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

# Time-Changed Generalized Fractional Skellam Process-II

Searching arXiv for recent 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 \(Y_\alpha\) [2510.26156]. 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 [2504.08374].

## 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 \(1,\dots,k\), rate vectors \((\lambda_1,\dots,\lambda_k)\) and \((\mu_1,\dots,\mu_k)\), and aggregate rates
\[
\Lambda=\sum_{j=1}^k \lambda_j,\qquad \bar\Lambda=\sum_{j=1}^k \mu_j.
\]
Its one-dimensional law is
\[
p(n,t)=\Pr\{S(t)=n\}=e^{-(\Lambda+\bar\Lambda)t}\left(\frac{\Lambda}{\bar\Lambda}\right)^{n/2}I_{|n|}\!\big(2t\sqrt{\Lambda\bar\Lambda}\big), \qquad n\in\mathbb Z,
\]
and its moment generating function is
\[
M_S(u,t)=\mathbb E[e^{uS(t)}]
=\exp\!\left\{\sum_{j=1}^k\big(\lambda_j(e^{uj}-1)+\mu_j(e^{-uj}-1)\big)t\right\}.
\]
This is an integer-valued Lévy process [2510.26156].

The generalized fractional Skellam process \(S^\alpha\), \(0<\alpha\le 1\), is defined by
\[
S^\alpha(t)=
\begin{cases}
S(Y_\alpha(t)), & 0<\alpha<1,\\
S(t), & \alpha=1,
\end{cases}
\]
where \(Y_\alpha\) is an inverse stable subordinator independent of \(S\). Its probability generating function is
\[
G_{S^\alpha}(u,t)
=E_{\alpha,1}\!\left(\sum_{j=1}^k\big(\lambda_j(u^j-1)+\mu_j(u^{-j}-1)\big)t^\alpha\right), \qquad |u|\le 1,
\]
with \(E_{\alpha,1}\) the Mittag–Leffler function [2510.26156].

A Lévy subordinator \(D_f\) is characterized by
\[
\mathbb E[e^{-sD_f(t)}]=e^{-tf(s)}, \qquad s>0,
\]
where the Bernstein function \(f\) has the Lévy–Khintchine representation
\[
f(s)=c_1+c_2s+\int_0^\infty (1-e^{-sx})\,\bar\nu(dx),
\]
with \(c_1,c_2\ge 0\). Its inverse, or first-passage time process, is
\[
H_f(t)=\inf\{x\ge 0:D_f(x)>t\}, \qquad t\ge 0.
\]
The TCGFSP-II is then
\[
\bar Z_f^\alpha(t)=S^\alpha(H_f(t)),
\]
and for \(\alpha=1\) it reduces to the time-changed generalized Skellam process-II (TCGSP-II),
\[
\bar Z_f(t)=S(H_f(t)).
\]
Initial conditions are
\[
\Pr\{\bar Z_f(0)=0\}=1,\qquad \Pr\{\bar Z_f(0)=n\}=0,\quad n\neq 0
\]
[2510.26156].

## 2. Distributional structure and transform formulas

The central transform formula for TCGFSP-II is its pgf:
\[
\mathbb E[u^{\bar Z_f^\alpha(t)}]
=\sum_{n=0}^\infty
\frac{\left(\sum_{j=1}^k(\lambda_j(u^j-1)+\mu_j(u^{-j}-1))\right)^n}{\Gamma(n\alpha+1)}
\,\mathbb E\!\left[H_f(t)^{n\alpha}\right],\qquad |u|\le 1.
\]
This exhibits the process as a moment mixture of the GFSP pgf under the random clock \(H_f(t)\) [2510.26156].

