Time-Changed Generalized Fractional Skellam Process-I

• TCGFSP-I is a time-changed generalized fractional Skellam process defined by evaluating a non-homogeneous generalized Skellam process at an inverse α-stable clock, resulting in over-dispersion and flexible dependence structures.
• It integrates deterministic, time-varying rate functions with signed jumps of sizes 1 through k, making it applicable for modeling high-frequency financial data and other count-based phenomena.
• Analytical tools such as Laplace and Wright-transform methods yield its probability generating function and moment formulas, supporting both simulation and structural analysis.

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 $\alpha=1$ (Tathe et al., 2024). 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 (Khandakar et al., 30 Oct 2025).

1. Terminology and position in the literature

The nomenclature surrounding TCGFSP-I varies across papers. In the non-homogeneous framework of (Tathe et al., 2024), 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 (Gupta et al., 2020).

Source	Name used	Construction
(Tathe et al., 2024)	TCGFSP-I $\equiv$ NGFSP	$S_\alpha(t)=S(Y_\alpha(t))$
(Khandakar et al., 30 Oct 2025)	TCGFSP-I	$Z_f^\alpha(t)=S^\alpha(D_f(t))=S(Y_\alpha(D_f(t)))$
(Gupta et al., 2020)	Type I space-fractional Skellam	$N_1(D_{\alpha_1}(t))-N_2(D_{\alpha_2}(t))$

Within the convention of (Tathe et al., 2024), 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 $S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$0 be fixed. A non-homogeneous generalized counting process (NGCP) $S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$1 has time-dependent intensities $S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$2 with cumulative rates

$S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$3

If $S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$4 and $S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$5 are independent NGCPs with intensities $S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$6 and $S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$7 and cumulative rates $S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$8 and

$S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$9

then the non-homogeneous generalized Skellam process is

$S_\alpha(t)=S(t)$0

It is convenient to write

$S_\alpha(t)=S(t)$1

The inverse $S_\alpha(t)=S(t)$2-stable clock is defined from an $S_\alpha(t)=S(t)$3-stable subordinator $S_\alpha(t)=S(t)$4 with Laplace transform

$S_\alpha(t)=S(t)$5

by

$S_\alpha(t)=S(t)$6

Its density $S_\alpha(t)=S(t)$7 satisfies

$S_\alpha(t)=S(t)$8

Assuming independence of $S_\alpha(t)=S(t)$9 and $\alpha=1$0, the TCGFSP-I is

$\alpha=1$1

These definitions and conventions are given in (Tathe et al., 2024).

The probability generating function of the driving NGSP is

$\alpha=1$2

Its state probabilities have the Bessel form

$\alpha=1$3

where $\alpha=1$4 is the modified Bessel function of the first kind (Tathe et al., 2024).

Conditioning on the inverse stable clock yields the pgf of TCGFSP-I:

$\alpha=1$5

Its probability mass function is the corresponding mixture

$\alpha=1$6

that is,

$\alpha=1$7

Using the Wright-function representation $\alpha=1$8, the same formula can be written as a Wright mixture (Tathe et al., 2024).

3. Governing equations and structural representations

The NGSP pgf satisfies

$\alpha=1$9

where the dot denotes time derivative. At the level of state probabilities, $1,\dots,k$0 satisfies a differential-difference equation with four terms, involving $1,\dots,k$1, $1,\dots,k$2, and $1,\dots,k$3, and coefficients depending on $1,\dots,k$4, $1,\dots,k$5, $1,\dots,k$6, and $1,\dots,k$7, with initial data $1,\dots,k$8 and $1,\dots,k$9 for $\equiv$0 (Tathe et al., 2024).

For the fractional process, the Caputo derivative appears through the inverse-subordinator identity

$\equiv$1

together with

$\equiv$2

The derivation uses the relations

$\equiv$3

and the resulting equation is the $\equiv$4-average of the NGSP differential-difference system (Tathe et al., 2024).

A basic analytical tool is the Laplace-transform link

$\equiv$5

valid for bounded measurable $\equiv$6. 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 $\equiv$7, the NGSP state probabilities satisfy

$\equiv$8

Weighted-sum representations are also available. The NGSP can be written as

$\equiv$9

where $S_\alpha(t)=S(Y_\alpha(t))$0 and $S_\alpha(t)=S(Y_\alpha(t))$1 are independent non-homogeneous Poisson processes with cumulative intensities $S_\alpha(t)=S(Y_\alpha(t))$2 and $S_\alpha(t)=S(Y_\alpha(t))$3. Correspondingly,

$S_\alpha(t)=S(Y_\alpha(t))$4

so TCGFSP-I is a weighted sum of independent non-homogeneous fractional Skellam processes (Tathe et al., 2024).

A further structural property is the martingale characterization:

$S_\alpha(t)=S(Y_\alpha(t))$5

is a martingale with respect to the natural filtration (Tathe et al., 2024).

4. Moments, over-dispersion, and dependence structure

For the NGSP,

$S_\alpha(t)=S(Y_\alpha(t))$6

$S_\alpha(t)=S(Y_\alpha(t))$7

and

$S_\alpha(t)=S(Y_\alpha(t))$8

From these expressions,

$S_\alpha(t)=S(Y_\alpha(t))$9

so the NGSP is over-dispersed (Tathe et al., 2024).

