Time-Changed Generalized Fractional Skellam Process-II

• TCGFSP-II is an integer-valued stochastic process derived from the generalized fractional Skellam process using successive inverse clocks.
• It employs inverse stable and Lévy subordinators to introduce non-Markovian, renewal-driven dynamics in complex jump processes.
• The model extends classical counting processes by combining two layers of temporal randomization and governing via generalized Caputo operators.

Searching arXiv for papers on the generalized fractional Skellam process and time-changed variants. The Time-Changed Generalized Fractional Skellam Process-II (TCGFSP-II) is an integer-valued stochastic process obtained by inverse-subordinator time-changing the generalized fractional Skellam process. In the explicit nomenclature of Tathe and Ghosh, it is defined by

$$\bar Z_f^\alpha(t)=S^\alpha(H_f(t)), \qquad t\ge 0,$$

where $S^\alpha(t)$ is the generalized fractional Skellam process (GFSP), $H_f$ is the inverse of an independent Lévy subordinator $D_f$ with Bernstein exponent $f$, and $S^\alpha$ itself is obtained from the generalized Skellam process $S$ by the inverse stable time change $Y_\alpha$ (Khandakar et al., 30 Oct 2025). The construction combines two layers of temporal randomization: the inverse stable clock that produces the GFSP, and the inverse Lévy clock that produces the “Process-II” dynamics. A distinct literature also uses “Type-II” for a common-clock fractional Skellam construction based on $S(D_\beta(Y_\alpha(t)))$; that alternative usage is related but not identical and requires separate treatment (Tathe et al., 11 Apr 2025).

1. Genealogy and defining construction

The generalized Skellam process (GSP) is the difference of two independent generalized counting processes,

$S(t)=M_1(t)-M_2(t),$

with fixed jump-size range $S^\alpha(t)$0, rate vectors $S^\alpha(t)$1 and $S^\alpha(t)$2, and aggregate rates

$S^\alpha(t)$3

Its one-dimensional law is

$S^\alpha(t)$4

and its moment generating function is

$S^\alpha(t)$5

This is an integer-valued Lévy process (Khandakar et al., 30 Oct 2025).

The generalized fractional Skellam process $S^\alpha(t)$6, $S^\alpha(t)$7, is defined by

$S^\alpha(t)$8

where $S^\alpha(t)$9 is an inverse stable subordinator independent of $H_f$0. Its probability generating function is

$H_f$1

with $H_f$2 the Mittag–Leffler function (Khandakar et al., 30 Oct 2025).

A Lévy subordinator $H_f$3 is characterized by

$H_f$4

where the Bernstein function $H_f$5 has the Lévy–Khintchine representation

$H_f$6

with $H_f$7. Its inverse, or first-passage time process, is

$H_f$8

The TCGFSP-II is then

$H_f$9

and for $D_f$0 it reduces to the time-changed generalized Skellam process-II (TCGSP-II),

$D_f$1

Initial conditions are

$D_f$2

(Khandakar et al., 30 Oct 2025).

2. Distributional structure and transform formulas

The central transform formula for TCGFSP-II is its pgf: $D_f$3 This exhibits the process as a moment mixture of the GFSP pgf under the random clock $D_f$4 (Khandakar et al., 30 Oct 2025).

An equivalent conditioning representation is

$D_f$5

where $D_f$6 is the density of the inverse subordinator $D_f$7. For the non-fractional reduction $D_f$8, the state probabilities are

$D_f$9

with $f$0 the GSP pmf. In the same case, the paper gives the explicit series

$f$1

(Khandakar et al., 30 Oct 2025).

These formulas are structurally identical to the mixture representation for inverse-subordinator time-changed Skellam processes in the convolution-derivative framework: $f$2 where $f$3 is the inverse-subordinator density associated with a Bernstein exponent $f$4 (Buchak et al., 2018).

The paper does not explicitly provide moment generating function or characteristic function formulas for TCGFSP-II. It states, however, that the conditioning representation

$f$5

applies formally, with $f$6 obtained from the GFSP transform at deterministic time $f$7 (Khandakar et al., 30 Oct 2025).

3. Moments, covariance structure, and dependence

Let

$f$8

and define

$f$9

where $S^\alpha$0 is the Beta function. Then, for $S^\alpha$1,

$S^\alpha$2

$S^\alpha$3

and

$S^\alpha$4

(Khandakar et al., 30 Oct 2025).

These formulas extend the classical inverse-subordinator Skellam identities, where

$S^\alpha$5

to the generalized fractional setting (Buchak et al., 2018).

Two negative results are explicit. First, the paper does not provide explicit factorial moment formulas for TCGFSP-II; it only notes that they may be derived from the pgf expansion by conditioning on $S^\alpha$6. Second, long-range dependence and short-range dependence are established for TCGFSP-I under specific moment asymptotics, but not for TCGFSP-II. No LRD claim is made there for the inverse-subordinated model (Khandakar et al., 30 Oct 2025).

The non-Markovian character is nevertheless intrinsic to the construction. The inverse stable subordinator $S^\alpha$7 has non-Markovian, non-stationary, non-independent increments, and inverse time change generally destroys the Lévy and Markov increment structure (Khandakar et al., 30 Oct 2025). This suggests that TCGFSP-II should be regarded as a renewal-driven integer-valued process rather than as a Lévy process.

4. Governing equations and generalized Caputo operators

The analytic core of Process-II models is the inverse-subordinator density $S^\alpha$8. Under Condition I, namely $S^\alpha$9 and absolute continuity of the tail $S$0, the density satisfies

$S$1

with boundary and initial data

$S$2

where $S$3 is Toaldo’s generalized Riemann–Liouville derivative and

$S$4

relates it to the generalized Caputo derivative $S$5 (Khandakar et al., 30 Oct 2025).

