1. Introduction
LRD time series increasingly gains applications to many fields of science and technologies; see, for example, Mandelbrot [1] 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 [2], Abuzeid et al. [3, 4], and Liao et al. [5]. Following [1, H11], [2], its ACF is given by
(1)ρ(τ)=r(τ;H)=σ22[(|τ|+1)2H-2|τ|2H+||τ|-1|2H],
where H is the Hurst parameter and σ2=Γ(1-2H)cos(Hπ)/Hπ. It implies three families of time series. In the case of H∈(0.5,1), ρ is nonintegrable, and a corresponding series is LRD. For H∈(0,0.5), r is integrable, and a corresponding series is short-range dependent (SRD). The case of H=0.5 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 [6] 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
(2)C(τ)=C(τ;H,α)=0.5σ2((|τ|α+1)2H-2(|τ|α)2H+||τ|α-1|2H),
where H∈(0.5,1) and α∈(0,1]. We call a process whose ACF follows (2) GfGn for simplicity because it takes fGn as a special case of C(τ;H,1)=ρ(τ;H). Without loss of generality, the following considers the normalized ACF by letting r(τ)=C(τ)/σ2. This paper aims at giving PSD of GfGn. The Fourier transform (FT) of r(τ) is treated as a generalized function over Schwartz space of test functions since r(τ) is nonintegrable.

2. PSD of GfGn
Denote
(3)r(τ)=0.5[r1(τ)-2r2(τ)+r3(τ)],
where r1=(|τ|a+1)2H, and r2=(|τ|α)2H, r3=||τ|a-1|2H. Denote Sm(ω)=F(rm), where F means FT and m=1,2,3. Then, FT of r(τ) is given by
(4)S(ω)=0.5[S1(ω)-2S2(ω)+S3(ω)].

Lemma 1 (see [<xref ref-type="bibr" rid="B7">7</xref>] or Gelfand and Vilenkin [<xref ref-type="bibr" rid="B8">8</xref>, Chapter 2]).
FT of |t|λ is expressed by
(5)F[|t|λ]=-sin(λπ2)Γ(λ+1)|ω|-λ-1,
where λ≠-1,-3,….

Corollary 2.
S
2
(
ω
)
equals -sin(Hαπ)Γ(2Hα+1)|ω|-2Hα-1.

Proof.
Note 2Hα≠-1,-3,…. Thus, doing F(|τ|2Hα) with (5) yields Corollary 2.

Lemma 3 (binomial series).
(
1
+
x
)
ν
and (1-x)ν can be expanded as
(6a)(1+x)ν=∑k=0∞(νk)xk=∑k=0∞Γ(ν+k)Γ(ν)Γ(1+k)xk for |x|<1,(6b)(1-x)ν=∑k=0∞(νk) (-x)k=∑k=0∞(-1)kΓ(ν+k)Γ(ν)Γ(1+k)xk for |x|<1,
where x and ν are real number, and (νk) is binomial coefficient [9].

Corollary 4.
r
1
(
τ
)
and r3(τ) for |τ|<1 can be expanded as
(7a)(1+|τ|α)2H=∑k=0∞(2Hk)|τ|αk=∑k=0∞Γ(2H+k)Γ(2H)Γ(1+k)|τ|αk,(7b)(1-|τ|α)2H=∑k=0∞(-1)k(2Hk)|τ|αk=∑k=0∞(-1)kΓ(2H+k)Γ(2H)Γ(1+k)|τ|αk.

Proof.
This corollary is straightforward from Lemma 3.

Corollary 5.
For |τ|α>1, r1(τ) and r3(τ) can be expanded as
(8a)(1+|τ|α)2H=|τ|2Hα∑k=0∞(2Hk)|τ|-αk=∑k=0∞Γ(2H+k)Γ(2H)Γ(1+k)|τ|α(2H-k),(8b)||τ|a-1|2H=|τ|2Hα∑k=0∞(-1)k(2Hk)|τ|-αk=∑k=0∞(-1)kΓ(2H+k)Γ(2H)Γ(1+k)|τ|α(2H-k).

Proof.
Since (1+|τ|α)2H=|τ|2Hα(1+|τ|-α)2H, according to (7a), (8a) results. Similarly, (8b) follows due to ||τ|a-1|2H=|τ|2Hα|1-|τ|-α|2H and (7b).

Corollary 6.
For |τ|<1,S1 and S3 are given by (6), respectively,
(6)S1(ω)=∑k=0∞-Γ(2H+k)Γ(αk+1)Γ(2H)Γ(1+k) ×sin(αkπ2)|ω|-αk-1,S3(ω)=∑k=0∞(-1)k+1Γ(2H+k)Γ(αk+1)Γ(2H)Γ(1+k) ×sin(αkπ2)|ω|-αk-1.

Proof.
Doing F[|τ|αk] term by term for (7a) and (7b) with Lemma 1 yields (6), respectively.

Corollary 7.
For |τ|α>1, S1 and S3 are given by (7), respectively,
(7)S1(ω)=∑k=0∞-Γ(2H+k)Γ[(α(2H-k)+1] Γ(2H)Γ(1+k) ×sin[α(2H-k)π2]|ω|-α(2H-k)-1,S3(ω)=∑k=0∞(-1)k+1Γ(2H+k)Γ[α(2H-k)+1]Γ(2H)Γ(1+k) ×sin[α(2H-k)π2]|ω|-α(2H-k)-1.

Proof.
Doing FTs for (8a) and (8b) based on Lemma 1 results in (7).

The following proposition is a consequence of Corollaries 2, 6, and 7.

Proposition 8.
PSD of GfGn is given by
(8)S(ω)={sin(Hαπ)Γ(2Hα+1)|ω|-2Hα-1+0.5∑k=0∞[(-1)k+1-1]Γ(2H+k)Γ(αk+1)Γ(2H)Γ(1+k) ×sin(αkπ2)|ω|-αk-1, |τ|<1,sin(Hαπ)Γ(2Hα+1)|ω|-2Hα-1+0.5∑k=0∞[(-1)k-1]Γ(2H+k)Γ[α(2H-k)+1]Γ(2H)Γ(1+k) ×sin[α(2H-k)π2]|ω|-α(2H-k)-1, |τ|α>1.

Considering the leading term of (8) results in the following proposition.

Proposition 9.
PSD of GfGn has the following approximate value:
(9)S(ω)≈sin(Hαπ)Γ(2Hα+1)|ω|-2Hα-1.

From (9), we can easily get the two notes below.

Note 1.
S
(
ω
)
is divergent at the origin for 0<2Hα+1<1, which is the LRD condition. This is the basic feature of LRD process.

Note 2.
Recall 2Hα+1>0. Then, the cases of 2Hα+1<1 and H∈(0.5,1) imply α∈(0,1]. This explains the range of α for GfGn from a view in the frequency domain.