Dynamical Processes and Systems of Fractional OrderView this Special Issue
Power Spectrum of Generalized Fractional Gaussian Noise
Recently, we introduced a type of autocorrelation function (ACF) to describe a long-range dependent (LRD) process indexed with two parameters, which takes standard fractional Gaussian noise (fGn for short) as a special case. For simplicity, we call it the generalized fGn (GfGn). This short paper gives the power spectrum density function (PSD) of GfGn.
LRD time series increasingly gains applications to many fields of science and technologies; see, for example, Mandelbrot  and references therein. In this regard, standard fGn introduced by Mandelbrot and van Ness is a widely used tool for modeling LRD time series; see, for example, Beran , Abuzeid et al. [3, 4], and Liao et al. . Following [1, H11], , its ACF is given by where is the Hurst parameter and . It implies three families of time series. In the case of , is nonintegrable, and a corresponding series is LRD. For , is integrable, and a corresponding series is short-range dependent (SRD). The case of corresponds to white noise. Note that statistics of LRD series substantially differ from SRD ones. From a practice view, SRD fGn may be less interesting in applications as can be seen from [1, 2]. This paper only considers LRD series unless otherwise stated.
Li  recently introduced an ACF form that is a generalization of ACF of fGn. Since ACF is an even function, we write ACF of GfGn by where and . We call a process whose ACF follows (2) GfGn for simplicity because it takes fGn as a special case of . Without loss of generality, the following considers the normalized ACF by letting . This paper aims at giving PSD of GfGn. The Fourier transform (FT) of is treated as a generalized function over Schwartz space of test functions since is nonintegrable.
2. PSD of GfGn
Denote where , and , . Denote , where means FT and . Then, FT of is given by
Corollary 2. equals .
Lemma 3 (binomial series). and can be expanded as where and are real number, and is binomial coefficient .
Corollary 4. and for can be expanded as
Proof. This corollary is straightforward from Lemma 3.
Corollary 5. For , and can be expanded as
Corollary 6. For and are given by (6), respectively,
Corollary 7. For and are given by (7), respectively,
Proposition 8. PSD of GfGn is given by
Considering the leading term of (8) results in the following proposition.
Proposition 9. PSD of GfGn has the following approximate value:
From (9), we can easily get the two notes below.
Note 1. is divergent at the origin for , which is the LRD condition. This is the basic feature of LRD process.
Note 2. Recall . Then, the cases of and imply . This explains the range of for GfGn from a view in the frequency domain.
We have derived PSD of GfGn. Its approximate expression has been given. The range of has been explained from a spectral view.
This work was supported in part by the National Natural Science Foundation of China under the Project Grant nos. 61272402, 61070214, and 60873264, and by the 973 Plan under the Project Grant no. 2011CB302800.
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