe that By Lemma 2, for sufficiently large m and some K 1 , K 2 > 0, the first sum is bounded from above by Note that by (11), for sufficiently large m, the term in (23) is bounded from above by In order to complete the proof of (22) we only need to show that Similarly to (20), we have Now from Lemma 7 it follows that for every k sufficiently large. Hence, Applying Berman's inequality, we get for t < k where .
For any 0 ≤ i ≤ J k , 0 ≤ j ≤ J t , 0 ≤ l ≤ L k i , 0 ≤ p ≤ L t j , and t < k, y k i s k i,l − y t j s t j,p = a k i + y k i lq k i − a t j + y t j pq t j ≥ S k − T t ≥ S k − T k−1 ≥ where the last inequality holds for k large enough, since S k+1 − T k T k+1 − T k ∼ 1, as k → ∞.