Fractional Generalizations of the Compound Poisson Process
]Neha Gupta
Department of Theoretical Statistics and Mathematics Unit,
Indian Statistical Institute Delhi, New Delhi - 110016, India.
]Aditya Maheshwari
Operations Management and Quantitative Techniques Area, Indian Institute of Management Indore, Indore 453556, India.

60G22, 60G51, 60G55.

This paper introduces the Generalized Fractional Compound Poisson Process (GFCPP), which claims to be a unified fractional version of the compound Poisson process (CPP) that encompasses existing variations as special cases. We derive its distributional properties, generalized fractional differential equations, and martingale properties. Some results related to the governing differential equation about the special cases of jump distributions, including exponential, Mittag-Leffler, Bernstéin, discrete uniform, truncated geometric, and discrete logarithm. Some of processes in the literature such as the fractional Poisson process of order $k$, Pólya-Aeppli process of order $k$, and fractional negative binomial process becomes the special case of the GFCPP. Classification based on arrivals by time-changing the compound Poisson process by the inverse tempered and the inverse of inverse Gaussian subordinators are studied. Finally, we present the simulation of the sample paths of the above-mentioned processes.

§ INTRODUCTION

The compound Poisson process (CPP) is one the most important stochastic models for count data with random jump events. It generalizes the classical unit jump size of the Poisson process to random jump size distribution. This model is distinguished by its ability to effectively represent real-world scenarios where the size or impact of events varies and it is particularly useful in situations where extreme events of random sizes play a crucial role. Therefore, it becomes a natural model for
wide range of applications in various sectors including insurance [10, 14], reliability [35], statistical physics [6], mining [18], evolutionary biology [13] and many more. Due to its wide applicability, it always remains a central of attraction of both theoretical and applied probabilists.

In this context, the generalizations of the CPP becomes an important problem to consider. Several attempts has been made to generalize the CPP by various authors and here we mention important results in the literature from fractional generalizations point of view. The first fractional generalization of the CPP was proposed by [24, 22], and a semi-Markov extension of the CPP was discussed by [28]. Later, in [3, 2], the authors studied alternative forms of the compound fractional Poisson processes and derived several important results about their time-changed versions, governing differential equations and limiting behaviour of the processes. These fractional forms (see [4]) are defined by time-changing the CPP by inverse stable subordinator and also by changing the jump size distribution. A multivariate extension of the fractional generalizations of the CPP is introduced in [5]. In [9], the Poisson subordinated CPP is considered. More recently, some fractional versions of the CPP by changing jump size and/or by  time-changing the CPP are examined in [31, 11, 17, 16].

After examining the existing literature on this topic, we felt a need for a unified fractional form of the CPP and therefore, we introduce a fractional generalization of the  CPP such that most of the studied CPP become a special case of the proposed process.  Our process is defined as

\begin{equation*}
Y_{f}(t): = \sum_{i=1}^{N_{f}(t)} X_i,
\end{equation*}

where $N_{f}(t)\stackrel{d}{=}N(E_f(t))$ is the Poisson process time-changed with an independent inverse subordinator $\{E_{f}(t)\}_{t\geq 0}$ (<ref>) and $X_i$'s$,\;i=1,2,\ldots,$ are the iid jumps with common distribution $F_X$. We call this process as the  generalized fractional CPP (GFCPP). It is to note that we have assumed the most generalized form on the count and jump size distributions and therefore, this formulation will serve as an all-encompassing process for any type of  fractional generalization of the CPP. We compute the distributional properties and the governing generalized fractional differential equation of the probability mass function (pmf) of the GFCPP. We also prove that the compensated GFCPP is a martingale with respect to its natural filtration.

The special cases of the jump distribution $X_i,i=1,2,\ldots$, namely, exponential, Mittag-Leffler, Bernstéin, discrete uniform, truncated geometric, and discrete logarithm, are investigated. We obtain their Laplace Transform (LT), governing fractional differential equations and time-changed representations. Specifically to mention that this approach generalize the fractional Poisson process of order $k$ ([30, 11]), the  Pólya-Aeppli process of order $k$ (see [7]), and  the fractional negative binomial process (see [34, 4]). The classification of the GFCPP based on arrivals, particularly,  tempered fractional Poisson process and inverse of inverse Gaussian time-change of the  Poisson process are worked out. We further discuss their special cases based on jump sizes and obtain the LT and governing fractional differential equations. Lastly, we present the simulations for the special cases of the aforementioned processes.

The paper is organized as follows. In Section <ref>, we present some preliminary definitions and results that are required for the rest of the paper. The GFCPP is examined in detailed in Section <ref>. The classification of the special cases of the GFCPP based on jump sizes and arrivals are discussed in Sections <ref> and <ref>, respectively. In Section <ref>, the simulations of the sample paths are presented.

§ PRELIMINARIES

In this section, we introduce some notations and results that will be used later.

§.§ Lévy subordinator and its inverse

A Lévy subordinator (hereafter referred to as the subordinator) $\{D_{f}(t)\}_{t\geq0}$ is a non-decreasing Lévy process and its Laplace transform (LT) (see <cit.>) has the form

\begin{equation*}%\label{subordinator-LT}
\mathbb{E}[e^{-s D_{f}(t)}]=e^{-tf(s)},
\;{\rm where}\;
f(s)=b s+\int_{0}^{\infty}(1-e^{-s x})\nu(dx),~b\geq0, s>0,
\end{equation*}

is the Bernstéin function (see [29] for more details).
Here $b$ is the drift coefficient and $\nu$ is a non-negative Lévy measure on positive half-line satisfying

\begin{equation*}
\int_{0}^{\infty}(x\wedge 1)\nu(dx)<\infty~~{\rm and}~~\nu([0,\infty))= \infty
\end{equation*}

which ensures that the sample paths of $\{D_{f}(t)\}_{t \geq 0}$ are almost surely $(a.s.)$  strictly increasing.
Also, the first-exit time of $\{D_f(t)\}_{t\geq0}$ is defined as

\begin{equation}\label{inverse-sub}
E_{f}(t)=\inf\{r\geq 0:D_{f}(r)>t\},
\end{equation}

which is the right-continuous inverse of the subordinator $\{D_f(t)\}_{t\geq 0}$.
The process $\{E_{f}(t)\}_{t \geq 0}$ is non-decreasing and its sample paths are continuous.
We list some special cases of strictly increasing subordinators. The following subordinators with Laplace exponent denoted by $f(s)$ are very often used in literature.

\begin{equation}\label{Levy_exponent}
f(s) =
\begin{cases}
s^{\alpha},\; 0<\alpha<1, & $(stable subordinator)$;\\
(s+\mu)^{\alpha}-\mu^{\alpha},\;\mu >0,\; 0<\alpha<1,&$(tempered stable  subordinator)$;\\
\delta(\sqrt{2s+\gamma^2}-\gamma),\; \gamma>0,\; \delta>0, & $(inverse Gaussian subordinator)$.
\end{cases}
\end{equation}

§.§ Compound Poisson Process

A compound Poisson process is a continuous-time process with iid jumps $X_i,i=1,2,\ldots$.
A compound Poisson process $\{Y(t)\}_{t \geq 0}$ is given by

\begin{equation}\label{CPP}
\end{equation}

where $X_i$'s follows $F_X$ distribution and jumps arrive  randomly according to an independent Poisson process $N(t)$ with intensity rate $\lambda >0$.
The LT of the  CPP $\{Y(t)\}_{t\geq 0}$ is given by

\begin{align}\label{LT:CPP}
\mathbb{E}[e^{-sY(t)}] = \mathbb{E}[e^{-\lambda t (1-\mathbb{E}[e^{-sX_1}])}].
\end{align}

The pmf $P(n,t)=\mathbb{P}[Y(t)=n]$ of the  CPP is given by

P(n,t)= \sum_{m=1}^{\infty}F_X^{*m}(n) \frac{e^{-\lambda t}(\lambda t)^m}{m!},

where $F_X^{*m}$ is the $m$-fold convolution of the density of $F_X$.

§.§ Generalized fractional derivatives

Let $f$ be a Bernstéin function with integral representation

$$ f(s) =  \int_{0}^{\infty} (1-e^{-x s}) \nu(dx),\;\; s>0.$$

We will use the generalized Caputo-Djrbashian (C-D) derivative with respect to the Bernstéin function $f$, which is defined on the space of absolutely continuous functions as follows (see [32], Definition 2.4)

\begin{equation}\label{Gen_Caputo_FD}
\mathcal{D}^{f}_{t}u(t)=b\frac{d}{dt}u(t)+\int \frac{\partial}{\partial t}u(t-s)\Bar{\nu}(s)ds,
\end{equation}

where $\Bar{\nu}(s) = a + \Bar{\nu}(s, \infty)$ is the tail of the Lévy measure.

The generalized Riemann-Liouville (R-L)
derivative according to the Bernstéin function $f$ as (see <cit.>)

\begin{equation}\label{gen_R-L}
\mathbb{D}^{f}_{t}u(t)=b\frac{d}{dt}u(t)+\frac{d}{dt}\int u(t-s)\Bar{\nu}(s)ds.
\end{equation}

The relation between $\mathbb{D}^{f}_{t}$ and $\mathcal{D}^{f}_{t}$ are given by (see [32], Proposition 2.7)

\begin{equation}\label{RElation_RL_C}
\mathbb{D}^{f}_{t}u(t)= \mathcal{D}^{f}_{t}u(t)+\Bar{\nu}(t) u(0).
\end{equation}

§.§ LRD for non-stationary process

Let $s>0$ be fixed and $t>s$. Suppose a stochastic process $\{X(t)\}_{t\geq0}$ has the correlation function Corr$(X(s),X(t))$ that satisfies

\begin{equation}
c_1(s)t^{-d}\leq\text{Corr}(X(s),X(t))\leq c_2(s)t^{-d},
\end{equation}

for large $t$, $d>0$, $c_1(s)>0$ and $c_2(s)>0$. In other words,

\begin{equation}
\lim\limits_{t\to\infty}\frac{\text{Corr}(X(s),X(t))}{t^{-d}}=c(s),
\end{equation}
for some $c(s)>0$ and $d>0.$  We say $\{X(t)\}_{t\geq0}$ has the LRD property if $d\in(0,1)$.

§ GENERALIZED FRACTIONAL COMPOUND POISSON PROCESSES

In this section, we introduce the generalized fractional compound Poisson process and study their properties.

