# On the Long-Range Dependence of Fractional Brownian Motion

Authors: Ming Li
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper clarifies that the fractional Brownian motion,BH(t), is of long-range dependence (LRD) for the Hurst parameter 0<H<1 except H=1/2. In addition, we note that the fractional Brownian motion is positively correlated for 0<H<1 except H=1/2. Moreover, we present a theorem to state that the differential or integral of a random function, X(t), may substantially change the statistical dependence of X(t). One example is that the differential of BH(t), in the domain of generalized functions, changes the LRD of BH(t) to be of short-range dependence (SRD) when 0<H<0.5.


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## Body

## 1. Introduction

Fractional Brownian motion (fBm) is widely used [1–10]. Its theory and applications attract the interests of researchers in various fields, ranging from telecommunications to biomedical engineering; see, for example, [11–44], simply citing a few.There is a set of statistical properties of fBm, such as nonstationarity and being nondifferentiable in the domain of ordinary functions [45]. Two properties, namely, nonstationarity and nondifferentiable property, are the basic properties of standard Brownian motion (Bm) [46–52], which is well known in the fields of time series as well as stochastic processes [53, 54]. As the substantial generalization of Bm, fBm has a property that Bm lacks, that is, its statistical dependence [1–4, 45]. The measure of the statistical dependence of fBm is characterized by the Hurst parameter H∈(0,1).Note that the fBm for the Hurst parameterH∈(0,1) and H≠1/2 is of LRD [11, 12, 45, 55, 56]. In addition, fBm is positively correlated for H∈(0,1) but H≠1/2 [57]. However, the LRD property of fBm may be sometimes conservatively expressed. For example, the LRD property of fBm was restricted by H∈(0.5,1) as can be seen from [58, page 2341] and [59, page 708]. For this reason, it may be meaningful to clarify, which this paper aims at.The remaining paper is organized as follows. In Section2, we describe that the range of H for fBm to be of LRD is H∈(0,1) and H≠1/2. Discussions are in Section 3, which is followed by conclusions.


## 2. FBm Is LRD for0<H<1 except H=0.5

In what follows, a random function in general is denoted byX(t) for t∈(0,∞). We denote BH(t) for t∈(0,∞) as fBm with H∈(0,1).Without generality losing, we assume thatX(t) is a random function with mean zero. The autocorrelation function (ACF) of X(t) is, for t,s∈(0,∞), denoted by
(1)CXX(t,s)=E[X(t)X(s)].
By LRD [1, 2], we mean that
(2)∫0∞CXX(t,s)dt=∞.
If
(3)∫0∞CXX(t,s)dt<∞,X(t) is of short-range dependence (SRD).Denote bySXX(ω,t) the power spectrum density function (PSD) of X(t). Denote by F the operator of the Fourier transform. Then [60–64],
(4)SXX(ω,t)=F[CXX(t,s)].
The LRD condition described in the frequency domain is expressed by
(5)limω→0SXX(ω,s)=∞.
The above expression implies the property of 1/f noise regarding random functions with LRD [1–4, 65–70]. On the other hand, X(t) is of SRD if
(6)limω→0SXX(ω,s)<∞.LetW-v be the Weyl integral of order v>0. Then, for random function X(t); see, for example, [71–75], one has
(7)W-vX(t)=1Γ(v)∫t∞(u-t)v-1X(u)du.
Thus, the fBm of the Weyl type is in the form:
(8)BH(t)-BH(0)=1Γ(H+1/2)×{∫-∞0[(t-u)H-0.5-(-u)H-0.5]dB(u)+∫0t(t-u)H-0.5dB(u)}.
Following [76], the PSD of the fBm of the Weyl type is expressed by
(9)SBHBH(ω,t)=1|ω|2H+1(1-21-2Hcos2ωt).
Therefore, we have the following theorem.Theorem 1.
FBm is of LRD forH∈(0,1) except H=1/2.Proof.
Becauselimω→0SBHBH(ω,t)=∞ for all t>0 and for H∈(0,1) except H=1/2, the theorem holds.As a matter of fact, fBm reduces to the standard Bm ifH=1/2. The PSD of BM, see [11], is given by
(10)SB1/2B1/2(t,ω)=1ω2(1-cos2ωt).
Thus,
(11)limω→0SB1/2B1/2(t,ω)=2t2≠∞.
From the theorem, we have the following corollary.Corollary 2.
FBm is not SRD forH∈(0,1).In passing, we mention that the ACF ofBH(t) of the Weyl type is in the form:
(12)CBHBH(t,s)=VH(H+1/2)Γ(H+1/2)×[|t|2H+|s|2H-|t-s|2H],
where VH is the strength of BH(t). It is given by
(13)VH=Var[BH(1)]=Γ(1-2H)cosπHπH.Following [57, page 4], we have the following remark.Remark 3.
The ACF of fBm is positively correlated forH∈(0,1) except H=1/2. That is, CBHBH(t,s)≥0 for t,s∈(0,∞). Figures 1 and 2 indicate the plots of CBHBH(t,s) for t,s=0,1,…,30 with H=0.2 and 0.4, respectively.Figure 1
ACF of fBm fort,s=0,1,…,30 and H=0.2.Figure 2
ACF of fBm fort,s=0,1,…,30 and H=0.4.


## 3. Discussions

LetGH(t) be the fractional Gaussian noise (fGn). Then, in the domain of generalized functions over the Schwartz space of test functions [45], we write
(14)GH(t)=dBH(t)dt.
Denote by CGHGH(τ,s) the ACF of GH(t). Then, for ε>0 [19, 45], one has
(15)CGHGH(τ;ε)=VHε2H-22×[(|τ|ε+1)2H+||τ|ε-1|2H-2|τε|2H].
From the contents in Section 2, we have the following theorem.Theorem 4.
LetX(t) be a random function. Then, the statistical dependence of dX(t)/dt may substantially differ from that of X(t), where the differential is in the domain of generalized functions.Proof.
To prove the theorem, we only need an example to show it. LetX(t)=BH(t). Then, dX(t)/dt=GH(t). It is well known that fGn is LRD when H∈(0.5,1) as CGHGH is nonintegrable if H∈(0.5,1). On the other hand, for H∈(0,0.5), the integral of CGHGH is zero. Hence, fGn is SRD when H∈(0,0.5). In passing, we note that CGHGH(τ;ε) changes its sign and becomes negative for some τ proportional to ε in this parameter domain [45, page 434]. Since BH(t) is LRD for H∈(0,1) except H=1/2, the statistical dependence of GH(t) substantially differs from that of BH(t). This completes the proof.From Theorem4, we immediately obtain the corollary below.Corollary 5.
LetX(t) be a random function. Then, the statistical dependence of D-1X(t) may substantially differ from that of X(t), where D-1 is the integral operator of order one.Proof .
LetX(t)=GH(t). Then, D-1X(t)=BH(t). Since BH(t) is LRD for H∈(0,0.5) while GH(t) is SRD when H∈(0,0.5), one sees that the statistical dependence of D-1X(t) substantially differs from that of X(t). Thus, Corollary 5 results.


## 4. Conclusions

We have clarified that fBm is LRD and positively correlated forH∈(0,1) except H=1/2. In addition, we have proved that the differential or integral of a random function may considerably change its statistical dependence.



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*Source: 842197-2013-05-30.xml*