At the level made explicit in the paper, the general inverse-subordination governing system is stated for the $S$6 reduction TCGSP-II. Its marginals $S$7 satisfy

$S$8

with

$S$9

For $Y_\alpha$0, $Y_\alpha$1, the generalized Caputo derivative reduces to the Caputo fractional derivative, recovering the fractional governing equation for the GFSP (Khandakar et al., 30 Oct 2025).

This inverse-subordinator mechanism sits within the broader convolution-derivative framework developed for time-changed Poisson and Skellam processes. For a Bernstein exponent $Y_\alpha$2, the generalized Caputo-type derivative is characterized by

$Y_\alpha$3

and, in the classical Skellam case,

$Y_\alpha$4

The corresponding bilateral pgf $Y_\alpha$5 obeys

$Y_\alpha$6

(Buchak et al., 2018).

5. Reductions and specific inverse subordinators

Several reductions organize the model hierarchy. Setting $Y_\alpha$7 yields the TCGSP-II,

$Y_\alpha$8

Setting $Y_\alpha$9 recovers the GFSP $S(D_\beta(Y_\alpha(t)))$0. Taking both $S(D_\beta(Y_\alpha(t)))$1 and $S(D_\beta(Y_\alpha(t)))$2 recovers the generalized Skellam process $S(D_\beta(Y_\alpha(t)))$3 (Khandakar et al., 30 Oct 2025).

For the inverse tempered stable subordinator, the Bernstein function is

$S(D_\beta(Y_\alpha(t)))$4

and if $S(D_\beta(Y_\alpha(t)))$5 denotes its inverse, then the pmf of

$S(D_\beta(Y_\alpha(t)))$6

satisfies

$S(D_\beta(Y_\alpha(t)))$7

For the integer case $S(D_\beta(Y_\alpha(t)))$8, the density satisfies a PDE with finitely many $S(D_\beta(Y_\alpha(t)))$9-derivatives, leading to an alternative differential representation involving the backward and forward shift operators $S(t)=M_1(t)-M_2(t),$0 and $S(t)=M_1(t)-M_2(t),$1 (Khandakar et al., 30 Oct 2025).

For the first-passage time of the inverse Gaussian subordinator, the Bernstein function is

$S(t)=M_1(t)-M_2(t),$2

If $S(t)=M_1(t)-M_2(t),$3 denotes the corresponding inverse Gaussian hitting time, then the pmf of

$S(t)=M_1(t)-M_2(t),$4

satisfies

$S(t)=M_1(t)-M_2(t),$5

and the paper also gives a second form involving $S(t)=M_1(t)-M_2(t),$6, the boundary value of the inverse Gaussian hitting-time density (Khandakar et al., 30 Oct 2025).

No general tail asymptotics are stated for TCGFSP-II beyond these operator-level descriptions. The source explicitly limits itself, in this inverse-subordinated branch, to distributional formulas and special-case governing equations rather than to a full asymptotic theory (Khandakar et al., 30 Oct 2025).

6. Terminological variants and the alternative common-clock “Type-II” model

The expression “TCGFSP-II” is not uniform across the fractional Skellam literature. In the explicit naming of Tathe and Ghosh it means

$S(t)=M_1(t)-M_2(t),$7

that is, inverse subordination of the GFSP by $S(t)=M_1(t)-M_2(t),$8 (Khandakar et al., 30 Oct 2025). In a separate line of work, however, “Type-II” is used in the natural sense of a common random clock shared by both branches of the Skellam difference. In that usage the model is identified with the generalized space-time fractional Skellam process (GSTFSP),

$S(t)=M_1(t)-M_2(t),$9

where both generalized counting components are evaluated at the same clock $S^\alpha(t)$00. The paper introducing GSTFSP does not itself use Type-I/II terminology, but the common-clock interpretation is exact (Tathe et al., 11 Apr 2025).

Under this alternative usage, the pgf takes the product-Mittag–Leffler form

$S^\alpha(t)$01

while the p.m.f. is given by a derivative series in $S^\alpha(t)$02 and $S^\alpha(t)$03. This common-clock model admits Caputo governing equations, explicit transition probabilities for small $S^\alpha(t)$04, formulas for $S^\alpha(t)$05-th arrival levels and first upcrossing times, tail asymptotics and upper bounds, the scaling limit

$S^\alpha(t)$06

and non-infinitely divisible one-dimensional distributions. It also admits weighted-sum representations and an explicit simulation strategy based on separate simulation of $S^\alpha(t)$07, $S^\alpha(t)$08, and the two generalized counting processes evaluated at the common clock (Tathe et al., 11 Apr 2025).

A third terminological specialization appears in the moderate-deviation literature for the classical $S^\alpha(t)$09 case, where “type 2” denotes a single inverse stable time change of the classical Skellam process,

$S^\alpha(t)$10

as opposed to “type 1,” which uses two independent fractional Poisson clocks. In that setting the type-2 model admits explicit large- and moderate-deviation principles, and the rate-function comparisons imply faster convergence to zero than in the type-1 model under the stated cases (Lee et al., 2022).

The principal consequence of this nomenclatural divergence is that “TCGFSP-II” may designate either an inverse-subordinator time change of the GFSP or a common-clock space-time fractional Skellam construction. Both are genuinely fractional Skellam models, but they are analytically distinct: the former is organized by generalized Caputo operators attached to $S^\alpha(t)$11, while the latter is organized by a shared random operational time $S^\alpha(t)$12.