---
title: Time-Changed Generalized Fractional Skellam Process-I
url: https://www.emergentmind.com/topics/time-changed-generalized-fractional-skellam-process-i-tcgfsp-i
type: topic
---

# Time-Changed Generalized Fractional Skellam Process-I

Searching arXiv for the cited papers to ground the article in current arXiv records.
Searching arXiv for “Non-Homogeneous Generalized Fractional Skellam Process”.
Time-Changed Generalized Fractional Skellam Process-I (TCGFSP-I) is, in the convention adopted in "Non-Homogeneous Generalized Fractional Skellam Process," the non-homogeneous generalized Skellam process evaluated at an independent inverse $\alpha$-stable clock, that is,
$$
S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,
$$
with $S_\alpha(t)=S(t)$ when $\alpha=1$ [2407.19227]. In this form, the process combines signed jumps of sizes $1,\dots,k$, deterministic time-varying rate functions, and inverse-stable temporal randomization. The resulting model is integer-valued, over-dispersed, and capable of exhibiting long memory. The label "TCGFSP-I" is not fully standardized across the Skellam-process literature: some papers use it for this inverse-stable non-homogeneous model, whereas others apply the same name to further subordinated generalized fractional Skellam processes [2510.26156].

## 1. Terminology and position in the literature

The nomenclature surrounding TCGFSP-I varies across recent papers. In the non-homogeneous framework of [2407.19227], the requested process is explicitly identified with the Non-homogeneous Generalized Fractional Skellam Process (NGFSP). In a later paper titled "Time-changed generalized fractional Skellam process," TCGFSP-I instead denotes a generalized fractional Skellam process further time-changed by an independent Lévy subordinator. Earlier work on Skellam type processes also used "Type I" for legwise independent stable subordination rather than for inverse-stable time change of the whole Skellam process [2003.09471].

| Source | Name used | Construction |
|---|---|---|
| [2407.19227] | TCGFSP-I $\equiv$ NGFSP | $S_\alpha(t)=S(Y_\alpha(t))$ |
| [2510.26156] | TCGFSP-I | $Z_f^\alpha(t)=S^\alpha(D_f(t))=S(Y_\alpha(D_f(t)))$ |
| [2003.09471] | Type I space-fractional Skellam | $N_1(D_{\alpha_1}(t))-N_2(D_{\alpha_2}(t))$ |

Within the convention of [2407.19227], the process is built from a non-homogeneous generalized Skellam process $S(t)$, not from two separately fractionalized counting components. This distinction matters: the paper states that the fractionalization is performed by time-changing the entire NGSP with the inverse stable clock.

## 2. Construction and one-dimensional distributions

Let $k\in\mathbb{N}$ be fixed. A non-homogeneous generalized counting process (NGCP) $M(t)$ has time-dependent intensities $\lambda_j(t)\ge 0$ with cumulative rates
$$
\Lambda_j(t)=\int_0^t \lambda_j(u)\,du<\infty,\qquad j=1,\dots,k.
$$
If $M_1(t)$ and $M_2(t)$ are independent NGCPs with intensities $\lambda_j(t)$ and $\gamma_j(t)$ and cumulative rates $\Lambda_j(t)$ and
$$
T_j(t)=\int_0^t \gamma_j(u)\,du,
$$
then the non-homogeneous generalized Skellam process is
$$
S(t)=M_1(t)-M_2(t), \qquad t\ge 0.
$$
It is convenient to write
$$
A(t)=\sum_{j=1}^k \Lambda_j(t), \qquad B(t)=\sum_{j=1}^k T_j(t).
$$
The inverse $\alpha$-stable clock is defined from an $\alpha$-stable subordinator $D_\alpha$ with Laplace transform
$$
\mathbb{E}[e^{-sD_\alpha(t)}]=e^{-t s^\alpha}, \qquad 0<\alpha<1,
$$
by
$$
Y_\alpha(t)=\inf\{x\ge 0:D_\alpha(x)>t\}.
$$
Its density $h_\alpha(x,t)$ satisfies
$$
\widetilde h_\alpha(x,s)=s^{\alpha-1}e^{-x s^\alpha}, \qquad h_\alpha(0+,t)=\frac{t^{-\alpha}}{\Gamma(1-\alpha)}.
$$
Assuming independence of $Y_\alpha$ and $S$, the TCGFSP-I is
$$
S_\alpha(t)=S(Y_\alpha(t)) \quad (0<\alpha<1).
$$
These definitions and conventions are given in [2407.19227].

