An Erdös–Révész type law of the iterated logarithm for reflected fractional Brownian motion

# An Erdös–Révész type law of the iterated logarithm for reflected fractional Brownian motion

K. Debicki Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.  and  K.M. Kosiński Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
July 15, 2019
###### Abstract.

Let be \textcolorblacka fractional Brownian motion with Hurst parameter . For the stationary storage process , \textcolorblack, we provide a tractable criterion for assessing whether, for any positive, non-decreasing function , equals 0 or 1. Using this criterion we find that, for a family of functions , such that , for some , . Consequently, with , for , and a.s. Complementary, we prove an Erdös–Révész type law of the iterated logarithm lower bound on , i.e., a.s., ; a.s., , where .

###### Key words and phrases:
Extremes of Gaussian fields, storage processes, fractional Brownian motion, law of the iterated logarithm
###### 2010 Mathematics Subject Classification:
Primary: 60F15, 60G70; Secondary: 60G22.

## 1. Introduction and Main Results

The analysis of properties of reflected stochastic processes, being developed in the context of classical Skorokhod problems and \textcolorblacktheir 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., [13, 14, 1, 2] or novel works on -reflected Gaussian processes [7, 12].

Consider a reflected (at 0) fractional Brownian motion with drift , given by the following formula

 (1) QBH(t)=BH(t)−ct+max(QBH(0),−infs∈[0,t](BH(s)−cs)),

where and is a fractional Brownian motion (fBm) with Hurst parameter , i.e., a centered Gaussian process with covariance function We focus on the investigation of the long-time behavior of the unique stationary solution of (1), which has the following representation

 (2) QBH(t)=sup−∞

With no loss of generality in the reminder of \textcolorblackthis paper we assume that the drift parameter . An important stimulus to analyze the distributional properties of and its functionals stems from \textcolorblackthe Gaussian fluid queueing theory, where the stationary buffer content process in a queue which is fed by and emptied with constant rate 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 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 . Our first contribution is an \textcolorblackanalog of the classical finding of Watanabe [18], where an asymptotic 0-1 type of behavior for centered stationary Gaussian processes was analyzed.

###### Theorem 1.

For all functions that are positive and nondecreasing on some interval , it follows that

 P(QBH(t)>f(t)\rm{i.o.})=0or1,

according as the integral

 If:=∫∞T1f(u)P(supt∈[0,f(u)]QBH(t)>f(u))du

is finite or infinite.

The exact asymptotics, as grows large, of the probability in \textcolorblackwas found by Piterbarg [14, Theorem 7]. \textcolorblackNamely, for any ,

 (3) P(supt∈[0,Tf(u)]QBH(t)>f(u))=√πa2Hb−12H2BHT(vf(u))2H−1Ψ(vf(u))(1+o(1)), as u→∞,

where , , is the distribution function of \textcolorblackthe unit normal law and the constants are given explicitly in Section 2. Since relation (3) also holds when , provided that , \textcolorblackwe have that for , as ,

 1f(u)P(supt∈[0,f(u)]QBH(t)>f(u))∼P(supt∈[0,1]QBH(t)>f(u)).

Theorem 1 provides a tractable criterion for settling the dichotomy of . For instance, \textcolorblacklet and

 (4) fp(s)=(2A2(logs+(1+CH−p)log2s))12(1−H),p∈R,H∈(0,1).

One \textcolorblackcan check that, as ,

 (5) 1fp(u)P(supt∈[0,fp(u)]QBH(t)>fp(u))=a2Hb−12√2H2BHA11−H2CH(ulog1−pu)−1(1+o(1)).

Hence, for any ,

###### Corollary 1.

For any ,

 limsupt→∞QBH(t)(logt)12(1−H)=(2A2)12(1−H)a.s.

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 for all .

Now consider the process defined as

 ξp(t)=sup{s:0≤s≤t,QBH(s)≥fp(s)}.

Since for , from Theorem 1 it follows that

 limt→∞ξp(t)=∞a.s.andlimsupt→∞(ξp(t)−t)=0a.s.