An equivalent conditioning representation is
\[
\mathbb E[u^{\bar Z_f^\alpha(t)}]
=\int_0^\infty G_{S^\alpha}(u,x)\,l_f(x,t)\,dx,
\]
where \(l_f(x,t)\) is the density of the inverse subordinator \(H_f(t)\). For the non-fractional reduction \(\alpha=1\), the state probabilities are
\[
\Pr\{\bar Z_f(t)=n\}
=\int_0^\infty p(n,x)\,l_f(x,t)\,dx,
\]
with \(p(n,x)\) the GSP pmf. In the same case, the paper gives the explicit series
\[
\Pr\{\bar Z_f(t)=n\}
=\sum_{m=\max(0,-n)}^\infty
\frac{\Lambda^{m+n}\bar\Lambda^m}{(m+n)!\,m!}\,
\mathbb E\!\left[e^{-(\Lambda+\bar\Lambda)H_f(t)}H_f(t)^{2m+n}\right],\qquad n\in\mathbb Z
\]
[2510.26156].

These formulas are structurally identical to the mixture representation for inverse-subordinator time-changed Skellam processes in the convolution-derivative framework:
\[
p_n(t)=\int_0^\infty
e^{-(\lambda_1+\lambda_2)\tau}
\left(\frac{\lambda_1}{\lambda_2}\right)^{n/2}
I_{|n|}\!\big(2\tau\sqrt{\lambda_1\lambda_2}\big)\,
l_\phi(t,\tau)\,d\tau,
\]
where \(l_\phi\) is the inverse-subordinator density associated with a Bernstein exponent \(\phi\) [1806.00277].

The paper does not explicitly provide moment generating function or characteristic function formulas for TCGFSP-II. It states, however, that the conditioning representation
\[
\mathbb E\big[M_{S^\alpha}(u;H_f(t))\big]
\]
applies formally, with \(M_{S^\alpha}(u;x)\) obtained from the GFSP transform at deterministic time \(x\) [2510.26156].

## 3. Moments, covariance structure, and dependence

Let
\[
m_1=\sum_{j=1}^k j(\lambda_j-\mu_j),\qquad
m_2=\sum_{j=1}^k j^2(\lambda_j+\mu_j),
\]
and define
\[
l_1=\frac{m_1}{\Gamma(1+\alpha)},\qquad
l_2=\frac{m_2}{\Gamma(1+\alpha)},\qquad
d=\alpha l_1^2 B(\alpha,1+\alpha),
\]
where \(B\) is the Beta function. Then, for \(0<s\le t<\infty\),
\[
\mathbb E[\bar Z_f^\alpha(t)]
= l_1\,\mathbb E[H_f(t)^\alpha],
\]
\[
\operatorname{Var}(\bar Z_f^\alpha(t))
=\mathbb E[H_f(t)^\alpha]\big(l_2-l_1^2\mathbb E[H_f(t)^\alpha]\big)
+2d\,\mathbb E[H_f(t)^{2\alpha}],
\]
and
\[
\operatorname{Cov}(\bar Z_f^\alpha(s),\bar Z_f^\alpha(t))
= l_2\,\mathbb E[H_f(s)^\alpha]
+d\,\mathbb E[H_f(s)^{2\alpha}]
-l_1^2\,\mathbb E[H_f(s)^\alpha]\mathbb E[H_f(t)^\alpha]
+l_1^2\alpha\,\mathbb E\!\left[H_f(t)^{2\alpha}B\!\left(\alpha,\alpha+1;\frac{H_f(s)}{H_f(t)}\right)\right]
\]
[2510.26156].

These formulas extend the classical inverse-subordinator Skellam identities, where
\[
\mathbb E[X_{II}(t)]=(\lambda_1-\lambda_2)\mathbb E[E_t],\qquad
\operatorname{Var}(X_{II}(t))=(\lambda_1+\lambda_2)\mathbb E[E_t]+(\lambda_1-\lambda_2)^2\operatorname{Var}(E_t),
\]
to the generalized fractional setting [1806.00277].

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 \(H_f(t)\). 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 [2510.26156].