The probability generating function of the driving NGSP is
$$
G_S(u,t)=\mathbb{E}[u^{S(t)}]
=\exp\!\Big(\sum_{j=1}^k \big(\Lambda_j(t)(u^j-1)+T_j(t)(u^{-j}-1)\big)\Big).
$$
Its state probabilities have the Bessel form
$$
p(n,t)=\Pr\{S(t)=n\}
=e^{-(A+B)}\Big(\frac{A}{B}\Big)^{|n|/2} I_{|n|}\!\big(2\sqrt{AB}\big), \qquad n\in\mathbb{Z},
$$
where $I_\nu$ is the modified Bessel function of the first kind [2407.19227].

Conditioning on the inverse stable clock yields the pgf of TCGFSP-I:
$$
G_{S_\alpha}(u,t)=\int_0^\infty G_S(u,x)\,h_\alpha(x,t)\,dx.
$$
Its probability mass function is the corresponding mixture
$$
p^{(\alpha)}(n,t)=\Pr\{S_\alpha(t)=n\}
=\int_0^\infty p(n,x)\,h_\alpha(x,t)\,dx,
$$
that is,
$$
p^{(\alpha)}(n,t)
=\int_0^\infty e^{-(A(x)+B(x))}
\Big(\frac{A(x)}{B(x)}\Big)^{|n|/2}
I_{|n|}\!\big(2\sqrt{A(x)B(x)}\big)\,h_\alpha(x,t)\,dx.
$$
Using the Wright-function representation $h_\alpha(x,t)=t^{-\alpha}M_\alpha(x/t^\alpha)$, the same formula can be written as a Wright mixture [2407.19227].

## 3. Governing equations and structural representations

The NGSP pgf satisfies
$$
\frac{\partial}{\partial t}G_S(u,t)
=
G_S(u,t)\sum_{j=1}^k\Big(\dot\Lambda_j(t)(u^j-1)+\dot T_j(t)(u^{-j}-1)\Big),
$$
where the dot denotes time derivative. At the level of state probabilities, $p(n,t)$ satisfies a differential-difference equation with four terms, involving $p(n-1,t)$, $p(n,t)$, and $p(n+1,t)$, and coefficients depending on $\lambda_j(t)$, $\gamma_j(t)$, $A(t)$, and $B(t)$, with initial data $p(0,0)=1$ and $p(n,0)=0$ for $n\neq 0$ [2407.19227].

For the fractional process, the Caputo derivative appears through the inverse-subordinator identity
$$
p^{(\alpha)}(n,t)=\int_0^\infty p(n,u)h_\alpha(u,t)\,du,
$$
together with
$$
\frac{d^\alpha}{dt^\alpha}p^{(\alpha)}(n,t)
=
\int_0^\infty h_\alpha(u,t)\,\frac{d}{du}p(n,u)\,du.
$$
The derivation uses the relations
$$
D_t^\alpha h_\alpha(u,t)=-\frac{\partial}{\partial u}h_\alpha(u,t),
\qquad
h_\alpha(0+,t)=\frac{t^{-\alpha}}{\Gamma(1-\alpha)},
$$
and the resulting equation is the $h_\alpha$-average of the NGSP differential-difference system [2407.19227].

A basic analytical tool is the Laplace-transform link
$$
\mathcal{L}_t\{\mathbb{E}[f(Y_\alpha(t))]\}(s)
=
s^{\alpha-1}\int_0^\infty f(x)e^{-x s^\alpha}\,dx,
$$
valid for bounded measurable $f$. This identity encodes the inverse-subordinator subordination relation and underlies transform-based derivations of pgf, pmf, and moment formulas.