Let, cf. (5),

 hp(t)=p(1fp(t)P(sups∈[0,fp(t)]QBH(s)>fp(t)))−1log2t.

The second contribution of this paper is an Erdös–Révész type of law of the iterated logarithm for the process . 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; \textcolorblacksee also Debicki and Kosiński [5] for extensions to order statistics.

###### Theorem 2.

If , then

 liminft→∞ξp(t)−thp(t)=−1  a.s.

If , then

 liminft→∞log(ξp(t)/t)hp(t)/t=−1  a.s.

Now, let us complementary put , where

 ηp(t)=inf{s≥t:QBH(s)≥fp(s)}.

Since

 P(ξp(t)−t≤−x)=P(sups∈(t−x,t]QBH(s)fp(s)<1)

and

 P(z−ηp(z)≤−x)=P(sups∈[z,z+x]QBH(s)fp(s)<1),

then it follows that

 (6) liminft→∞ξp(t)−thp(t)=liminfz→∞z−ηp(z)hp(z).

Theorem 2 shows that for big enough, there exists an in (as well as in by (6)) such that and that the length \textcolorblack of the interval is the smallest possible. This shines new light on \textcolorblackresults, which are intrinsically connected with Gumbel limit theorems; see, e.g., [11], where the function 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 as extremes of a Gaussian field related to the fractional Brownian motion . Furthermore, \textcolorblackin Section 4 we prove lemmas, which constitute building blocks of the proofs of the main results.

## 2. Properties of the storage process

\textcolor

black In this section we introduce some notation and state some properties of the supremum of the process as derived in [14, 10]. We begin with the relation

 (7) P(supt∈[0,T]QBH(t)>u)=P⎛⎜⎝sups∈[0,T/u]τ≥0Zu(s,τ)>u1−H⎞⎟⎠,for anyT>0,

where, with ,

 Zu(s,τ):=BH(u(τ+s))−BH(su)τHuHν(τ)

is a Gaussian field. Note that the self-similarity property of implies that the field has the same distribution for any . Thus, we do not use as an additional parameter in the \textcolorblackfollowing notation whenever it is not needed; \textcolorblacklet . Furthermore, the field is stationary in , but not in . The variance of the field equals and has a single maximum point at

 τ0=H1−H.

 σZ(τ)=1A−B2A2(τ−τ0)2+O((τ−τ0)3),

as , where

 A =11−H(H1−H)−H=ν(τ0), B =H(H1−H)−H−2=ν′′(τ0).

Let us define the correlation function of the process as follows

 ru,u′ (s,τ,s′,τ′):=EZu(s,τ)Z′u(s′,τ′)ν(τ)ν(τ′) (8) =|us−u′s′|2H2(uτu′τ′)H(∣∣∣1+uτ(us−u′s′)∣∣∣2H−∣∣∣1+(uτ−u′τ′)(us−u′s′)∣∣∣2H+∣∣∣1−u′τ′(us−u′s′)∣∣∣2H−1).

By series expansion we find for any fixed and with ,

 |ru,u′(s,τ,s′,τ′)|≤|us−u′s′|2H(uτu′τ′)H2H|2H−1||us−u′s′|−2(uτu′τ′),2H≠1,

provided that and are sufficiently small. For , we have since the increments of Brownian motion on disjoint intervals are independent. Therefore,

 (9) r∗(t):=sup∣∣∣us−u′s′u∣∣∣,∣∣∣us−u′s′u′∣∣∣≥tτ1<τ,τ′<τ2u,u′,s,s′>0|ru,u′(s,τ,s′,τ′)|≤Kt−λ,

for , sufficiently large and some positive constant depending only on , and . Similarly, from (8) it follows that for any fixed there exists such that

 (10) 0<δ≤inf∣∣∣us−u′s′u∣∣∣,∣∣∣us−u′s′u′∣∣∣≤M|τ−τ∗|,|τ′−τ∗|≤mru,u′(s,τ,s′,τ′)≤sup∣∣∣us−u′s′u∣∣∣,∣∣∣us−u′s′u′∣∣∣≤M|τ−τ∗|,|τ′−τ∗|≤mru,u′(s,τ,s′,τ′)≤1−δ<1,

for sufficiently small .

