2.1. Generalized counting process

Let $\mathbb{N}_0$ be the set of non-negative integers and let $\Lambda=\lambda_{1}+\lambda_{2}+\cdots+\lambda_{k}$ . The state probabilities $p(n,t)=\mathbb{P}\{M(t)=n\}$ of a GCP are given by (see [Reference Di Crescenzo, Martinucci and Meoli14])

(7) \begin{equation}p(n,t)=\sum_{\Omega(k,n)}\biggl(\prod_{j=1}^{k}\frac{(\lambda_{j}t)^{x_{j}}}{x_{j}!}\biggr){\mathrm{e}}^{-\Lambda t},\quad n\ge 0,\end{equation}

where $\Omega(k,n)\,:\!=\{(x_{1},x_{2},\ldots,x_{k})\,:\,\sum_{j=1}^{k}jx_{j}=n,\ x_{j}\in\mathbb{N}_0\}$ .

Its PGF $G(u,t)=\mathbb{E}(u^{M(t)})$ is given by (see [Reference Kataria and Khandakar21])

(8) \begin{equation}G(u,t)=\exp\biggl(-\sum_{j=1}^{k}\lambda_{j}(1-u^{j})t\biggr), \quad |u|\le 1.\end{equation}

The following limiting result holds for the GCP (see [Reference Kataria and Khandakar21, Equation (17)]):

(9) \begin{equation}\lim\limits_{t\to \infty}\frac{M(t)}{t}=\sum_{j=1}^{k}j\lambda_{j}\ \text{in probability}.\end{equation}

2.2. The subordinator and its inverse

Let

(10) \begin{equation}f(s)=c_{1}+c_{2}s+\int_{0}^{\infty}\left(1-{\mathrm{e}}^{-sx}\right)\overline{\nu}(\mathrm{d}x), \quad s\gt0,\ c_{1}\ge0,\ c_{2}\geq0\end{equation}

be a Bernštein function and let $\overline{\nu}(\!\cdot\!)$ be a non-negative Lévy measure on $(0,\infty)$ such that $\displaystyle\int_0^\infty (x\wedge1)\,\overline{\nu}(\mathrm{d}x)\lt\infty$ . A subordinator $\{D_f(t)\}_{t\ge0}$ is a one-dimensional Lévy process characterized by the following Laplace transform (see [Reference Applebaum1, Section 1.3.2]):

\begin{equation*}\mathbb{E}\!\left({\mathrm{e}}^{-sD_f(t)}\right)={\mathrm{e}}^{-tf(s)}.\end{equation*}

Its first passage time is called the inverse subordinator and is defined as

(11) \begin{equation}Y_f (t)\,:\!=\inf\{x\ge0\,:\, D_f (x)\gt t\}, \quad t\ge0.\end{equation}

Let $h_{f}(t,x)$ be the density of $\{Y_{f}(t)\}_{t\ge0}$ . Its Laplace transform is given by (see [Reference Toaldo34])

(12) \begin{equation}\tilde{h}_{f}(s,x)=\int_{0}^{\infty}{\mathrm{e}}^{-st}h_{f}(t,x)\,\mathrm{d}t=\frac{f(s)}{s}\,{\mathrm{e}}^{-xf(s)},\end{equation}

provided that the following condition holds.

Condition I. We have that $\overline{\nu}(0,\infty)=\infty$ and that the tail $\nu(x)=c_{1}+\overline{\nu}(x,\infty)$ of the Lévy measure $\overline{\nu}(\!\cdot\!)$ is absolutely continuous.

Toaldo [Reference Toaldo34] discussed a new type of differential operator that generalizes the Caputo fractional derivative and Riemann–Liouville fractional derivative. Let $u(\!\cdot\!)$ be an absolutely continuous function. Its generalized Caputo derivative associated with the Bernštein function f is defined as (see [Reference Toaldo34, Definition 2.4])

(13) \begin{equation}^f{\mathcal D}_t u(t)\,:\!= c_{2} \frac{{\mathrm{d}}}{{\mathrm{d}} t}u(t)+\int_{0}^{t}\frac{\partial }{\partial t }u(t-x)\nu (x)\,\mathrm{d}x.\end{equation}

It is related to the generalized Riemann–Liouville derivative ${^f}\mathscr{D}_t$ through the following relation (see [Reference Toaldo34, Proposition 2.7]):

(14) \begin{equation}{^f}\mathscr{D}_t u(t)= {^f}\mathcal{D}_t u(t)+\nu(t) u(0).\end{equation}

If $|u(t)|\leq C {\mathrm{e}}^{s_0t}$ , where C and $s_0$ are some constants, then the Laplace transform of generalized Caputo derivative is given by (see [Reference Toaldo34, Lemma 2.5])