The paper also derives recurrence relations. For $n\ge 1$, the NGSP state probabilities satisfy
$$
p(n,t)=\frac{1}{n}\sum_{j=1}^k j\Big(\Lambda_j(t)\,p(n-j,t)-T_j(t)\,p(n+j,t)\Big).
$$
Weighted-sum representations are also available. The NGSP can be written as
$$
S(t)\overset{d}= \sum_{j=1}^k j\,S_j(t),
$$
where $S_j(t)=N_{1j}(t)-N_{2j}(t)$ and $N_{1j},N_{2j}$ are independent non-homogeneous Poisson processes with cumulative intensities $\Lambda_j(t)$ and $T_j(t)$. Correspondingly,
$$
S_\alpha(t)\overset{d}= \sum_{j=1}^k j\,S_j(Y_\alpha(t)),
$$
so TCGFSP-I is a weighted sum of independent non-homogeneous fractional Skellam processes [2407.19227].

A further structural property is the martingale characterization:
$$
S(t)-\sum_{j=1}^k j\big(\Lambda_j(t)-T_j(t)\big)
$$
is a martingale with respect to the natural filtration [2407.19227].

## 4. Moments, over-dispersion, and dependence structure

For the NGSP,
$$
\mathbb{E}[S(t)] = \sum_{j=1}^k j\big(\Lambda_j(t)-T_j(t)\big),
$$
$$
\operatorname{Var}(S(t)) = \sum_{j=1}^k j^2\big(\Lambda_j(t)+T_j(t)\big),
$$
and
$$
\operatorname{Cov}(S(s),S(t))
=
\sum_{j=1}^k j^2\big(\Lambda_j(s\wedge t)+T_j(s\wedge t)\big).
$$
From these expressions,
$$
\operatorname{Var}(S(t))-\mathbb{E}[S(t)]
=
\sum_j \big[(j^2-j)\Lambda_j(t)+(j^2+j)T_j(t)\big] >0,
$$
so the NGSP is over-dispersed [2407.19227].

For TCGFSP-I,
$$
\mathbb{E}[S_\alpha(t)]
=
\sum_{j=1}^k j\,\mathbb{E}\big[\Lambda_j(Y_\alpha(t))-T_j(Y_\alpha(t))\big],
$$
and the variance and covariance contain two distinct contributions: one from the randomized operational time through expectations of $\Lambda_j(Y_\alpha(t))+T_j(Y_\alpha(t))$, and another from the variance and covariance of the time-changed cumulative rates themselves. The paper states
$$
\operatorname{Var}(S_\alpha(t))-\mathbb{E}[S_\alpha(t)]\ge 0,
$$
hence the NGFSP is also over-dispersed [2407.19227].

For non-homogeneous Weibull-type rates,
$$
\Lambda_j(t)=\Big(\frac{t}{b_{1j}}\Big)^{c_{1j}}, \qquad
T_j(t)=\Big(\frac{t}{b_{2j}}\Big)^{c_{2j}},
$$
with $c_{1j}\ge c_{2j}\ge 0$, the NGSP correlation satisfies
$$
\operatorname{Corr}(S(s),S(t))
=
\sqrt{\frac{\sum_{j=1}^k j^2(\Lambda_j(s)+T_j(s))}
{\sum_{j=1}^k j^2(\Lambda_j(t)+T_j(t))}}
\sim C_0\, t^{-d/2},
$$
where
$$
d:=\min_j c_{2j}.
$$
Accordingly, the NGSP exhibits long-range dependence if $d\in(0,2)$ and short-range dependence if $d\in(2,\infty)$ [2407.19227].

For TCGFSP-I, the inverse subordinator has slowly decaying covariance and introduces long memory. The paper states that precise LRD/SRD classification for the NGFSP depends on the rate functions $\{\Lambda_j,T_j\}$ [2407.19227]. This suggests that the inverse-stable clock supplies the memory mechanism, while the non-homogeneous rates determine its observable decay regime.

## 5. Passage times, increments, and alternative fractional variants

Arrival and passage-time distributions are available in explicit series-integral form. For the NGSP, if
$$
\tau_n:=\inf\{t\ge 0:S(t)=n\},
$$
then
$$
F_{\tau_n}(t)=\Pr(\tau_n\le t)
=
\sum_{x=n}^\infty e^{-(A+B)}
\Big(\frac{A}{B}\Big)^{|x|/2} I_{|x|}(2\sqrt{AB}).
$$
For the first upcrossing time
$$
T_n:=\inf\{s\ge 0:S(s)\ge n\},
$$
one has
$$
\Pr\{T_n>t\}
=
\sum_{x=0}^{n-1} e^{-(A+B)}
\Big(\frac{A}{B}\Big)^{|x|/2} I_{|x|}(2\sqrt{AB}).
$$
The corresponding TCGFSP-I formulas are obtained by mixing these expressions against $h_\alpha(u,t)$ [2407.19227].

