On the Long-Range Dependence of Fractional Brownian Motion

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.

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][47][48][49][50][51][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][2][3][4]45].The measure of the statistical dependence of fBm is characterized by the Hurst parameter  ∈ (0, 1).
The remaining paper is organized as follows.In Section 2, we describe that the range of  for fBm to be of LRD is  ∈ (0, 1) and  ̸ = 1/2.Discussions are in Section 3, which is followed by conclusions.
Without generality losing, we assume that () is a random function with mean zero.The autocorrelation function (ACF) of () is, for ,  ∈ (0, ∞), denoted by By LRD [1,2], we mean that If () is of short-range dependence (SRD).Denote by   (, ) the power spectrum density function (PSD) of ().Denote by  the operator of the Fourier transform.Then [60][61][62][63][64], The LRD condition described in the frequency domain is expressed by The above expression implies the property of Let  −V be the Weyl integral of order V > 0.Then, for random function (); see, for example, [71][72][73][74][75], one has Thus, the fBm of the Weyl type is in the form:  Following [76], the PSD of the fBm of the Weyl type is expressed by Therefore, we have the following theorem.
As a matter of fact, fBm reduces to the standard Bm if  = 1/2.The PSD of BM, see [11], is given by Thus, lim From the theorem, we have the following corollary.
In passing, we mention that the ACF of   () of the Weyl type is in the form: where   is the strength of   ().It is given by Following [57, page 4], we have the following remark.

Discussions
Let   () be the fractional Gaussian noise (fGn).Then, in the domain of generalized functions over the Schwartz space of test functions [45], we write Denote by      (, ) the ACF of   ().Then, for  > 0 [19,45], one has From the contents in Section 2, we have the following theorem.
Theorem 4. Let () be a random function.Then, the statistical dependence of ()/ may substantially differ from that of (), where the differential is in the domain of generalized functions.
From Theorem 4, we immediately obtain the corollary below.
Corollary 5. Let () be a random function.Then, the statistical dependence of  −1 () may substantially differ from that of (), where  −1 is the integral operator of order one.

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