The non-Markovian character is nevertheless intrinsic to the construction. The inverse stable subordinator \(Y_\alpha(t)\) has non-Markovian, non-stationary, non-independent increments, and inverse time change generally destroys the Lévy and Markov increment structure [2510.26156]. 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 \(l_f(x,t)\). Under Condition I, namely \(\bar\nu(0,\infty)=\infty\) and absolute continuity of the tail \(\nu(\cdot)\), the density satisfies
\[
{}^{f}\mathbb D_t\,l_f(x,t)=-\frac{\partial}{\partial x}l_f(x,t),
\]
with boundary and initial data
\[
l_f(x/c_2,t)=0,\qquad l_f(0,t)=\nu(t),\qquad l_f(x,0)=\delta(x),
\]
where \(^{f}\mathbb D_t\) is Toaldo’s generalized Riemann–Liouville derivative and
\[
{}^{f}\mathbb D_t w(t)= {}^{f}\mathscr D_t w(t)+\nu(t)w(0)
\]
relates it to the generalized Caputo derivative \(^{f}\mathscr D_t\) [2510.26156].

At the level made explicit in the paper, the general inverse-subordination governing system is stated for the \(\alpha=1\) reduction TCGSP-II. Its marginals \(\bar p_f(n,t)=\Pr\{\bar Z_f(t)=n\}\) satisfy
\[
{}^{f}\mathscr D_t \bar p_f(n,t)
=\Lambda\big(\bar p_f(n-1,t)-\bar p_f(n,t)\big)
-\bar\Lambda\big(\bar p_f(n,t)-\bar p_f(n+1,t)\big),\qquad n\in\mathbb Z,
\]
with
\[
\bar p_f(n,0)=
\begin{cases}
1, & n=0,\\
0, & n\neq 0.
\end{cases}
\]
For \(f(s)=s^\alpha\), \(0<\alpha<1\), the generalized Caputo derivative reduces to the Caputo fractional derivative, recovering the fractional governing equation for the GFSP [2510.26156].

This inverse-subordinator mechanism sits within the broader convolution-derivative framework developed for time-changed Poisson and Skellam processes. For a Bernstein exponent \(\phi\), the generalized Caputo-type derivative is characterized by
\[
\mathcal L\{\mathcal D_t^\phi f(t)\}(s)
=\phi(s)\tilde f(s)-\frac{\phi(s)}{s}f(0),
\]
and, in the classical Skellam case,
\[
\mathcal D_t^\phi p_n(t)
=\lambda_1\big(p_{n-1}(t)-p_n(t)\big)+\lambda_2\big(p_{n+1}(t)-p_n(t)\big),
\qquad n\in\mathbb Z.
\]
The corresponding bilateral pgf \(G(z,t)=\sum_{n\in\mathbb Z}p_n(t)z^n\) obeys
\[
\mathcal D_t^\phi G(z,t)
=\big[\lambda_1(z-1)+\lambda_2(z^{-1}-1)\big]G(z,t),\qquad G(z,0)=1
\]
[1806.00277].

## 5. Reductions and specific inverse subordinators

Several reductions organize the model hierarchy. Setting \(\alpha=1\) yields the TCGSP-II,
\[
\bar Z_f(t)=S(H_f(t)).
\]
Setting \(H_f(t)=t\) recovers the GFSP \(S^\alpha(t)\). Taking both \(\alpha=1\) and \(H_f(t)=t\) recovers the generalized Skellam process \(S(t)\) [2510.26156].