For TCGFSP-I,

$Z_f^\alpha(t)=S^\alpha(D_f(t))=S(Y_\alpha(D_f(t)))$0

and the variance and covariance contain two distinct contributions: one from the randomized operational time through expectations of $Z_f^\alpha(t)=S^\alpha(D_f(t))=S(Y_\alpha(D_f(t)))$1, and another from the variance and covariance of the time-changed cumulative rates themselves. The paper states

$Z_f^\alpha(t)=S^\alpha(D_f(t))=S(Y_\alpha(D_f(t)))$2

hence the NGFSP is also over-dispersed (Tathe et al., 2024).

For non-homogeneous Weibull-type rates,

$Z_f^\alpha(t)=S^\alpha(D_f(t))=S(Y_\alpha(D_f(t)))$3

with $Z_f^\alpha(t)=S^\alpha(D_f(t))=S(Y_\alpha(D_f(t)))$4, the NGSP correlation satisfies

$Z_f^\alpha(t)=S^\alpha(D_f(t))=S(Y_\alpha(D_f(t)))$5

where

$Z_f^\alpha(t)=S^\alpha(D_f(t))=S(Y_\alpha(D_f(t)))$6

Accordingly, the NGSP exhibits long-range dependence if $Z_f^\alpha(t)=S^\alpha(D_f(t))=S(Y_\alpha(D_f(t)))$7 and short-range dependence if $Z_f^\alpha(t)=S^\alpha(D_f(t))=S(Y_\alpha(D_f(t)))$8 (Tathe et al., 2024).

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 $Z_f^\alpha(t)=S^\alpha(D_f(t))=S(Y_\alpha(D_f(t)))$9 (Tathe et al., 2024). 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

$N_1(D_{\alpha_1}(t))-N_2(D_{\alpha_2}(t))$0

then

$N_1(D_{\alpha_1}(t))-N_2(D_{\alpha_2}(t))$1

For the first upcrossing time

$N_1(D_{\alpha_1}(t))-N_2(D_{\alpha_2}(t))$2

one has

$N_1(D_{\alpha_1}(t))-N_2(D_{\alpha_2}(t))$3

The corresponding TCGFSP-I formulas are obtained by mixing these expressions against $N_1(D_{\alpha_1}(t))-N_2(D_{\alpha_2}(t))$4 (Tathe et al., 2024).

Increment processes are also studied. For

$N_1(D_{\alpha_1}(t))-N_2(D_{\alpha_2}(t))$5

the marginal pmf again has a Bessel form with delayed cumulative rates. For the fractional increments,

$N_1(D_{\alpha_1}(t))-N_2(D_{\alpha_2}(t))$6

the distribution is

$N_1(D_{\alpha_1}(t))-N_2(D_{\alpha_2}(t))$7

and the increment pmf satisfies an analogous fractional differential-integral equation (Tathe et al., 2024).

An important companion model in the same paper is the alternative non-homogeneous generalized fractional Skellam process

$N_1(D_{\alpha_1}(t))-N_2(D_{\alpha_2}(t))$8

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 $N_1(D_{\alpha_1}(t))-N_2(D_{\alpha_2}(t))$9 (Tathe et al., 2024). 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 $S(t)$0,

$S(t)$1

so the TCGFSP-I reduces to the NGSP. If $S(t)$2 and $S(t)$3, the NGSP reduces to the generalized Skellam process; for $S(t)$4 it reduces to the classical Skellam process. If $S(t)$5, or equivalently $S(t)$6 for all $S(t)$7, then $S(t)$8 and the NGSP pmf is symmetric around $S(t)$9; the same symmetry is inherited by the TCGFSP-I mixture (Tathe et al., 2024).

The paper provides simulation procedures. The inverse stable subordinator $S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$00 is simulated via Kanter’s method for stable increments and inversion to obtain $S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$01. The TCGFSP-I is then simulated by first generating $S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$02, then simulating the two NGCPs at the random times $S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$03 using time-dependent rates such as Gompertz-Makeham or Weibull, and finally taking their difference (Tathe et al., 2024).

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 (Tathe et al., 2024).

The model sits within a broader Skellam-process program on arXiv. The homogeneous generalized fractional Skellam process $S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$04 was studied earlier as the GFSP, where integral pmf representations, Mittag-Leffler pgf, long-range dependence, and non-infinite divisibility were established (Kataria et al., 2021). The generalized space-time fractional Skellam process

$S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$05

extends the same inverse-time-change mechanism to a space-fractional base process (Tathe et al., 11 Apr 2025). A later paper then used the exact label TCGFSP-I for the further subordinated model

$S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$06

deriving pgf, mgf, factorial moments, covariance asymptotics, a law-of-the-iterated-logarithm variant, and LRD criteria under moment assumptions on $S_\alpha(t)=S(Y_\alpha(t)), \qquad 0<\alpha<1,$07 (Khandakar et al., 30 Oct 2025). 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.