Let $N_{f}(t)\stackrel{d}{=}N(E_f(t))$ be the Poisson process time-changed with  the inverse subordinator $\{E_{f}(t)\}_{t \geq 0}$ (see [23]) and let
$X_i,\;i=1,2,\ldots,$ be the iid jumps with common
distribution $F_X$. The process defined by

\begin{equation}\label{GFCPP}
Y_{f}(t) := \sum_{i=1}^{N_{f}(t)} X_i,t\geq0,
\end{equation}

is called the generalized fractional compound Poisson process (GFCPP).

Using (<ref>), we obtain the LT of the  pmf of the GFCPP  $\{Y_{f}(t)\}_{t\geq0}$. It is given by

\begin{align}\label{LT:GCFPP}
\mathbb{E}[e^{-sY_{f}(t)}] &= \mathbb{E}\left[\mathbb{E}\left[\sum_{i=1}^{N_f(t)} X_i|E_{f}(t)\right]\right]= \mathbb{E}[e^{-\lambda E_{f}(t) (1-\mathbb{E}[e^{-sX_1}])}].
\end{align}

It is clear from the above LT that we can express the GFCPP as a time-changed CPP
$$ Y_{f}(t)\stackrel{d}{=}Y(E_f(t)),t\geq0,$$
where  $Y(t)=\sum_{i=0}^{N(t)} X_i$ denotes the CPP.

Next, we present the governing generalized fractional differential equation of the pmf of the GFCPP.

The pmf $P_{f}(n,t)=\mathbb{P}[Y_f(t)=n]$  satisfies following fractional differential equation

\begin{align}\label{Diff_Gcpp}
\mathcal{D}^{f}_{t} P_{f}(n,t) &= -\lambda  P_{f}(n,t) +\lambda \int_{-\infty}^{\infty} P_{f}(n-x,t) F_X(x) dx.
\end{align}

Let $h_{f}(y,t)$ be the probability density function (pdf) of inverse subordinator $\{E_f(t)\}_{t \geq 0}$ and $P(n,t)$ be pmf of CPP. Using conditional argument, we have that

\mathbb{P}(Y_{f}(t) \in dz)=\int_{0}^{\infty} \mathbb{P}(Y(x) \in dz) h_{f}(x,t) dy.

We now take generalized Riemann-Liouville
derivative given by (<ref>) on both sides of above equation, we get

\begin{align}
\mathbb{D}^{f}_{t} P_{f}(n,t)&=  \int_{0}^{\infty}p(n,x)\mathbb{D}^{f}_{t}h_f(x,t) dx.\nonumber\\
&=-\int_{0}^{\infty}P(n,y)\frac{\partial}{\partial x}h_f(x,t)dx \nonumber\\
&= -P(n,x)h_f(x,t)|_{0}^{\infty} + \int_{0}^{\infty} \frac{\partial}{\partial x}P(n,x)h_f(x,t) dx. \label{rl-pmf1}
\end{align}

It is known that (see [25]) the distribution of CPP satisfies the following partial differential equation

\begin{equation}\label{diff-cpp}
\frac{\partial}{\partial y}P(n,x) = \lambda P(n,x)+ \lambda \int_{-\infty}^{+\infty}P(n-u,y)F_X(u)du.
\end{equation}

Substituting (<ref>) in (<ref>) and subsequently using the relation (<ref>), we obtain the result mentioned in (<ref>). This completes the proof.

Further, we discuss some distributional properties of the GFCPP.

The mean, variance and covariance of GFCPP is given by:

* $ \mathbb{E}[Y_f(t)] =\lambda \mathbb{E}[E_{f}(t)]\mathbb{E}[X_1]$

* ${\rm Var}[Y_f(t)] = \lambda \mathbb{E}[E_f(t)]\mathbb{E}[X_1^2]+\lambda^2(\mathbb{E}[X_1])^2{\rm Var}[E_f(t)]$

* ${\rm Cov}[Y_f(t),Y_f(s)]  = \lambda \mathbb{E}[X_1^2]\mathbb{E}[E_f(s)]+\lambda^2(\mathbb{E}[X_1])^2{\rm Cov}[E_f(t),E_f(s)].$

Using the conditional argument and independence of $N_f(t)$ and $X_i$, we have that

\mathbb{E}[Y_f(t)]= \mathbb{E}[N_f(t)]\mathbb{E}[X_1]=\lambda \mathbb{E}[E_f(t)]\mathbb{E}[X_1].

The variance of $\{Y_f(t)\}_{t \geq 0}$ can be written as (see   [20])

{\rm Var}[Y_f(t)] = {\rm Var}[X_1]\mathbb{E}[N_f(t)]+\mathbb{E}[X_1]{\rm Var}[N_f(t)].

Next, we compute the ${\rm Cov}[Y_f(t),Y_f(s)]$,  $s\leq t$,

\begin{align*}
{\rm Cov}[Y_f(t),Y_f(s)]&= \mathbb{E}\left[\sum_{k=1}^{\infty}X_k^2 \mathbb{I}\{N_f(s)\geq k\}\right]+\mathbb{E}\left[ \sum_{i\neq j}\sum X_i X_k \mathbb{I}\{N_f(s)\geq k, N_f(t)\geq i\}\right]\\
& \hspace{3cm}-(\mathbb{E}[X_1])^2\mathbb{E}[N_f(s)]\mathbb{E}[N_f(t)]\\
&= \mathbb{E}[X_1^2]\sum_{k=1}^{\infty}\mathbb{P}(N_f(s)\geq k)\\
&\hspace{3cm}+(\mathbb{E}[X_1])^2\left[\sum_{i=1}^{\infty}\sum_{k=1}^{\infty}\mathbb{P}(N_f(s)\geq k, N_f(t)\geq i)-\mathbb{P}(N_f(s)\geq k)\right]\\
&=  \mathbb{E}[X_1^2] \mathbb{E}[N_f(s)]+(\mathbb{E}[X_1])^2\mathbb{E}[N_f(s) N_f(t)]\\
&= {\rm Var}[X_1]\mathbb{E}[N_f(s)]+(\mathbb{E}[X_1])^2{\rm Cov}[N_f(t),N_f(s)].
\end{align*}

The expression for the covariance of time-changed Poisson process, that is ${\rm Cov}[N_f(t),N_f(s)]$, is derived in [20]. Substituting the same in the above equation, we get the desired result.

Next, we prove the martingale property for the compensated GCFPP. Consider the compensated GCFPP defined by

\begin{equation*}
M_{f}(t):=Y_{f}(t)-\lambda E_f(t) \mathbb{E}[X_1],\;\; t\geq 0.
\end{equation*}

Let $\mathbb{E}[X_i]<\infty, i=1,2,\ldots$.  The compensated GCFPP $\{M_{f}(t)\}_{t\geq 0} $ is a martingale with respect to a natural filtration $\mathscr{F}_{t}= (N_f(s), s\leq t) \vee \sigma(E_{f}(s), s\leq t)$.

It is to note that (see [11]) the compensated time-changed Poisson process $Q(t):=N_f(t)-\lambda E_{f}(t)$ is a martingale with respect to the filtration $\mathscr{F}_{t}  = (N_f(s), s\leq t) \vee \sigma(E_{f}(s), s\leq t)$. We have that

\begin{align*}
\mathbb{E}[M_{f}(t)-M_{f}(s)|\mathscr{F}_{s}] &=\mathbb{E}\left[\sum_{i=1}^{N_{f}(t)} X_i - \lambda \mathbb{E}[X_1]E_f(t) -\left(\sum_{i=1}^{N_{f}(s)} X_i-\lambda \mathbb{E}[X_1] E_f(s) \right) \Bigg| \mathscr{F}_{s} \right]\\
&= \mathbb{E}\left[\left(\sum_{i=N_{f}(s)+1}^{N_{f}(t)} X_i - \lambda a (E_f(t)-E_f(s))\right)\Bigg|\mathscr{F}_{s}\right]\\
& = \mathbb{E}\left[\mathbb{E}\left[\sum_{i=N_{f}(s)+1}^{N_{f}(t)} X_i\Bigg| \mathscr{F}_{t}\right] \Bigg| \mathscr{F}_{s} \right] - \lambda a \mathbb{E}\left[E_f(t)-E_f(s)\Bigg| \mathscr{F}_{s}\right]\\
& = \mathbb{E}\left[a (N_f(t)-N_f(s))-  \lambda a (E_f(t)-E_f(s))| \mathscr{F}_{s} \right]\\
&=a\mathbb{E}\left[Q(t)-Q(s)| \mathscr{F}_{s}\right]\\
\end{align*}

since the $\{Q(t)\}_{t\geq0}$ is a $\mathscr{F}_{t}$-martingale. This completes the proof.

§ GENERALIZED TIME-FRACTIONAL COMPOUND POISSON PROCESS

In the previous section, we discussed some important properties and results related to the GFCPP. Note that Definition <ref> assumes general distribution $F_X$ on jump size. In this section, we consider several special cases of distribution on the jump size $X_i,i=1,2,\ldots$ of the GFCPP  and study their properties. We call this process as generalized time-fractional CPP (GTFCPP) and it can also be expressed as the CPP time-changed  with the inverse subordinator $\{Y(E_f(t))\}_{t \geq 0}$, where $\{Y(t)\}_{t \geq 0}$ and $\{E_f(t)\}_{t \geq 0}$ are independent with specific jump distribution in the CPP.

§.§ GTFCPP with exponential jumps
In this subsection, we assume that the jumps $X_i,\;i=1,2,\ldots,$ are exponentially distributed with parameter $\eta>0$ and denote it by

\begin{equation}\label{Time_CFPP}
Y_f^\eta(t) := \sum_{i=1}^{N_{f}(t)} X_i, t\geq0.
\end{equation}

Using (<ref>), we obtain the LT of the density $P^{\eta}_{f}(x,t)$, given by

\begin{align}\label{LT_GTFCPP}
\mathcal{L}[P^{\eta}_{f}(x,t)] = \mathbb{E}[e^{-sY_f^\eta(t)}] = \mathbb{E}[e^{-\lambda E_{f}(t) \frac{s}{s+\eta}}].
\end{align}

Next, we derive the differential equation associated with the density $P^{\eta}_{f}(x,t)$ of the GTFCPP with exponential jumps $\{Y_f^\eta(t)\}_{t\geq 0}$.