For the inverse tempered stable subordinator, the Bernstein function is
\[
f_2(s)=(\eta+s)^\theta-\eta^\theta,\qquad 0<\theta<1,\ \eta>0,
\]
and if \(L_{\eta,\theta}(t)\) denotes its inverse, then the pmf of
\[
\bar Z_{f_2}(t)=S(L_{\eta,\theta}(t))
\]
satisfies
\[
(\eta + d/dt)^\theta \bar p_{f_2}(n,t)
= (\eta^\theta-\Lambda-\bar\Lambda)\bar p_{f_2}(n,t)
+\Lambda \bar p_{f_2}(n-1,t)+\bar\Lambda \bar p_{f_2}(n+1,t)
- t^{-\theta}E_{1,1-\theta}^{1-\theta}(-\eta t)\,p(n,0)
+\big[p(n,x)l_{\eta,\theta}(x,t)\big]_{x=0}.
\]
For the integer case \(m=1/\theta\ge 2\), the density satisfies a PDE with finitely many \(x\)-derivatives, leading to an alternative differential representation involving the backward and forward shift operators \(B\) and \(F\) [2510.26156].

For the first-passage time of the inverse Gaussian subordinator, the Bernstein function is
\[
f_3(s)=\delta(\sqrt{2s+\gamma^2}-\gamma),\qquad \delta,\gamma>0.
\]
If \(H(t)\) denotes the corresponding inverse Gaussian hitting time, then the pmf of
\[
\bar Z_{f_3}(t)=S(H(t))
\]
satisfies
\[
\delta(\gamma^2+2d/dt)^{1/2}\bar p_{f_3}(n,t)
=(\delta\gamma-\Lambda-\bar\Lambda)\bar p_{f_3}(n,t)
+\Lambda \bar p_{f_3}(n-1,t)+\bar\Lambda \bar p_{f_3}(n+1,t)
-\delta\gamma\,\operatorname{Erf}(\gamma\sqrt{t/2})\,\bar p_{f_3}(n,0),
\]
and the paper also gives a second form involving \(h(0,t)\), the boundary value of the inverse Gaussian hitting-time density [2510.26156].

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 [2510.26156].

## 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
\[
\bar Z_f^\alpha(t)=S^\alpha(H_f(t)),
\]
that is, inverse subordination of the GFSP by \(H_f\) [2510.26156]. 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),
\[
\mathcal S_\beta^\alpha(t)=M_{1,\beta}^\alpha(t)-M_{2,\beta}^\alpha(t)=S(D_\beta(Y_\alpha(t))),
\]
where both generalized counting components are evaluated at the same clock \(E(t)=D_\beta(Y_\alpha(t))\). The paper introducing GSTFSP does not itself use Type-I/II terminology, but the common-clock interpretation is exact [2504.08374].

Under this alternative usage, the pgf takes the product-Mittag–Leffler form
\[
\mathcal G_{\mathcal S_\beta^\alpha}(u,t)
=
E_{\alpha,1}\!\left(-\Big[\sum_{j=1}^k(1-u^j)\lambda_j\Big]^\beta t^\alpha\right)
E_{\alpha,1}\!\left(-\Big[\sum_{j=1}^k(1-u^{-j})\mu_j\Big]^\beta t^\alpha\right),
\]
while the p.m.f. is given by a derivative series in \(\Lambda\) and \(T\). This common-clock model admits Caputo governing equations, explicit transition probabilities for small \(\delta\), formulas for \(n\)-th arrival levels and first upcrossing times, tail asymptotics and upper bounds, the scaling limit
\[
\frac{\mathcal S_\beta^\alpha(t)}{t^{\alpha/\beta}}
\Rightarrow
(Y_\alpha(1))^{1/\beta}D_\beta(1)\sum_{j=1}^k j(\lambda_j-\mu_j),
\]
and non-infinitely divisible one-dimensional distributions. It also admits weighted-sum representations and an explicit simulation strategy based on separate simulation of \(Y_\alpha\), \(D_\beta\), and the two generalized counting processes evaluated at the common clock [2504.08374].

A third terminological specialization appears in the moderate-deviation literature for the classical \(k=1\) case, where “type 2” denotes a single inverse stable time change of the classical Skellam process,
\[
Z_{\nu,\underline\lambda}(t)=S_{\underline\lambda}(L_\nu(t)),
\]
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 [2208.06376].

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 \(H_f\), while the latter is organized by a shared random operational time \(D_\beta(Y_\alpha(t))\).

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