### 2.1. Asymptotics

Due to the following lemma, while analyzing tail asymptotics of the supremum of , we can restrict the considered domain of to a strip with .

###### Lemma 1 (Piterbarg [14], Lemma 2 and 4).

There exists a positive constant such that for any ,

 (11) P⎛⎜ ⎜⎝sups∈[0,T]|τ−τ0|≥logv/vAZ(s,τ)>v⎞⎟ ⎟⎠≤CTv2/Hexp(−12v2−blog2v),

where . Furthermore, for any , with , as ,

 P⎛⎜ ⎜⎝sups∈[0,T]|τ−τ0|≤logv/vAZ(s,τ)>v⎞⎟ ⎟⎠=√πa2Hb−12H2BHTv2H−1Ψ(v)(1+o(1)),

where

 HBH=limT→∞T−1Eexp(supt∈[0,T](√2BH(t)−t2H))∈(0,∞),

is the so-called Pickands’ constant. This holds also for , with .

Hüsler and Piterbarg [9, Corollary 2] showed that the above actually holds true for depending on such that , for any and .

### 2.2. Discretization

Let and . For a fixed and some , let us define a discretization of the set as follows

 sl =lq(v),0≤l≤L,L=[T/q(v)],q(v)=θv−1H, τn =τ0+nq(v),0≤|n|≤N,N=[τ∗(v)/q(v)].

Along the same lines as in [10, Lemma 6] we get the following lemma.

###### Lemma 2.

There exist positive constants , such that, for any and ,

 P ⎛⎜ ⎜⎝max0≤l≤L0≤|n|≤NAZ(sl,τn)≤v−θH2v,sups∈[0,T]τ∈J(v)AZ(s,τ)>v⎞⎟ ⎟⎠≤K1v2H−1Ψ(v)θH2exp(−θ−H/K2).

Finally, it is possible to approximate tail asymptotics of the supremum of on the strip by maximum taken over discrete time points. The proof of the following lemma follows line-by-line \textcolorblackthe same as the proof of [14, Lemma 4] and thus we omit it. Similar result can be found in, e.g., [10, Lemma 7].

###### Lemma 3.

For any , as ,

 P⎛⎜⎝max0≤l≤L0≤|n|≤NAZ(sl,τn)>v⎞⎟⎠=√πa2Hb−12(HθBH)2Tv2H−1Ψ(v)(1+o(1)),

where .

It follows easily that as , so that the above asymptotics \textcolorblackis the same as in Lemma 1 when the discretization parameter decreases to zero so that the number of discretization points grows to infinity.

## 3. Auxiliary Lemmas

We begin with some auxiliary lemmas that are later needed in the proofs. The first lemma is \textcolorblacka slightly modified version of [11, Theorem 4.2.1].

###### Lemma 4 (Berman’s inequality).

Suppose \textcolorblackthat are normal \textcolorblackrandom variables with \textcolorblackcorrelation matrix and similarly with \textcolorblackcorrelation matrix . Let \textcolorblack, \textcolorblackand be real numbers, . Then,

 P(n⋂j=1{ξj≤uj}) −P(n⋂j=1{ηj≤uj}) ≤12π∑1≤i

The following lemma is a general form of the Borel-Cantelli lemma; cf. [17].

###### Lemma 5 (Borel-Cantelli lemma).

Consider a sequence of \textcolorblackevents . If

 ∞∑k=0P(Ek)<∞,

then . Whereas, if

 ∞∑k=0P(Ek)=∞andliminfn→∞∑1≤k≠t≤nP(EkEt)(∑nk=1P(Ek))2≤1,

then .

###### Lemma 6.

For any , there exist positive constants and depending only on and such that

 P(supSfp(u))du)+KS−ρ,

for any , with and being some universal positive constant.

###### Proof.

Let be some positive constant. For the reminder of the proof let and be two positive constants depending only on and that may differ from line to line. For any put , , , and

 sk =tk−1+εxk−1,yk=fp(sk),tk=sk+yk,xk=fp(tk), (12) Ik =(sk,tk],vk=Ax1−Hk=vfp(tk),~Ik=Ikxk=(~sk,~tk],|~Ik|=ykxk.