The pdf $P^{\eta}_{f}(x,t)$ of $\{Y_f^\eta(t)\}_{t\geq 0}$ satisfy following fractional differential equation

\begin{align}\label{Diff_GFCPP_expo}
\eta \mathcal{D}^{f}_{t} P^{\eta}_{f}(x,t) &= -\left[\lambda +\mathcal{D}^{f}_{t}\right]\frac{\partial}{\partial x}  P^{\eta}_{f}(x,t).
\end{align}

with following conditions

P^{\eta}_{f}(x,0)=0, \; \;
\mathbb{P}(Y_f^\eta (t) >0)= 1-\mathbb{E}[e^{-\lambda E_f(t)}].

Consider the subordinated form of the pdf $P^{\eta}_{f}(x,t)$

P^{\eta}_{f}(x,t)=\int_{0}^{\infty} P(x,y)h_f(y,t) dy,

where $h_{f}(y,t)$ is pdf of inverse subordinator $\{E_f(t)\}_{t \geq 0}$.
Taking generalized Riemann-Liouville
derivative given by (<ref>), we get

\begin{align}
\mathbb{D}^{f}_{t} P^{\eta}_{f}(x,t)&=  \int_{0}^{\infty} P_{Y}(x,y)\mathbb{D}^{f}_{t}h_f(y,t) dy.\nonumber\\
&=-\int_{0}^{\infty} P_{Y}(x,y)\frac{\partial}{\partial y}h_f(y,t)dy\nonumber\\
&= - P_{Y}(x,y)h_f(y,t)|_{0}^{\infty} + \int_{0}^{\infty} \frac{\partial}{\partial y} P_{Y}(x,y)h_f(y,t) dy.\label{eq:thm41}
\end{align}

Note that (see [2]) the pdf $P_{Y}(x,t)$ of CPP satisfies the following equation

\eta \frac{\partial}{\partial t}P_{Y}(x,t)=-\left[\lambda +  \frac{\partial}{\partial t} \right]\frac{\partial}{\partial x}P_{Y}(x,t).

Substituting the above equation in (<ref>)  and using (<ref>), we obtain the desired result.

As a special case of Theorem <ref>, the mean and covariance of the GTFCPP with exponential jumps $\{Y_f^\eta(t)\}_{t\geq 0}$ can be found as follows

\begin{align*}
\mathbb{E}[Y_f^\eta(t)] &=\frac{\lambda}{\eta} \mathbb{E}[E_{f}(t)];\\
%{\rm Var}[Y_f^\eta(t)] &= \frac{3\lambda}{\eta^2}\mathbb{E}[E_{f}(t)]-\frac{2\lambda^2}{\eta^2}\mathbb{E}[E_{f}(t)]^2+\frac{2}{\eta^2}\mathbb{E}[E_{f}(t)^2];\\
{\rm Var}[Y_f^{\eta}f(t)] &= \frac{2\lambda}{\eta^2} \mathbb{E}[E_f(t)]+\frac{\lambda^2}{\eta^2}{\rm Var}[E_f(t)];\\
{\rm Cov}[Y_f^{\eta}(t),Y_f^{\eta}(s)] &= \frac{\lambda}{\eta^2} \mathbb {E}[E_f(s)]+\frac{1}{\eta^2}{\rm Cov}[N_f(t),N_f(s)],\;\; s<t.
\end{align*}

§.§  GTFCPP with Mittag-Leffler jumps
We now define the process

\begin{equation*}
Y_f^{\beta,\eta}(t) := \sum_{i=1}^{N_{f}(t)} X_i, t\geq0,
\end{equation*}

where jump size $X_i,\;i=1,2,\ldots,$ are the Mittag-Leffler distributed random variables with parameter $\beta$ and $\eta$ and having the pdf as $q_{\beta,\eta}(x,t)= \eta x^{\beta-1}E_{\beta,\beta}(-\eta x^{\beta}), \; \; \beta \in(0,1], \; \eta >0$.

The LT of the $\{Y_f^{\eta} (t)\}_{t\geq 0}$ is given by

\begin{equation}\label{LT:CPP_ML}
\mathcal{L}[P^{\beta, \eta}_f(x,t)] =  \mathbb{E}[e^{-\lambda E_{f}(t) \frac{s^\beta}{s^\beta+\eta}}].
\end{equation}

We now derive a time-changed representation of the GTFCPP with Mittag-Leffler jumps $\{Y_f^{\beta,\eta}(t)\}_{ t\geq 0}$.

Consider an $\beta$-stable subordinator $\{D_\beta(t)\}_{t\geq0}$ time-changed with   an independent GTFCPP with exponential jump $\{Y_f^{\eta} (t)\}_{t\geq 0}$, i.e.

$$ Y^{\eta,\beta}_{f}(t) \stackrel{d}{=} D_\beta(Y_f^{\eta} (t)),\; \beta \in (0,1),\; \eta > 0, t\geq0.$$

The LT of the pdf of $\{Y^{\eta,\beta}_{f}(t)\}_{t \geq 0}$ is

\mathbb{E}[e^{-sD_\beta(Y_f^{\eta} (t))}] = \mathbb{E}[\mathbb{E}[e^{-sD_\beta(Y_f^{\eta} (t))}|Y_f^{\eta} (t)]]= \mathbb{E}[e^{ -s^{\beta}y_{f}^{\eta}(t)}].

Using (<ref>), we get

\mathbb{E}[e^{-sD_\beta(Y_f^{\eta} (t))}]=\mathbb{E}[e^{-\lambda E^{f}(t) \frac{s^\beta}{s^\beta+\eta}}].

Comparing the above equation with the LT (<ref>) of the $\{Y_f^{\beta,\eta}(t)\}_{t\geq 0}$, we get the desired result.

Next, we present the fractional differential equation for the pmf of the GTFCPP with Mittag-Leffler jumps $\{Y_f^{\beta,\eta}(t)\}_{t\geq 0}.$

The pdf $P^{\beta,\eta}_f(x,t)$ of $\{Y_f^{\beta,\eta}(t)\}_{t\geq 0}$ satisfy following fractional differential equation

\begin{align*}
\eta \mathcal{D}^{f}_{t} P^{\beta,\eta}_f(x, t) &= -\left[\lambda +\mathcal{D}^{f}_{t}\right] \mathcal{D}^{\beta}_xP^{\beta,\eta}_f(x, t),
\end{align*}

where $\mathcal{D}^{\beta}_t$ denotes the C-D fractional derivative (a special case of (<ref>)) with the following conditions

P^{\beta,\eta}_f(x, 0)=0, \; \;
\mathbb{P}[Y_f^{\beta,\eta} (t) >0]= 1-\mathbb{E}[e^{-\lambda E_f(t)}].

Writing the pdf $P^{\beta,\eta}_f(x,t)$ using the conditional probability approach and then taking a generalized R-L fractional derivative,  we have that

\begin{align}
\mathbb{D}^{f}_{t} P^{\beta,\eta}_f(x,t)&=  \int_{0}^{\infty} l_{\beta}(x,y)\mathbb{D}^{f}_{t}P_{f}^{\eta}(y,t) dy, \;\;(\text{using \eqref{Diff_GFCPP_expo}})\nonumber\\
&=-\frac{1}{\eta}\left[\lambda +\mathcal{D}^{f}_{t}\right]\int_{0}^{\infty}l_\beta(x,y)\frac{\partial}{\partial y}P_{f}^{\eta}(y,t) dy\nonumber\\
&= -\frac{1}{\eta}\left[\lambda +\mathcal{D}^{f}_{t}\right] \left(-l_{\beta}(x,y) P_{f}^{\eta}(y,t)|_{0}^{\infty} + \int_{0}^{\infty} \frac{\partial}{\partial y} l_\beta(x,y)P_{f}^{\eta}(y,t) dy\right),\label{eq:thm43}
\end{align}

where $l_{\beta}(x,t)$ is the pdf of $\beta$-stable subordinator which satisfies the equation

\begin{equation}\label{eq:thm43-2}
\mathcal{D}^{\beta}_{x}l_{\beta}(x,t)=-\frac{\partial}{\partial t}l_{\beta}(x,t),\;\;l_{\beta}(x,0)=\delta(x).\end{equation}

We obtain the desired result by substituting (<ref>) and   (<ref>) in (<ref>).

§.§  GTFCPP with Bernstéin jumps

In this subsection, we assume
that the jump size $X_i, \; i=1,2,\ldots$ of the GFCPP are  distributed as follows

\begin{equation}\label{Dis_Bern}
\mathbb{P}(X_i >t)= \mathbb{E}[e^{\eta E_g(t)}].
\end{equation}

Note  that the above distribution also occurs as the distribution of the inter-arrivals between the consecutive jumps of the time-changed Poisson process $\{N(E_g(t))\}_{t \geq 0}$, where $\{E_g(t)\}_{t \geq 0}$ is an independent inverse subordinator with Bernstéin function $g$ (see [25] for more details). Now, we define the process

\begin{equation*}
Y_f^{g}(t):= \sum_{i=1}^{N_{f}(t)} X_i, t\geq 0.
\end{equation*}

It is called as the  GTFCPP with Bernstéin jumps.

The LT of the $\{Y_{f}^{g}(t)\}_{t\geq 0}$ is given by

\begin{align}\label{LT_TSFCPP-1}
\mathbb{E}[e^{-sY_{f}^{g}(t)}]= \mathbb{E}\left[e^{-\lambda E^{f}(t) \frac{g(s)}{g(s)+\eta}}\right],
\end{align}

$$\mathbb{E}[e^{-sX_1}]=  \frac{\eta}{g(s)+\eta}.

Next, we obtain the time changed representation and the governing fractional differential equation of the GTFCPP with Bernstéin jumps $\{Y_{f}^{g}(t)\}_{t\geq 0}$ in the following Theorem.

Let $\{D_{g}(t)\}_{t \geq 0}$ be a Lévy subordinator with Bernstéin function $g$ and $\{Y_{f}^{\eta}(t)\}_{t \geq 0}$ be the GTFCPP with exponential jumps (<ref>), independent of $\{D_{g}(t)\}_{t \geq 0}.$ Then

\begin{equation}\label{relation_tscfpp-1}
\end{equation}

The pdf $P_{f}^{g}(x,t)$ of $\{Y_{f}^{g}(t)\}_{t\geq 0}$ satisfies the following equation

\begin{align*}
\eta \mathcal{D}^{f}_{t} P_{f}^{g}(x,t) &= -\left[\lambda +\mathcal{D}^{f}_{t}\right]\mathcal{D}^{g}_{x} P_{f}^{g}(x,t),\;
\end{align*}

with conditions

P_{f}^{g}(x,0)=0, \; \;
\mathbb{P}(Y_f^g (t) >0)= 1-\mathbb{E}[e^{-\lambda E_f(t)}].