Increment processes are also studied. For
$$
I(t,v)=S(t+v)-S(v),
$$
the marginal pmf again has a Bessel form with delayed cumulative rates. For the fractional increments,
$$
I_\alpha(t,v)=S_\alpha(t+v)-S_\alpha(v)=S(Y_\alpha(t)+v)-S(v),
$$
the distribution is
$$
\Pr\{I_\alpha(t,v)=\ell\}
=
\int_0^\infty \Pr\{I(u,v)=\ell\}\,h_\alpha(u,t)\,du,
$$
and the increment pmf satisfies an analogous fractional differential-integral equation [2407.19227].

An important companion model in the same paper is the alternative non-homogeneous generalized fractional Skellam process
$$
\widetilde S^\alpha(t)=\widetilde M_1^\alpha(t)-\widetilde M_2^\alpha(t),
$$
built from two independent NHGFCPs driven by a non-homogeneous time-fractional Poisson clock. Its one-dimensional distributions are given in closed series form through the Prabhakar-Mittag-Leffler function, in contrast to the integral-mixture form of $S_\alpha(t)$ [2407.19227]. This distinction is often useful in applications: the original TCGFSP-I offers a direct inverse-stable time-change representation, whereas the alternative variant offers closed-form series expressions.

## 6. Special cases, simulation, and broader connections

Several limiting and structural reductions are immediate. As $\alpha\to 1$,
$$
S_\alpha(t)\to S(t),
$$
so the TCGFSP-I reduces to the NGSP. If $\lambda_j(t)\equiv \lambda_j$ and $\gamma_j(t)\equiv \gamma_j$, the NGSP reduces to the generalized Skellam process; for $k=1$ it reduces to the classical Skellam process. If $A(t)=B(t)$, or equivalently $\Lambda_j(t)=T_j(t)$ for all $j$, then $\mathbb{E}[S(t)]=0$ and the NGSP pmf is symmetric around $0$; the same symmetry is inherited by the TCGFSP-I mixture [2407.19227].

The paper provides simulation procedures. The inverse stable subordinator $Y_\alpha(t)$ is simulated via Kanter’s method for stable increments and inversion to obtain $Y_\alpha$. The TCGFSP-I is then simulated by first generating $Y_\alpha(t_i)$, then simulating the two NGCPs at the random times $Y_\alpha(t_i)$ using time-dependent rates such as Gompertz-Makeham or Weibull, and finally taking their difference [2407.19227].

An application is given to a high-frequency financial data set. The reported modeling advantages are threefold: multi-size jumps with size-specific intensities, non-homogeneous rates reflecting intraday or temporal variability, and dependence with long memory induced by the inverse stable subordinator. The paper specifically notes that the inverse-stable clock yields Mittag-Leffler inter-arrivals, which fit high-frequency trade durations better than exponential inter-arrivals [2407.19227].

The model sits within a broader Skellam-process program on arXiv. The homogeneous generalized fractional Skellam process $S_\alpha(t)=S(Y_\alpha(t))$ was studied earlier as the GFSP, where integral pmf representations, Mittag-Leffler pgf, long-range dependence, and non-infinite divisibility were established [2107.08307]. The generalized space-time fractional Skellam process
$$
S_\beta^\alpha(t)=S(D_\beta(Y_\alpha(t)))=S_\beta(Y_\alpha(t))
$$
extends the same inverse-time-change mechanism to a space-fractional base process [2504.08374]. A later paper then used the exact label TCGFSP-I for the further subordinated model
$$
Z_f^\alpha(t)=S^\alpha(D_f(t))=S(Y_\alpha(D_f(t))),
$$
deriving pgf, mgf, factorial moments, covariance asymptotics, a law-of-the-iterated-logarithm variant, and LRD criteria under moment assumptions on $D_f(t)$ [2510.26156]. These parallel usages show that TCGFSP-I designates a family of closely related but not identical constructions, all centered on generalized Skellam dynamics combined with inverse-stable or Lévy-subordinator time changes.

Source: https://www.emergentmind.com/topics/time-changed-generalized-fractional-skellam-process-i-tcgfsp-i