From this construction, it is easy to see that the intervals are disjoint. Furthermore, , and , for any and sufficiently large . Note that, for any , as grows large, therefore if is the smallest number of intervals needed to cover , then . Moreover, since is bounded by the constant not depending on and , it follows that, for any .

Now let us introduce a discretization of the set as in Section 2.2. That is, for some , define grid points

 sk,l =~sk+lqk,0≤l≤Lk,Lk=[(1−ε)/qk],qk=θv−1Hk, τk,n =τ0+nqk,0≤|n|≤Nk,Nk=[τ∗(vk)/qk].

Since is an increasing function, it easily follows that,

 P ≤P⎛⎜⎝T(S,ε)⋂k=0⎧⎪⎨⎪⎩max0≤l≤Lk0≤|n|≤NkAZxk(sk,l,τk,n)≤vk⎫⎪⎬⎪⎭⎞⎟⎠ ≤T(S,ε)∏k=0P⎛⎜⎝max0≤l≤Lk0≤|n|≤NkAZxk(sk,l,τk,n)≤vk⎞⎟⎠+∑0≤t

where the last inequality follows from Berman’s inequality with

 Ck,t=∑0≤l≤Lk0≤p≤Lt∑|n|≤Nk|m|≤Nt|rxk,xt(sk,l,τk,n,st,p,τt,m)|√1−r2xk,xt(sk,l,τk,n,st,p,τt,m)exp⎛⎝−12(v2k+v2t)1+|rxk,xt(sk,l,τk,n,st,p,τt,m)|⎞⎠.

Estimate of .

Note that we can use the fact that has the same distribution as for any . Since the process is stationary \textcolorblackwith respect to the first variable, from Lemma 3, for any , sufficiently large and small ,

 P1 ≤exp⎛⎜⎝−T(S,ε)∑k=0P⎛⎜⎝max0≤l≤Lk0≤|n|≤NkAZxk(sk,l,τk,n)>vk⎞⎟⎠⎞⎟⎠ ≤exp⎛⎝−(1−ε4)T(S,ε)∑k=0P(sup(s,τ)∈~Ik×J(vk)AZ(s,τ)>vk)⎞⎠

Then, by (7) combined with (3),

 P1 ≤exp⎛⎜⎝−(1−ε2)T(S,ε)∑k=0P⎛⎜⎝sups∈~Ikτ≥0AZ(s,τ)>vk⎞⎟⎠⎞⎟⎠ =exp⎛⎜⎝−(1−ε2)T(S,ε)∑k=0P⎛⎜⎝supt∈[0,ykxkfp(tk)]QBH(t)>fp(tk)⎞⎟⎠⎞⎟⎠ ≤exp⎛⎝−(1−ε)T(S,ε)∑k=0P(supt∈[0,fp(tk)]QBH(t)>fp(tk))fp(sk)fp(tk)⎞⎠ ≤exp(−1−ε1+ε∫TS+fp(S)1fp(u)P(supt∈[0,fp(u)]QBH(t)>fp(u))du).

Estimate of .

For any , , , we have

 xksk,l−xtst,p =(sk+xklqk)−(st+xtpqt) =k−1∑i=t(yi+εxi)+xklqk−xtpqt≥k−1∑i=t(yi+εxi)−xt(1−ε) ≥(yt+εxt)(k−t)−xt(1−ε)≥xt(k−t)ε,

where the last inequality holds provided that with sufficiently large. Therefore, c.f. (9),

 r∗k,t:=sup0≤l≤Lk,0≤p≤Lt|n|≤Nk,|m|≤Nt|rxk,xt(sk,l,τk,n,st,p,τt,m)|≤r∗((k−t)ε)≤K(k−t)−λ≤min(1,λ)/4.

Moreover, from (10) it follows that, for any , there exists a constant depending only on such that for sufficiently large ,

 sup0≤l≤Lk,0≤p≤Lt|n|≤Nk,|m|≤Nt|rxk,xt(sk,l,τk,n,st,p,τt,m)|≤ζ<