The relation (<ref>) can be proved  by taking the LT of $\{D_{g}(Y_{f}^{\eta}(t))\}_{t \geq 0}$, which is given by

$$\mathbb{E}[e^{-sD_{g}(Y_{f}^{\eta}(t))}]= \mathbb{E}[e^{-\lambda E^{f}(t) \frac{^{g(s)}}{g(s)+\eta}}]$$

This is equal to the LT given in (<ref>). Hence by the uniqueness of LT, the result follows. Now, we express the pdf $P_{f}^{g}(x,t)$ using the conditional probability approach  and  then take a generalized R-L fractional derivative on both sides. We have that

\begin{align}
\mathbb{D}^{f}_{t} P_{f}^{g}(x,t)&=  \int_{0}^{\infty} l_{g}(x,y)\mathbb{D}^{f}_{t}P_{f}^{\eta}(y,t) dy, (\text{from \eqref{Diff_GFCPP_expo}})\nonumber\\
&=-\frac{1}{\eta}\left[\lambda +\mathcal{D}^{f}_{t}\right]\int_{0}^{\infty}l_g(x,y)\frac{\partial}{\partial y}P_{f}^{\eta}(y,t) dy\nonumber\\
&= -\frac{1}{\eta}\left[\lambda +\mathcal{D}^{f}_{t}\right] \left(-l_{g}(x,y) P_{f}^{\eta}(y,t)|_{0}^{\infty} + \int_{0}^{\infty} \frac{\partial}{\partial y} l_g(x,y)P_{f}^{\eta}(y,t) dy\right),\label{eq:thm44}
\end{align}
where $l_{g}(x,t)$ is the density of Lévy subordinator which satisfies the equation (see [32])
$$\mathcal{D}^{f}_{x}l_{g}(x,t)=-\frac{\partial}{\partial t}l_{g}(x,t),\;\;l_{g}(x,0)=\delta(x).$$

We substitute the above equation in (<ref>) and subsequently use (<ref>) to get the desired result.

§.§ Generalized fractional Poisson process of order $k$

In this subsection, we assume that jumps
$X_i,\;i=1,2,\ldots,$ are iid discrete uniform random variables such that $\mathbb{P}(X_i = j) = \frac{1}{k},\; j=1,2,\ldots, k$ and $\{N_f(t)\}_{t \geq 0}$ be time-changed Poisson process with intensity rate of the Poisson process $\{N(t)\}_{t\geq 0}$ is assumed to be  $k\lambda$. The process (<ref>) can be written as

\begin{equation}\label{CFPPOK}
Y_{f}^{k}(t): = \sum_{i=1}^{N_{f}(t)} X_i, t\geq 0,
\end{equation}

is called the generalized fractional Poisson process of order $k$ (GFCPPoK). This process was first defined and studied in [11].

The time-changed representation of GFCPPoK is given as (see [11])

Y_f^{k}(t) \stackrel{d}{=} N^k(E_f(t)), t\geq 0,

where $\{N^k(t)\}_{t \geq 0}$ is Poisson process of order $k$ (PPoK).

The pmf $P^{k}_{f}(n,t)=\mathbb{P}[Y_f^{k}(t)=n]$ of $\{Y_{f}^{k}(t)\}_{t \geq 0}$ satisfy following fractional differential-difference equation (see <cit.>),

\begin{align*}
\mathcal{D}^{f}_{t} P^{k}_{f}(n,t) &= -k \lambda \left(1-\frac{1}{k}\sum_{j=1}^{n \wedge k} B^{j}\right)P^{k}_{f}(n,t),\; n>0,\\
\mathcal{D}^{f}_{t}P^{k}_{f}(0,t)  &= - k \lambda P^{k}_{f}(0,t),\nonumber
\end{align*}

where $B$ is the backward shift operator i.e. $B[P(n,t)]=P(n-1,t)$.

§.§ Generalized Pólya-Aeppli process of order $k$

Consider the GTFCPP with $X_i$'s $i=1,2,\ldots$ as iid truncated geometrically distributed random variables with success probability $1-\rho$ and pmf given by

\mathbb{P}[X_i=j] = \frac{1-\rho}{1-\rho^k}\rho^{j-1},\;\; j=1,2,\ldots,k, \;\; \rho \in [0,1).

The LT of $ X_1$ given by

\mathbb{E}[e^{-sX_1}] = \frac{(1-\rho)e^{-s}}{(1-\rho^k)} \frac{1-\rho^k e^{-s^k}}{1-\rho{e^{-s}}}.

Note that when $k \rightarrow \infty $ the truncated geometric distribution approaches the geometric distribution starting at $1$ and success probability $1-\rho$.
We denote the process  as

\begin{equation}\label{tsfcpp}
Y_{f}^{\rho,k}(t) := \sum_{i=1}^{N_{f}(t)} X_i, t\geq 0.
\end{equation}

This is called the generalized Pólya-Aeppli process of order $k$ (GPAPoK).
The LT of the $\{Y_{f}^{\rho,k}(t)\}_{t \geq 0}$ is given by

\begin{align*}
\mathbb{E}[e^{-sY_{f}^{\rho}(t) }]= \mathbb{E}[e^{-\lambda E^{f}(t) (1-\mathbb{E}[e^{-sX_1}])}]=\mathbb{E}\left[e^{-\lambda E^{f}(t)\left(1- \frac{(1-\rho)e^{-s}}{(1-\rho^k)} \frac{1-\rho^k e^{-s^k}}{1-\rho{e^{-s}}}\right) }\right].
\end{align*}

The time-changed representation of GPAPoK is

Y_f^{\rho,k}(t) \stackrel{d}{=}N^k_A(E_f(t)), t\geq 0,

where $\{N^k_A(t)\}_{t\geq0}$ is Pólya-Aeppli process of order $k$ (PAPoK) (see [7]). Further, we derive the differential equation for the GPAPoK.

The pmf $P^{\rho,k}_{f}(n,t)=\mathbb{P}[Y_f^{\rho,k}(t)=n]$ satisfy following fractional differential equation

\begin{align*}
\mathcal{D}^{f}_{t} P^{\rho,k}_{f}(n,t) &= -\lambda \left(1-\frac{1-\rho}{1-\rho^k}\sum_{j=1}^{n \wedge k}\rho^{j-1}B^j\right)  P^{\rho,k}_{f}(n,t), \; \; n=1,2, \ldots,\\
\mathcal{D}^{f}_{t} P^{\rho,k}_{f}(0,t) &= -\lambda P^{\rho,k}_{f}(0,t),
\end{align*}

with an  initial condition
Writing the  pmf $P^{\rho,k}_{f}(n,t)$ using the condition probability approach, we have that

P^{\rho,k}_{f}(n,t)=\int_{0}^{\infty} P_{\rho,k}(n,y)h_f(y,t) dy,

where the $P_{\rho,k}(n,t)$ is the pmf of PAPoK $\{N^k_A(t)\}_{t\geq0}$.
Taking generalized Riemann-Liouville
derivative (<ref>) on both sides of the above equation, we get

\begin{align}
\mathbb{D}^{f}_{t} P^{\rho, k}_{f}(n,t)&=  \int_{0}^{\infty} P_{\rho,k}(n,y)\mathbb{D}^{f}_{t}h_f(y,t) dy,\;\;(\text{see in [32]})\nonumber \\
&=-\int_{0}^{\infty} P_{\rho,k}(n,y)\frac{\partial}{\partial y}h_f(y,t)dy\nonumber\\
&= - P_{\rho,k}(n,y)h_f(y,t)|_{0}^{\infty} + \int_{0}^{\infty} \frac{\partial}{\partial y} P_{\rho,k}(n,y)h_f(y,t) dy.\label{eq:thrm45}
\end{align}

We know that (see [27]) the pmf  $P_{\rho,k}(n,t)$ of  the Pólya-Aeppli process of order $k$ $\{N^k_A(t)\}_{t\geq 0}$ satisfies the following differential equation

\frac{\partial}{\partial t}P_{\rho,k}(n,t)=-\lambda \left(1-\frac{1-\rho}{1-\rho^k}\sum_{j=1}^{n \wedge k}\rho^{j-1}B^j \right)P_{\rho,k}(n,t),\;\; n\geq 1.

Substituting the above equation in (<ref>)  and using (<ref>), we obtain the desired result.

§.§ Fractional negative binomial process

Let $X_{i}$'s $i=1,2,\ldots$ be a sequence of iid random variables with discrete logarithmic distribution, given by

\mathbb{P}[X_i=n]= \frac{-1}{\log (1-q)}\frac{q^n}{n},\;\; n\geq 1,\; q \in(0,1).

We denote the process as

\begin{equation*}
Y^{q}_{f}(t): = \sum_{i=0}^{N_f(t)}X_i, t\geq 0.
\end{equation*}

It is also known (see [4]) as the fractional negative binomial process with parameter $(1-q, \frac{\lambda}{\log{1-q}})$.
The LT of the $\{Y^{q}_{f}(t)\}_{t \geq 0}$ is given by

\begin{align*}
\mathbb{E}[e^{-sY^{q}_{\lambda}(t) }]= \mathbb{E}[e^{-\lambda E^{f}(t) (1-(s+\mu)^\alpha -\mu^\alpha))}].
\end{align*}

§ CLASSIFICATIONS BASED ON ARRIVALS

In this section, we work out special cases of the GFCPP, defined in (<ref>), by taking particular cases of time-changed Poisson process $\{N_f(t)\}_{t\geq 0}$. More specifically, we study two types of inverse subordinator $\{E_f(t)\}_{t \geq 0}$, namely the inverse tempered  $\alpha$-stable subordinator (ITSS) and the inverse of the inverse Gaussian (IG) subordinator. The distribution of jumps is assumed in a general sense. Further, some results are mentioned by taking special cases of the jump distributions of $X_i,i=1,2,\ldots$.

