On the Long-Range Dependence of Fractional Brownian Motion

Abstract

This paper clarifies that the fractional Brownian motion, 𝐵_𝐻(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 𝐵_𝐻(𝑡), in the domain of generalized functions, changes the LRD of 𝐵_𝐻(𝑡) to be of short-range dependence (SRD) when 0<𝐻<0.5.

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 parameter 𝐻∈(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 Section 2, 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 for 0<𝐻<1 except 𝐻=0.5

In what follows, a random function in general is denoted by 𝑋(𝑡) for 𝑡∈(0,∞). We denote 𝐵_𝐻(𝑡) for 𝑡∈(0,∞) as fBm with 𝐻∈(0,1).

Without generality losing, we assume that 𝑋(𝑡) is a random function with mean zero. The autocorrelation function (ACF) of 𝑋(𝑡) is, for 𝑡,𝑠∈(0,∞), denoted by 𝐶_𝑋𝑋(𝑡,𝑠)=𝐸[𝑋(𝑡)𝑋(𝑠)]. (1) By LRD [1, 2], we mean that ^∞₀𝐶_𝑋𝑋(𝑡,𝑠)𝑑𝑡=∞. (2) If ^∞₀𝐶_𝑋𝑋(𝑡,𝑠)𝑑𝑡<∞, (3)𝑋(𝑡) is of short-range dependence (SRD).

Denote by 𝑆_𝑋𝑋(𝜔,𝑡) the power spectrum density function (PSD) of 𝑋(𝑡). Denote by 𝐹 the operator of the Fourier transform. Then [60–64], 𝑆_𝑋𝑋(𝜔,𝑡)=𝐹𝐶_𝑋𝑋(𝑡,𝑠). (4) The LRD condition described in the frequency domain is expressed by lim_𝜔→0𝑆_𝑋𝑋(𝜔,𝑠)=∞. (5) 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 lim_𝜔→0𝑆_𝑋𝑋(𝜔,𝑠)<∞. (6)

Let 𝑊^−𝑣 be the Weyl integral of order 𝑣>0. Then, for random function 𝑋(𝑡); see, for example, [71–75], one has 𝑊^−𝑣𝑋(𝑡)=1Γ(𝑣)^∞_𝑡(𝑢−𝑡)^𝑣−1𝑋(𝑢)𝑑𝑢. (7) Thus, the fBm of the Weyl type is in the form: 𝐵_𝐻(𝑡)−𝐵_𝐻(0)=1Γ()×𝐻+1/2_0−∞(𝑡−𝑢)^𝐻−0.5−(−𝑢)^𝐻−0.5(𝑢)+𝑑𝐵_𝑡0(𝑡−𝑢)^𝐻−0.5𝑑𝐵(𝑢). (8) Following [76], the PSD of the fBm of the Weyl type is expressed by 𝑆_{𝐵𝐻𝐵𝐻}()=1𝜔,𝑡|𝜔|^2𝐻+11−2^1−2𝐻cos2𝜔𝑡. (9) Therefore, we have the following theorem.

Theorem 1. FBm is of LRD for 𝐻∈(0,1) except 𝐻=1/2.

Proof. Because lim_𝜔→0𝑆_{𝐵𝐻𝐵𝐻}(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 if 𝐻=1/2. The PSD of BM, see [11], is given by 𝑆_{𝐵1/2𝐵1/2}()=1𝑡,𝜔𝜔²(1−cos2𝜔𝑡). (10) Thus, lim_𝜔→0𝑆_{𝐵1/2𝐵1/2}(𝑡,𝜔)=2𝑡²≠∞. (11) From the theorem, we have the following corollary.

Corollary 2. FBm is not SRD for 𝐻∈(0,1).

In passing, we mention that the ACF of 𝐵_𝐻(𝑡) of the Weyl type is in the form: 𝐶_{𝐵𝐻𝐵𝐻}()=𝑉𝑡,𝑠^𝐻()Γ()×𝑡|𝐻+1/2𝐻+1/2|^2𝐻+|𝑠|^2𝐻−|𝑡−𝑠|^2𝐻, (12) where 𝑉_𝐻 is the strength of 𝐵_𝐻(𝑡). It is given by 𝑉_𝐻=Var𝐵_𝐻(1)=Γ(1−2𝐻)cos𝜋𝐻.𝜋𝐻 (13)

Following [57, page 4], we have the following remark.

Remark 3. The ACF of fBm is positively correlated for 𝐻∈(0,1) except 𝐻=1/2. That is, 𝐶_{𝐵𝐻𝐵𝐻}(𝑡,𝑠)≥0 for 𝑡,𝑠∈(0,∞). Figures 1 and 2 indicate the plots of 𝐶_{𝐵𝐻𝐵𝐻}(𝑡,𝑠) for 𝑡,𝑠=0,1,…,30 with 𝐻=0.2 and 0.4, respectively.

Figure 1 

ACF of fBm for 𝑡,𝑠=0,1,…,30 and 𝐻=0.2.

Figure 2 

ACF of fBm for 𝑡,𝑠=0,1,…,30 and 𝐻=0.4.

3. Discussions

Let 𝐺_𝐻(𝑡) be the fractional Gaussian noise (fGn). Then, in the domain of generalized functions over the Schwartz space of test functions [45], we write 𝐺_𝐻(𝑡)=𝑑𝐵_𝐻(𝑡).𝑑𝑡 (14) Denote by 𝐶_{𝐺𝐻𝐺𝐻}(𝜏,𝑠) the ACF of 𝐺_𝐻(𝑡). Then, for 𝜀>0 [19, 45], one has 𝐶_{𝐺𝐻𝐺𝐻}()=𝑉𝜏;𝜀^𝐻𝜀^2𝐻−22×|𝜏|𝜀+1^2𝐻+|𝜏|𝜀−1^2𝐻𝜏−2𝜀^2𝐻. (15) From the contents in Section 2, we have the following theorem.

Theorem 4. Let 𝑋(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. Let 𝑋(𝑡)=𝐵_𝐻(𝑡). Then, 𝑑𝑋(𝑡)/𝑑𝑡=𝐺_𝐻(𝑡). It is well known that fGn is LRD when 𝐻∈(0.5,1) as 𝐶_{𝐺𝐻𝐺𝐻} is nonintegrable if 𝐻∈(0.5,1). On the other hand, for 𝐻∈(0,0.5), the integral of 𝐶_{𝐺𝐻𝐺𝐻} is zero. Hence, fGn is SRD when 𝐻∈(0,0.5). In passing, we note that 𝐶_{𝐺𝐻𝐺𝐻}(𝜏;𝜀) changes its sign and becomes negative for some 𝜏 proportional to 𝜀 in this parameter domain [45, page 434]. Since 𝐵_𝐻(𝑡) is LRD for 𝐻∈(0,1) except 𝐻=1/2, the statistical dependence of 𝐺_𝐻(𝑡) substantially differs from that of 𝐵_𝐻(𝑡). This completes the proof.

From Theorem 4, we immediately obtain the corollary below.

Corollary 5. Let 𝑋(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 . Let 𝑋(𝑡)=𝐺_𝐻(𝑡). Then, 𝐷⁻¹𝑋(𝑡)=𝐵_𝐻(𝑡). Since 𝐵_𝐻(𝑡) is LRD for 𝐻∈(0,0.5) while 𝐺_𝐻(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 for 𝐻∈(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.

Acknowledgments

This work was supported in part by the 973 plan under the project Grant no. 2011CB302800 and by the National Natural Science Foundation of China under the project Grants nos. 61272402, 61070214, and 60873264.