## 2 Fractional Gaussian Noise and LRD Model  
Let \(X(t)\) be a random fractional Gaussian noise (FGN) series, then its auto-correlation function (ACF) denoted by \(r(\tau;\varepsilon,H)\) is given by [18],  
\[r(\tau;\varepsilon,H)=\frac{\sigma^{2}\varepsilon^{2H-2}}{2}[(\frac{|\tau|}{ \varepsilon}+1)^{2H}-2(\frac{|\tau|}{\varepsilon})^{2H}+(\frac{|\tau|}{ \varepsilon}-1)^{2H}] \tag{1}\]  
where \(0<H<1\) is the Hurst exponent, and \(\sigma^{2}=(H\pi)^{-1}\Gamma(1-2H)\cos(H\pi)\) is the intensity of FGN [19; 20]. Parameter \(\varepsilon>0\) is utilized by regularizing fractional Brownian motion (FBM) so that the regularized FBM is differentiable (see ref. [18]). Without loss of generality, let \(\varepsilon=1\), the ACF form of the FGN in the discrete case is expressed by,  
\[r(\tau;H)=\frac{\sigma^{2}}{2}[(|\tau|+1)^{2H}-2(|\tau|)^{2H}+(|\tau|-1)^{2H}] \tag{2}\]  
Generally, the random fractional Gaussian noise (FGN) series correspond to three subclasses of the time series, and the subclasses can be defined by the specific Hurst exponent \(H\): (a) LRD process with \(0.5<H<1\); (b) short-range dependence process (SRD) with \(0<H<0.5\); and (c) anti-persistence process (the time series displays a random walk behavior) with \(H=0.5\). Thus, the LRD time series is special case of the FGN.  
For a stochastic process \(x(t)\), its ACF is denoted by \(r_{xx}(\tau)=E[x(t)x(t+\tau)]\), and \(\tau\) denotes the time lag. When \(x(t)\) exhibits LRD characteristic, its ACF is non-integrable, that is \(\int_{0}^{\infty}r_{xx}(\tau)=\infty\), otherwise, \(x(t)\) exhibits SRD characteristic if the integral of ACF is finite, that is \(\int_{0}^{\infty}r_{xx}(\tau)<\infty\). From the perspective of asymptotic approximation, equivalently, two almost equivalent definitions with ACF and two-sided power spectral density (PSD) \(S_{x}(\tau)\) are as follow,  
\[r_{xx}(\tau)\sim c_{1}|\tau|^{-\beta},\ for\ (\tau\rightarrow\infty),\ 0<\beta<1 \tag{3}\]  
\[S_{x}(\tau)\sim c_{2}|\tau|^{-\alpha},\ for\ \tau\to 0,\ 0<\alpha<1 \tag{4}\]  
where \(c_{1}\) and \(c_{2}\) are constants. Note that the parameter \(\beta\) is the exponent of the LRD. As a matter of fact, the parameter \(\beta\) can be computed by Hurst \(H\), i.e., \(\beta=2-2H\) or \(H=1-\beta/2\).  
The \(f\)-ARIMA as a typical fractional order characteristic (FOC) model, which includes all three cases is the fractional ARIMA (\(p\), \(d\), \(q\)) process, i.e.,\(f\)-ARIMA (\(p\), \(d\), \(q\)), the \(f\)-ARIMA (\(p\), \(d\), \(q\)) formulation is described by [21; 22],  
\[\Phi(B)(1-B)^{d}x_{t}=\Theta(B)\varepsilon_{t} \tag{5}\]  
where \(0<d<0.5\), \(\varepsilon_{t}\) is a Gaussian white noise, \(B\) is the backshift operator, i.e., \(Bx_{t}=x_{t-1}\), \(Bx^{2}_{t}=x_{t-2},\cdots\). The function \(\Phi(B)\) and \(\Theta(B)\) are polynomials of the backshift operator \(B\)i.e., \(\Phi(B)=1-\phi_{1}B^{-1}-\phi_{2}B^{-2}-\cdots-\phi_{p}B^{-p}\), and \(\Theta(B)=1-\theta_{1}B^{-1}-\theta_{2}B^{-2}-\cdots-\theta_{q}B^{-q}\). Since \(d\) denotes fractional value, the operator \(\Delta^{d}=(1-B)^{d}\) is defined by binomial series, we have,  
\[\Delta^{d}=(1-B)^{d}=\sum_{k=0}^{\infty}\left(\begin{array}{c}d\\ k\end{array}\right)(-B)^{k}=\sum_{k=0}^{\infty}\frac{\Gamma(d+1)}{\Gamma(k+1) \Gamma(d-k+1)}(-B)^{k} \tag{6}\]  
where \(\Gamma\) is the Gamma function.  
It can be seen from the Equations (5) and (6) that \(d\) is a parameter to measure LRD feature, which can be calculated by the Hurst exponent; i.e., \(d=H-0.5\). Usually, the larger the Hurst exponent value, the stronger the long-range persistence. Besides, for the Hurst parameter \(H\), several estimators can be used to estimate the value of the \(H\), including the absolute value method (AVM), rescaled range (R/S) analysis method, Abry--Veitch, Whittle estimator, periodogram method, etc. The detailed formulas or estimation methodologies are presented in [23; 24]. In this paper, we will employ the R/S method to investigate the LRD characteristic of a bearing degradation time series.