§.§ Tempered fractional CPP

Consider the inverse subordinator (<ref>) associated with tempered stable Bernstéin function (<ref>) $f(s)= (\mu+s)^\alpha-\mu^\alpha,\; \alpha \in (0,1], \mu>0$, denoted by $\{E_{\alpha,\mu}(t)\}_{t\geq0}$. Let $N_{\alpha,\mu}(t):=N(E_{\alpha,\mu}(t)),t\geq 0$ be the tempered fractional Poisson process (TFPP)  (studied in [12]). The process (<ref>) is defined by

\begin{align}\label{TTSFCPP}
Y_{\alpha,\mu}(t):= \sum_{i=1}^{N_{\alpha,\mu}(t)}X_i, t\geq 0,
\end{align}

is called tempered fractional CPP (TFCPP) with  $X_i,\;i=1,2,\ldots,$ be the iid jumps having common
distribution $F_X $.

The pmf $P_{\alpha,\mu}(n,t)=\mathbb{P}[Y_{\alpha,\mu}(t)=n]$ satisfy following tempered fractional differential equation (see [25])

\begin{align*}%\label{Diff_TTScpp}
\mathcal{D}^{\alpha,\mu}_{t} P_{\alpha, \mu}(n,t) &= -\lambda  P_{\alpha,\mu}(n,t) +\lambda \int_{-\infty}^{\infty} P_{\alpha,\mu}(n-x,t) F_X(x) dx,
\end{align*}

where $\mathcal{D}^{\alpha, \mu}_t$ denotes the tempered C-D fractional derivative (a special case of (<ref>)).
We next give some distributional results for the TFCPP.

The mean, variance, and covariance of the TFCPP $\{Y_{\alpha,\mu}(t)\}_{t\geq 0}$ is given by

* $ \mathbb{E}[Y_{\alpha, \mu}(t)]  =\lambda \mathbb{E}[X_1]\sum_{n=0}^{\infty}\frac{\mu^{\alpha}\gamma(\mu t; \alpha(1+n))}{\Gamma(\alpha(1+n))};$

* ${\rm Var}[Y_{\alpha, \mu}(t)] = \lambda \mathbb{E}[X_1^2]\sum_{n=0}^{\infty}\frac{\mu^{\alpha}\gamma(\mu t; \alpha(1+n))}{\Gamma(\alpha(1+n))}+\lambda^2(\mathbb{E}[X_1])^2{\rm Var}[E_f(t)]$

* ${\rm Cov}[Y_{\alpha, \mu}(t),Y_{\alpha, \mu}(s)]  = \lambda \mathbb{E}[X_1^2]\sum_{n=0}^{\infty}\frac{\mu^{\alpha}\gamma(\mu s; \alpha(1+n))}{\Gamma(\alpha(1+n))}+\lambda^2(\mathbb{E}[X_1])^2{\rm Cov}[E_f(t),E_f(s)].$

where $\gamma(a;b)$ is an incomplete gamma function.
The results follows from Theorem (<ref>) by substituting the value of $\mathbb{E}[E_{\alpha,\mu}(t)]$ (see [26])

\mathbb{E}[E_{\alpha, \mu}(t)]=\sum_{n=0}^{\infty}\frac{\mu^{\alpha}\gamma(\mu t; \alpha(1+n))}{\Gamma(\alpha(1+n))}.\qedhere

Let $\mathbb{E}[X_i]=0, i=1,2,\ldots$, then the correlation of the process is given by

{\rm Corr}[Y_{\alpha, \mu}(t), Y_{\alpha, \mu}(s)]=\sqrt{\frac{\mathbb{E}[E_{\alpha, \mu}(s)]}{\mathbb{E}[E_{\alpha, \mu}(t)]}},

$$ \mathbb{E}[E_{\alpha, \mu}(t)] \sim \frac{t}{\alpha \mu^{\alpha-1}}, \text{as}\; {t\to\infty}, \text{(see in [26])}.

Using (<ref>), we obtain the correlation function of $Y_{\alpha, \mu}(t)$ and $Y_{\alpha, \mu}(s)$. It exhibits LRD property, i.e.

\lim\limits_{t\to\infty}\frac{{\rm Corr}[Y_{\alpha, \mu}(t), Y_{\alpha, \mu}(s)]}{t^{-1/2}}  \sim  \mu^{(\alpha-1)/2}\alpha^{1/2} \sqrt{\mathbb{E}[E_{\alpha,\mu}(s)]}.

Next, we discuss special cases for the TFCPP, where $X_i$ follows some particular type of distribution.

When $X_i,\;i=1,2,\ldots,$ follow exponential distribution with parameter $\eta >0$. The  process  $\{Y_f^\eta(t)\}_{t \geq 0}$ (<ref>) can be represented in the following notation

\begin{align}\label{tem_cpp_ex}
Y_{\alpha, \mu}^\eta(t):= \sum_{i=1}^{N_{\alpha, \mu}(t)} X_i ,t\geq 0.
\end{align}

This process $\{Y_{\alpha, \mu}^\eta(t)\}_{t \geq 0}$ can also be written in the time-changed representation as $\{Y(N_{\alpha, \mu}(t))\}_{t \geq 0}$.
The pdf $P_{\alpha, \mu}^{\eta}(x,t)$ satisfies the following equation

\begin{align*}%\label{Diff_G_FPPok}
\eta \mathcal{D}^{\alpha, \mu}_{t} P_{\alpha, \mu}^{\eta}(x,t) &= -\left[\lambda +\mathcal{D}^{\alpha, \mu}(t)\right]\frac{\partial}{\partial x}  P_{\alpha, \mu}^{\eta}(x,t).
\end{align*}

where $ \mathcal{D}^{\alpha, \mu}_{t}$ is the tempered C-D derivative of order $\alpha \in (0,1)$ with tempering parameter $\mu>0$ is defined by

\mathcal{D}^{\alpha, \mu}_{t}g(t) =  \frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int_{0}^{t}\frac{g(u)du}{(t-u)^{\alpha}} - \frac{g(0)}{\Gamma(1-\alpha)}\int_{t}^{\infty}e^{-\mu r}\alpha r^{-\alpha-1}dr.

with conditions

P_{\alpha, \mu}^{\eta}(x,0)=0, \; \;
\mathbb{P}(Y_{\alpha, \mu}^\eta (t) >0)= 1-e^{-\lambda t}.

When $\mu=0$, the tempered $\alpha$-stable subordinator reduces to the $\alpha$-stable subordinator. Then the process (<ref>) becomes the time-fractional compound Poisson process, as defined in [2].
The mean and covariance are given by

\begin{align*}
\mathbb{E}[Y_{\alpha}^{\eta}(t)] &=\frac{\lambda t^\alpha}{\eta \Gamma(1+\alpha)};\\
%{\rm Var}[Y_f^\eta(t)] &= \frac{3\lambda}{\eta^2}\mathbb{E}[E_{f}(t)]-\frac{2\lambda^2}{\eta^2}\mathbb{E}[E_{f}(t)]^2+\frac{2}{\eta^2}\mathbb{E}[E_{f}(t)^2];\\
{\rm Cov}[Y_{\alpha}^{\eta}(t),Y_{\alpha}^{\eta}(s)] &= \frac{2\lambda s^{\alpha}}{\eta^2 \Gamma(1+\alpha)} +\frac{\lambda^2}{\eta^2}{\rm Cov}[E_\alpha(t),E_\alpha(s)],\;\; s<t.
\end{align*}

$X_i,\; i=1,2,\ldots$ are iid random variables with tempered Mittag-Leffler distribution ([19])

f_{\beta, \eta, \nu}(x)= \lambda e^{-\nu x}\sum_{n=0}^{\infty} (-1)^n(\lambda-\nu^\beta)^n \frac{x^{\beta(n+1)-1}}{\Gamma(\beta(n+1))}, \; \lambda >\nu^{\beta}, x>0.

Then, process $\eqref{GFCPP}$ is defined as

\begin{align*}
Y_{\alpha,\mu}^{\beta, \eta, \nu}(t): = \sum_{i=1}^{N_{\alpha,\mu}(t)}X_{i},t\geq 0,
\end{align*}

where $N_{\alpha,\mu}(t) = N(E_{\alpha, \mu}(t))$ is tempered fractional Poisson process with rate parameter $\lambda>0$ (see [12]). It is called a  TFCPP with tempered Mittag-Leffler jumps.
The LT of pdf of $\{Y_{\alpha,\mu}^{\beta, \eta, \nu}(t)\}_{t \geq 0}$ is given by

\begin{align*}
\mathbb{E}[e^{-sY_{\alpha,\mu}^{\beta, \eta, \nu}(t) }]= \mathbb{E}[e^{-\lambda E_{\alpha, \mu}(t)\frac{(s+\nu)^\beta-\nu^\beta}{\eta+(s+\nu)^\beta-\nu^\beta}}],
\end{align*}

\mathbb{E}[e^{-sX_1}]= \frac{\eta}{\eta+(s+\nu)^\beta-\nu^\beta}.

An alternate representation of $\{Y_{\alpha,\mu}^{\beta, \eta, \nu}(t)\}_{ t \geq 0}$ is given by  time-changing the tempered $\beta$-stable subordinator $\{D_{\beta, \nu}(t)\}_{t \geq 0}, \; \beta \in (0,1]$ with $\{Y_{\alpha, \mu}^{\eta}(t)\}_{t \geq 0}$, i.e.

Y_{\alpha,\mu}^{\beta, \eta, \nu}(t) \stackrel{d}{=} D_{\beta, \nu}(Y_{\alpha, \mu}^{\eta}(t)).

The pdf $P_{\alpha,\mu}^{\beta, \eta, \nu}(x,t)$ satisfies the following fractional differential equation

\begin{align*}%\label{Diff_STCFPP}
\eta \mathcal{D}^{\alpha, \mu}_{t} P_{\alpha,\mu}^{\beta, \eta, \nu}(x,t) &= -\left[\lambda +\mathcal{D}^{\alpha, \mu}_{t}\right]\mathcal{D}^{\beta, \nu}_{x} P_{\alpha,\mu}^{\beta, \eta, \nu}(x,t)\;
\end{align*}

with initial condition

P_{\alpha,\mu}^{\beta, \eta, \nu}(x,0)=0, \; \;
\mathbb{P}(Y_{\alpha,\mu}^{\beta, \eta, \nu} (t) >0)= 1-\mathbb{E}[e^{-\lambda E_{\alpha, \mu}(t)}].

Here, we discuss the particular values of parameter of the above-introduced processes.

* For $\nu=0$, then the process  is called time-changed  $\beta$-stable process, i.e. $\{D_\beta(Y_{\alpha, \mu}^{\eta}(t))\}_{t \geq 0}$.

* For $\mu=0$, the process behaves as time-changed tempered $\beta$-stable subordinator, i.e. $\{D_{\beta, \nu}(Y_{\alpha}^{\eta}(t))\}_{t \geq 0}$

* When $\mu=0,\; \nu=0$, then the process behaves as time-changed in $\beta$-stable subordinator $\{D_{\beta}(t)\}_{t \geq 0}$ with $\{Y_{\alpha}^{\eta}(t)\}_{t \geq 0}$, i.e. $\{D_{\beta}(Y_{\alpha}^{\eta}(t))\}_{t \geq 0}$ (see [2]).

Let $X_i, i=1,2,\ldots,$ are as distributed in (<ref>). Then the process

\begin{equation*}
Y_{\alpha,\mu}^{g}(t) := \sum_{i=1}^{N_{\alpha,\mu}(t)} X_i,t\geq 0,
\end{equation*}

is called TFCPP with Bernstéin jumps.

The subsequent result follow from Theorem (<ref>) as a particular case. This process has a time-changed representation, i.e.

\begin{equation*}
Y_{\alpha,\mu}^{g}(t) \stackrel{d}{=}D_{g}(Y_{\alpha,\mu}^{\eta}(t)).
\end{equation*}

The pdf $P_{\alpha,\mu}^{g}(x,t)$ of $\{Y_{\alpha,\mu}^{g}(t)\}_{t \geq 0}$ satisfies the following equation

\begin{align*}
\eta \mathcal{D}^{\alpha,\mu}_{t} P_{\alpha,\mu}^{g}(x,t) &= -\left[\lambda +\mathcal{D}^{\alpha,\mu}_{t}\right]\mathcal{D}^{g}_{x} P_{\alpha,\mu}^{g}(x,t)\;
\end{align*}

with conditions

P_{\alpha,\mu}^{g}(x,0)=0, \; \;
\mathbb{P}(Y_{\alpha,\mu}^g (t) >0)= 1-\mathbb{E}[e^{-\lambda E_{\alpha,\mu}(t)}].

Let $X_i,i=1,2,\ldots $ be iid truncated geometrically distributed random variables. Using (<ref>), we define the process

Y_{\alpha,\mu}^{\rho,k}(t): = \sum_{i=1}^{N_{\alpha,\mu}(t)} X_i,t\geq 0.

It  is called as the  tempered fractional PAPoK.
Next, we mentioned the particular case of the  Theorem (<ref>) for the tempered fractional PAPoK. The pmf $P^{\rho,k}_{\alpha,\mu}(n,t)=\mathbb{P}[Y_{\alpha,\mu}^{\rho,k}(t)=n]$ satisfy following fractional differential equation

\begin{align*}
\mathcal{D}^{\alpha,\mu}_{t} P^{\rho,k}_{\alpha,\mu}(n,t) &= -\lambda \left(1-\frac{1-\rho}{1-\rho^k}\sum_{j=1}^{n \wedge k}\rho^{j-1}B^j\right)  P^{\rho,k}_{\alpha,\mu}(n,t),\\
\mathcal{D}^{\alpha,\mu}_{t} P^{\rho,k}_{\alpha,\mu}(0,t) &= -\lambda P^{\rho,k}_{\alpha,\mu}(0,t).
\end{align*}

The time-changed representation of $ \{Y_{\alpha,\mu}^{\rho,k}(t)\}_{t \geq 0}$ by time-changing in PAPoK $\{N^{k}_A(t)\}_{t \geq 0}$ with $\{E_{\alpha,\mu}(t)\}_{t \geq 0}$ such that

Y_{\alpha,\mu}^{\rho,k}(t) \stackrel{d}{=}N^k_A(E_{\alpha,\mu}(t)) ,t\geq 0.

When $\mu=0$, then $\{Y_{\alpha, 0}^{\rho,k}(t)\}_{t \geq 0}$ is called fractional FPAPok, defined in [15].

Let $X_i,\;i=1,2,\ldots,$ be the discrete uniform distributed random variables. From equation (<ref>), reduces the tempered fractional PPoK $\{Y_{\alpha, \mu}^k(t)\}_{t \geq 0}$, which is introduced and studied in [11].

§.§ Inverse IG fractional CPP

Consider the inverse subordinator (<ref>) associated with inverse Gaussian  Bernstéin function (<ref>) $f(s)=\delta(\sqrt{2s+\gamma^2}-\gamma)$, denoted by $\{E_{\delta,\gamma}(t)\}_{t\geq 0}$. Let $N_{\delta,\gamma}(t):=N(E_{\delta,\gamma}(t)),t\geq 0$ be the inverse IG process (see [33]). The process (<ref>) defined by

\begin{align}\label{def:IG-FCPP}
Y_{\delta,\gamma}(t):= \sum_{i=1}^{N_{\delta,\gamma}(t)}X_i,t\geq 0,
\end{align}

is called as the inverse IG fractional CPP with  $X_i,\;i=1,2,\ldots,$ be the iid jumps having common distribution $F_X $.

Let $\{D_{\delta,\gamma}(t)\}_{t \geq 0}$ be IG Lévy process with the LT (see [8])

\begin{equation*}
\mathbb{E}(e^{-s D_{\delta,\gamma}(t)}) = e^{-t \delta(\sqrt{2s+\gamma^2}-\gamma)}.
\end{equation*}

The Lévy measure $\nu_{\delta,\gamma}$ corresponding to the inverse Gaussian  subordinator is given by (see [8])

\nu_{\delta,\gamma}(dx)= \frac{\delta}{\sqrt{2\pi x^3}}e^{-\gamma^2x/2}\mathbb{I}_{\{x>0\}}dx.

Further, we define convolution type fractional derivative or non-local operator corresponding to the inverse of IG subordinator, we have

\begin{align*}
\Bar{\nu}_{\delta,\gamma}(s)  = \nu_{\delta,\gamma}(s,\infty) &= \int_{s}^{\infty}\frac{\delta}{\sqrt{2\pi x^3}}e^{-\gamma^2x/2}du, \; s>0,\nonumber\\
&= \sqrt{\frac{2}{\pi}}\delta s^{-1/2}e^{-\gamma^2s/2}-\frac{\delta \gamma}{\sqrt{\pi}}\Gamma{\left(1/2;\gamma^2s/2\right)},
\end{align*}

where $\Gamma(a;b)= \int_{b}^{\infty} u^{a-1}e^{-u}dt$ is the upper incomplete gamma function.
Using (<ref>), the generalized C-D fractional derivative corresponding to the IG subordinator is of the form

\begin{equation}\label{caputo_IG}
\mathcal{D}^{\delta, \gamma}_t V(t)= \frac{d}{dt}\int_{0}^{t} \ v(s)\left( \sqrt{\frac{2}{\pi}}\delta (t-s)^{-1/2}e^{-\gamma^2s/2}-\frac{\delta \gamma}{\sqrt{\pi}}\Gamma{\left(1/2,\gamma^2(t-s)/2\right)}\right)ds- v(0)\Bar{\nu}_{\delta,\gamma}(t).
\end{equation}

The generalized R-L
derivative corresponding to the inverse of IG subordinator is

\begin{equation}\label{RL_IG}
\mathbb{D}^{\delta, \gamma}_t v(t)=
\frac{d}{dt}\int_{0}^{t} \ v(s)\left( \sqrt{\frac{2}{\pi}}\delta (t-s)^{-1/2}e^{-\gamma^2(t-s)/2}-\frac{\delta \gamma}{\sqrt{\pi}}\Gamma{\left(1/2,\gamma^2(t-s)/2\right)}\right)ds.
\end{equation}

The pdf  $h_{\delta,\gamma}(x,t)$ of $\{E_{\delta,\gamma}(t)\}_{t \geq 0}$ solves the following fractional differential equation

\begin{align}\label{DDE_IIGS}
\mathbb{D}^{\delta, \gamma}_{t}h_{\delta,\gamma}(x,t) &= -\frac{\partial}{\partial x}h_{\delta,\gamma}(x,t), x>0,
\shortintertext{ with initial condition}
h_{\delta,\gamma}(x,0) & =\delta(x),~~
h_{\delta,\gamma}(0,t)  =\nu_G(t). \nonumber
\end{align}

Taking LT on both sides of (<ref>), it yields

\begin{align*}
\mathcal{L}_{t}\{\mathbb{D}^{\delta, \gamma}_t v(t)\}&= s \mathcal{L}_{t}(v(t))\left[\mathcal{L}_{t}\left(\sqrt{\frac{2}{\pi}}\delta (t)^{-1/2}e^{-\gamma^2t/2}\right)- \mathcal{L}_{t}\left(\frac{\delta \gamma}{\sqrt{\pi}}\Gamma{\left(1/2,\gamma^2t/2\right)}\right)\right]\nonumber\\
& =  \delta s \mathcal{L}_{t}(v(t))\left[\frac{2}{\sqrt{(2s+\gamma^2)}}-\frac{\gamma}{s}\frac{\sqrt{(2s+\gamma^2)}-\gamma}{\sqrt{(2s+\gamma^2)}}\right]\nonumber\\
& = \mathcal{L}_{t}(v(t)) \delta(\sqrt{2s+\gamma^2}-\gamma).
\end{align*}

Now,  applying  the LT with respect to $x$ on the both sides of (<ref>), we get

\mathbb{D}^{\delta, \gamma}_{t}\mathcal{L}_{x}[h_{\delta,\gamma}(x,t)](y)= -y\mathcal{L}_{x}[h_{\delta,\gamma}(x,t)](y)-h_{\delta,\gamma}(0,t).

Again, taking the LT with respect to $t$, we have that

\delta(\sqrt{2s+\gamma^2}-\gamma)  \mathcal{L}_{t}[\mathcal{L}_{x}[h_{\delta,\gamma}(x,t)](y)](s)  = - y \mathcal{L}_{t}[\mathcal{L}_{x}[h_{\delta,\gamma}(x,t)](y)](s)+\frac{\delta(\sqrt{2s+\gamma^2}-\gamma)}{s}.

We obtain

\mathcal{L}_{t}[\mathcal{L}_{x}[h_{\delta,\gamma}(x,t)](y)](s)= \frac{\delta(\sqrt{2s+\gamma^2}-\gamma)}{s\left( y+\delta(\sqrt{2s+\gamma^2}-\gamma)\right)},

which is the Laplace–Laplace transform of the pdf $h_{\delta,\gamma}(x,t)$ (see <cit.>). This completes the proof.

The pmf $P_{\delta,\gamma}(n,t)=\mathbb{P}[Y_{\delta,\gamma}(t)=n]$ satisfy following fractional differential equation

\begin{align*}
\mathcal{D}^{\delta, \gamma}_{t} P_{\delta, \gamma}(n,t) &= -\lambda  P_{\delta, \gamma}(n,t) +\lambda \int_{-\infty}^{\infty} P_{\delta, \gamma}(n-x,t) F_X(x) dx.
\end{align*}

The proof is similar to the proof of Theorem <ref> and hence it is omitted here.

Let $X_i,\;i=1,2,\ldots,$ are exponentially distributed with parameter $\eta$ in (<ref>). Then, the process $\{Y_f^\eta(t)\}_{t \geq 0}$ can be written as

\begin{align}\label{IIG_cpp_ex}
Y_{\delta, \gamma}^\eta(t):= \sum_{i=1}^{N_{\delta, \gamma}(t)} X_i,~t\geq 0.
\end{align}

This process $\{Y_{\delta, \gamma}^\eta(t)\}_{t \geq 0}$ can also represented as $\{Y(E_{\delta, \gamma}(t))\}_{t \geq 0}$.
The pdf $P_{\delta, \gamma}^{\eta}(x,t)$ satisfies the following equation

\begin{align*}
\eta \mathcal{D}^{\delta, \gamma}_{t} P_{\delta, \gamma}^{\eta}(x,t)(x,t) &= -\left[\lambda +\mathcal{D}^{\delta, \gamma}(t)\right]\frac{\partial}{\partial x}  P_{\delta, \gamma}^{\eta}(x,t),
\end{align*}

where $ \mathcal{D}^{\delta, \gamma}_{t}$ fractional derivative (<ref>), with initial condition

P_{\delta, \gamma}^{\eta}(x,0)=0, \; \;
\mathbb{P}(Y_{\delta, \gamma}^\eta (t) >0)= 1-e^{-\lambda t}.

Let $X_i,i=1,2,\ldots$ be Mittag-Leffler distributed random variables with parameter $0<\beta<1$ and $ \eta>0$. Then, we define new process $\{Y_{\delta, \gamma}^{\beta, \eta}(t)\}_{ t\geq 0}$ such as

Y_{\delta, \gamma}^{\beta, \eta}(t):= \sum_{i=1}^{N_{\delta,\gamma}(t)}X_{i},~t\geq 0.

The LT of  $\{Y_{\delta, \gamma}^{\beta, \eta}(t)\}_{t \geq 0}$ is given by

\begin{align*}
\mathbb{E}[e^{-sY_{\delta, \gamma}^{\beta, \eta}(t) }]= \mathbb{E}\left[e^{-\lambda E_{\delta, \gamma}(t)\frac{s^{\beta}}{\eta+s^{\beta}}}\right].
\end{align*}

The process $Y_{\delta, \gamma}^{\beta, \eta}(t)$ can be represented in terms of the $\beta$-stable subordinator, denoted by $D_{\beta}(t)$, time-changed with independent $\{Y_{\delta, \gamma}^{\eta}(t)\}_{t\geq0}$ (<ref>), i.e.

Y_{\delta, \gamma}^{\beta, \eta}(t) \stackrel{d}{=} D_{\beta}(Y_{\delta, \gamma}^{\eta}(t)),~t\geq 0.

The pdf $P_{\delta, \gamma}^{\beta, \eta}(x,t)$ satisfies the following fractional differential equation

\begin{align*}%\label{Diff_IIGCFPP}
\eta \mathcal{D}^{\delta, \gamma}_{t} P_{\delta, \gamma}^{\beta, \eta}(x,t) &= -\left[\lambda +\mathcal{D}^{\delta, \gamma}_{t}\right]\mathcal{D}^{\beta}_{x} P_{\delta, \gamma}^{\beta, \eta}(x,t)\;
\end{align*}

with initial conditions

P_{\delta, \gamma}^{\beta, \eta}(x,0)=0, \; \;
\mathbb{P}(Y_{\delta, \gamma}^{\beta, \eta}(t) >0)= 1-\mathbb{E}[e^{-\lambda E_{\delta, \gamma}(t)}].

Let $X_i,i=1,2,\ldots$ are iid random variables with LT

\mathbb{E}[e^{-sX_1}]=  \frac{\eta}{ \theta(\sqrt{2s+\chi^2}-\chi)+\eta}.

Note that the distribution of $X_i$'s coincides with the distribution of inter-arrival times of inverse IG subordinated Poisson renewal process. The new stochastic process $\{Y_{\delta, \gamma}^{\theta, \chi}(t)\}_{t \geq 0}$ can be  defined as

\begin{align*}
Y_{\delta, \gamma}^{\theta, \chi}(t): = \sum_{i=1}^{N_{\delta,\gamma}(t)}X_{i},~t\geq 0.
\end{align*}

The LT of pdf of $\{Y_{\delta, \gamma}^{\theta, \chi}(t)\}_{t \geq 0}$ is given by

\begin{align*}
\mathbb{E}[e^{-sY_{\delta, \gamma}^{\theta, \chi}(t) }]= \mathbb{E}\left[e^{-\lambda E_{\delta, \gamma}(t)\frac{\theta(\sqrt{2s+\chi^2}-\chi)}{\eta+\theta(\sqrt{2s+\chi^2}-\chi)}}\right].
\end{align*}

The process $\{Y_{\delta, \gamma}^{\theta, \chi}(t)\}_{t \geq 0}$ can be represented in terms of the IG subordinator, denoted by $\{D_{\theta, \chi}(t)\}_{t \geq 0}$, time-changed with independent $\{Y_{\delta, \gamma}^{\eta}(t)\}_{t\geq0}$ (<ref>), i.e.

Y_{\delta, \gamma}^{\theta, \chi}(t) \stackrel{d}{=} D_{\theta, \chi}(Y_{\delta, \gamma}^{\eta}(t)),~t\geq 0.

The pdf $P_{\delta, \gamma}^{\theta, \eta, \chi}(x,t)$ satisfies the following fractional differential equation

\begin{align*}%\label{Diff_IIGCFPP}
\eta \mathcal{D}^{\delta, \gamma}_{t} P_{\delta, \gamma}^{\theta, \eta, \chi}(x,t) &= -\left[\lambda +\mathcal{D}^{\delta, \gamma}_{t}\right]\mathcal{D}^{\theta, \chi}_{x} P_{\delta, \gamma}^{\theta, \eta, \chi}(x,t)\;
\end{align*}

with initial condition

P_{\delta, \gamma}^{\theta, \eta, \chi}(x,0)=0, \; \;
\mathbb{P}(Y_{\delta, \gamma}^{\theta, \chi}(t) >0)= 1-\mathbb{E}[e^{-\lambda E_{\delta, \gamma}(t)}].

Let $X_i, i=1,2,\cdot,$ are distributed as mentioned in (<ref>). Then the process $Y_{\delta, \gamma}^{g}$ is defined from the equation (<ref>),

\begin{equation}\label{Time_IIGFPP_Bern}
Y_{\delta, \gamma}^{g}(t) = \sum_{i=1}^{N_{\delta, \gamma}(t)} X_i,
\end{equation}

is called inverse IG CPP with Bernstéin jumps.

These are the following results from Theorem (<ref>) in the subsection(<ref>). This process has a time-changed representation, i.e.

\begin{equation}
Y_{\delta, \gamma}^{g}(t) \stackrel{d}{=}D_{g}(Y_{\delta, \gamma}^{\eta}(t)).
\end{equation}

The density $P_{\delta, \gamma}^{g}(x,t)$ satisfies the following equation

\begin{align}
\eta \mathcal{D}^{\delta, \gamma}_{t} P_{\delta, \gamma}^{g}(x,t) &= -\left[\lambda +\mathcal{D}^{\delta, \gamma}_{t}\right]\mathcal{D}^{g}_{x} P_{\delta, \gamma}^{g}(x,t)\;
\end{align}

with conditions

P_{\delta, \gamma}^{g}(x,0)=0, \; \;
\mathbb{P}(Y_{\delta, \gamma}^g (t) >0)= 1-\mathbb{E}[e^{-\lambda E_{\delta, \gamma}(t)}].

Let $X_i,\;i=1,2,\ldots,$ be the iid discrete uniform random variables and $\{N_{\delta, \gamma}(t)\}_{ t \geq 0}$ be the time-changed Poisson process with inverse IG subordinator. Then, the process (<ref>) is  defined,

\begin{equation*}
Y_{\delta, \gamma}^{k}(t) := \sum_{i=1}^{N_{\delta, \gamma}(t)} X_i,,~t\geq 0
\end{equation*}

and called as the inverse IG fractional CPP of order $k$. The pmf $P^{k}_{\delta, \gamma}(n,t)=\mathbb{P}[Y_{\delta, \gamma}^{k}(t)=n]$ satisfy following fractional differential-difference equation is given by

\begin{align*}
\mathcal{D}^{\delta, \gamma}_{t} P^{k}_{\delta, \gamma}(n,t) &= -k \lambda \left(1-\frac{1}{k}\sum_{j=1}^{n \wedge k} B^{j}\right)P^{k}_{\delta, \gamma}(n,t),\; n>0,\\
\mathcal{D}^{\delta, \gamma}_{t}P^{k}_{\delta, \gamma}(0,t)  &= - k \lambda P^{k}_{\delta, \gamma}(0,t).
\end{align*}

Let $X_i,i=1,2,\ldots,$ be iid truncated geometrically distributed random variables. From the equation (<ref>), we define the process

Y_{\delta, \gamma}^{\rho,k}(t): = \sum_{i=1}^{N_{\delta, \gamma}(t)} X_i,,~t\geq 0

is called inverse IG fractional PAPoK.
The following results can be obtained as a particular case of the Theorem (<ref>). The pmf $P^{\rho,k}_{\delta, \gamma}(n,t)=\mathbb{P}[Y_{\delta, \gamma}^{\rho,k}(t)=n]$ satisfy following fractional differential equation

\begin{align*}
\mathcal{D}^{\delta, \gamma}_{t} P^{\rho,k}_{\delta, \gamma}(n,t) &= -\lambda \left(1-\frac{1-\rho}{1-\rho^k}\sum_{j=1}^{n \wedge k}\rho^{j-1}B^j\right)  P^{\rho,k}_{\delta, \gamma}(n,t),\\
\mathcal{D}^{\delta, \gamma}_{t} P^{\rho,k}_{\delta, \gamma}(0,t) &= -\lambda P^{\rho,k}_{\delta, \gamma}(0,t).
\end{align*}

We can express the the process $ \{Y_{\delta, \gamma}^{\rho,k}(t)\}_{t \geq 0}$ in a time-changed form as follows

Y_{\delta, \gamma}^{\rho,k}(t)\stackrel{d}{=}N^k_A(E_{\delta, \gamma}(t)),~t\geq 0.

We now summarize the results obtained in this section in the following table.

Summary of results obtained in Section <ref>

Jump Size Distributions     DDE     Time-changed Representations
Exponential     $\eta \mathcal{D}^{\alpha, \mu}_{t}  = -\left[\lambda +\mathcal{D}^{\alpha, \mu}(t)\right]\frac{\partial}{\partial x} $     $Y(E_{\alpha, \mu}(t))$

Mittag-Leffler (ML)      $\eta \mathcal{D}^{\alpha, \mu}_{t}  = -\left[\lambda +\mathcal{D}^{\alpha, \mu}_{t}\right]\mathcal{D}^{\beta}_{x} $     $D_{\beta}(Y_{\alpha, \mu}^{\eta}(t))$

Tempered ML      $\eta \mathcal{D}^{\alpha, \mu}_{t}  = -\left[\lambda +\mathcal{D}^{\alpha, \mu}_{t}\right]\mathcal{D}^{\beta, \nu}_{x} $     $D_{\beta, \nu}(Y_{\alpha, \mu}^{\eta}(t))$

Inter-times Bernstéin      $\eta \mathcal{D}^{\alpha,\mu}_{t}  = -\left[\lambda +\mathcal{D}^{\alpha,\mu}_{t}\right]\mathcal{D}^{g}_{x}$     $D_{g}(Y_{\alpha,\mu}^{\eta}(t))$

Truncated geometric      $\mathcal{D}^{\alpha, \mu}_{t} = -\lambda \left(1-\frac{1-\rho}{1-\rho^k}\sum_{j=1}^{n \wedge k}\rho^{j-1}B^j\right)$     $N_A^k(E_{\alpha,\mu}(t))$

Discrete uniform      $\mathcal{D}^{\alpha,\mu}_{t} =- \left[{\left( \mu + k \lambda \left(1-\frac{1}{k}\sum_{j=1}^{n \wedge k} B^{j}\right)\right)}^{\alpha} - \mu^{\alpha}\right]$     $N^k(E_{\alpha,\mu}(t))$

Summary of results obtained in Section <ref>

Jump Size Distributions     DDE     Time-changed Representations
Exponential     $\eta \mathcal{D}^{\delta, \gamma}_{t}  = -\left[\lambda +\mathcal{D}^{\delta, \gamma}(t)\right]\frac{\partial}{\partial x} $     $Y(N_{\delta, \gamma}(t))$

Mittag-Leffler (ML)      $\eta \mathcal{D}^{\delta, \gamma}_{t}  = -\left[\lambda +\mathcal{D}^{\delta, \gamma}_{t}\right]\mathcal{D}^{\beta}_{x} $     $D_{\beta}(Y_{\delta, \gamma}^{\eta}(t))$

Inter-times inverse of IGS      $\eta \mathcal{D}^{\delta, \gamma}_{t}  = -\left[\lambda +\mathcal{D}^{\delta, \gamma}_{t}\right]\mathcal{D}^{\theta, \chi}_{x} $     $D_{\theta, \chi}(Y_{\delta, \gamma}^{\eta}(t))$

Inter-times Bernstéin      $\eta \mathcal{D}^{\delta, \gamma}_{t}  = -\left[\lambda +\mathcal{D}^{\delta, \gamma}_{t}\right]\mathcal{D}^{g}_{x}$     $D_{g}(Y_{\delta, \gamma}^{\eta}(t))$

Truncated geometric      $\mathcal{D}^{\delta, \gamma}_{t} = -\lambda \left(1-\frac{1-\rho}{1-\rho^k}\sum_{j=1}^{n \wedge k}\rho^{j-1}B^j\right)$     $N_A^k(E_{\delta, \gamma}(t))$

Discrete uniform      $\mathcal{D}^{\delta, \gamma}_{t} =- {\left[] k \lambda \left(1-\frac{1}{k}\sum_{j=1}^{n \wedge k} B^{j}\right)\right]}^{\alpha}$     $N^k(E_{\delta, \gamma}(t))$

§ SIMULATIONS

In this section, we simulate the sample trajectories of the special cases of the  GFCPP $\{Y_f(t)\}_{t\geq 0}$ (<ref>).
First, we reproduce an algorithm for generating the sample paths for CPP $\{Y(t)\}_{t\geq 0}$ (<ref>) with the jump size distribution $F_X$.

Simulation of the CPP
[1]

$\lambda>0 $, $T\geq0$, and $\theta$ (parameter of $F_X$).
$Y(t)$,  simulated sample paths for the CPP with jump size distribution $F_X$.

Initialisation : $t=0$, $Y=0$ and $v=0$.
$t < T$
generate a uniform random variable $U \sim U(0, 1)$.
set $t \leftarrow t +\left[-\frac{1}{\lambda}\log{U}\right]$.
generate an independent random variable $X$ with distribution $F_X$ with parameter $\theta$.
set $v \leftarrow v + X$ and append in $Y$.
return $Y.$

Here, $Y$ denotes the number of events that occurred up to time $T>0$.

Now, we present algorithm for time-changed by IIGS and ITSS  processes as we use the corresponding algorithms mentioned above for respective subordinators.

Simulation for time-changed CPP with IIGS (ITSS)
[1]

Parameter $\delta$ and $\gamma$ for respective IIGS $E_{\delta,\gamma}(t)$ (ITSS $E_{\alpha, \mu}(t)$), $\lambda>0$, $T\geq0$, $\eta>0$.
The sample paths of the subordinated Poisson process $Y(E_{\delta,\gamma}(t_i)),\ 1 \leq i \leq n.$

Initialisation :
set $dt = \frac{T}{n}$ and choose $n+1$ uniformly spaced time points $0=t_0, t_1, \dots, t_n=T$ with $dt=t_1-t_0$.

generate the values $E_{\delta,\gamma}(t_i),\ 1 \leq i \leq n,$ ( $E_{\alpha,\mu}(t)$) for the IIGS (ITSS) at time points $t_1, t_2, ..., t_n$ using the Algorithms from [21]

use the values $E_{\delta,\gamma}(t_i),\ 1 \leq i \leq n,$ generated in Step 2, as time points and compute the number of events of the subordinated CPP $Y(E_{\delta,\gamma}(t_i)),\ 1 \leq i \leq n,$ using Algorithm <ref>.
return	  $Y(E_{\delta,\gamma}(t_i))$

Using Algorithm <ref> and <ref>, we generate the sample paths for the chosen set of parameters which is given below

Exponential    Mittag-Leffler     Discrete Uniform    Discrete logarithm
CPP (Fig <ref>)     $\eta=2,\lambda=4$     $\eta=2, \lambda=4, \alpha=0.9$     $\lambda=4, k=5$     $\lambda=4, q=0.5$

CPP-IIGN (Fig <ref>)   $\eta=2,\lambda=4, \delta=0.3, \gamma=1$    $\eta=2,\lambda=4, \delta=0.3, \gamma=1, \beta=0.9$    $\lambda=4, \delta=0.3, \gamma=1, k=5$    $\lambda=4, \delta=0.3, \gamma=1, q=0.5$

CPP-ITSS (Fig <ref>)   $\eta=2,\lambda=4, \alpha=0.7, \mu=2$    $\eta=2,\lambda=4, \alpha=0.7, \mu=2, \beta= 0.9$   $\lambda=4, \alpha=0.7, \mu=2, k=5$    $\lambda=4, \alpha=0.7, \mu=2, q= 0.5$

Table of parameters for the generated sample paths

CPP with exponential, Mittag-Leffler, discrete uniform, discrete logarithmic jumps sample trajectories

Inverse IG fractional CPP with exponential, Mittag-Leffler, discrete uniform, discrete logarithmic jumps sample trajectories

Tempered fractional CPP exponential, Mittag-Leffler, discrete uniform, discrete logarithmic jumps sample trajectories

$Y_f(t)$Generalized fractional compound Poisson process (GFCPP)
$Y_f^{\eta}(t)$GFCPP with exponential jumps
$Y_f^{\beta, \eta}(t)$GFCPP with Mittag-Leffler jumps
$Y_f^g(t)$GFCPP with Bernstéin jumps
$Y_f^k(t)$Generalized fractional Poisson process of order $k$ (GFPPoK)
$Y_f^{\rho, k}(t)$Generalized fractional Pólya Aeppli process of order $k$ (GPAPoK)
$Y_f^q(t)$Generalized fractional negative binomial process (GFNBP)
$Y_{\alpha, \mu}(t)$Tempered fractional CPP (TFCPP)
$Y_{\alpha, \mu}^{\eta}(t)$TFCPP with exponential jump
$Y_{\alpha, \mu}^{\beta,\eta, \nu}(t)$TFCPP with tempered Mittag-Leffler jumps
$Y_{\alpha, \mu}^{g}(t)$TFCPP with Bernstéin jumps
$Y_{\alpha, \mu}^{\rho,k}(t)$Tempered fractional PAPoK
$Y_{\alpha, \mu}^{k}(t)$Tempered fractional PPoK
$Y_{\delta, \gamma}(t)$Inverse IG fractional CPP
$Y_{\delta, \gamma}^k(t)$Inverse IG fractional PPoK
$Y_{\delta, \gamma}^{\rho,k}(t)$Inverse IG fractional PAPoK
$Y_{\delta, \gamma}^{\eta}(t)$Inverse IG fractional CPP with exponential jumps
$Y_{\delta, \gamma}^{\beta,\eta}(t)$Inverse IG fractional CPP with Mittag-Leffler jumps
$Y_{\delta, \gamma}^{\theta, \chi}(t)$Inverse IG fractional CPP with IG type jumps
$Y_{\delta, \gamma}^{g}(t)$Inverse IG fractional CPP with Bernstéin jumps

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