(15) \begin{equation}\mathcal{L}\bigl(^f\mathcal{D}_t u(t);s\bigr)=f(s)\tilde{u}(s)-\frac{f(s)}{s}u(0), \quad s\gt s_0.\end{equation}

For $f(s)=s^{\beta}$ with $\beta\in (0,1)$ , the derivative (13) reduces to (see [Reference Toaldo34, Remark 2.6])

\begin{equation*}^f\mathcal{D}_t u(t)=\frac{\mathrm{d}^{\beta}}{\mathrm{d}t^{\beta}}u(t),\end{equation*}

where $\dfrac{\mathrm{d}^{\beta}}{\mathrm{d}t^{\beta}}$ is the Caputo fractional derivative defined as (see [Reference Kilbas, Srivastava and Trujillo24])

(16) \begin{equation}\frac{\mathrm{d}^{\beta}}{\mathrm{d}t^{\beta}}u(t)\,:\!=\left\{\begin{array}{ll}\dfrac{1}{\Gamma{(1-\beta)}}\displaystyle\int^t_{0} (t-s)^{-\beta}u'(s)\,\mathrm{d}s,&\;\; 0\lt\beta\lt1,\\\\u'(t),&\;\;\beta=1.\end{array}\right.\end{equation}

Its Laplace transform is given by (see [Reference Kilbas, Srivastava and Trujillo24, Equation (5.3.3)])

(17) \begin{equation}\mathcal{L}\Big(\frac{\mathrm{d}^{\beta}}{\mathrm{d}t^{\beta}}u(t);s\Big)=s^{\beta}\tilde{u}(s)-s^{\beta-1}u(0), \quad s\gt0.\end{equation}

2.3. The stable subordinator and its inverse

A stable subordinator $\{D_{\beta}(t)\}_{t\ge0}$ for $0\lt\beta\lt1$ is a non-decreasing Lévy process whose Laplace transform is given by $\mathbb{E}({\mathrm{e}}^{-sD_{\beta}(t)})={\mathrm{e}}^{-ts^{\beta}}$ for $s\gt0$ . Its associated Bernštein function is $f_{2}(s)=s^{\beta}$ . Its first passage time $\{Y_{\beta}(t)\}_{t\ge0}$ , which is defined as $Y_{\beta}(t)\,:\!=\inf\{x\ge 0\,:\,D_{\beta}(x)\gt t\}$ , is called the inverse stable subordinator.

Let $h_{\beta}(t,x)$ be the density of $\{Y_{\beta}(t)\}_{t\ge0}$ . Its Laplace transform is given by

(18) \begin{equation}\tilde{h}_{\beta}(s,x)=s^{\beta-1}{\mathrm{e}}^{-xs^{\beta}}.\end{equation}

The mean and variance of $\{Y_{\beta}(t)\}_{t\ge0}$ are (see [Reference Leonenko, Meerschaert, Schilling and Sikorskii25])

(19) \begin{align}\mathbb{E}\!\left(Y_{\beta}(t)\right)=\frac{t^{\beta}}{\Gamma(\beta+1)},\end{align}

(20) \begin{align}\operatorname{Var}\!\left(Y_{\beta}(t)\right)=\left(\frac{2}{\Gamma(2\beta+1)}-\frac{1}{\Gamma^{2}(\beta+1)}\right)t^{2\beta}.\end{align}

For fixed $t_1$ and large $t_2$ , the following asymptotic result holds for the covariance:

(21) \begin{equation}\operatorname{Cov}\!\left(Y_{\beta}(t_1),Y_{\beta}(t_2)\right)\sim\frac{t_{1}^{2\beta}}{\Gamma(2\beta+1)}.\end{equation}

2.4. Fractional negative binomial processes

The density function g(x, t) of a gamma subordinator $\{Z(t)\}_{t\ge0}$ is given by

\begin{equation*}g(x,t)=\frac{a^{bt}}{\Gamma(bt)}x^{bt-1}{\mathrm{e}}^{-ax}, \quad x\gt0,\end{equation*}

where $a\gt0$ and $b\gt0$ . Its associated Bernštein function is (see [Reference Applebaum1, p. 55])

(22) \begin{equation}f_{1}(s)=b\log(1+s/a), \quad s\gt0.\end{equation}

The space–time-fractional negative binomial process is defined as (see [Reference Kataria and Khandakar22])

\begin{equation*}\mathcal{N}^{\alpha}_{\beta}(t)\,:\!= N^{\alpha,\beta}(Z(t)),\quad t\ge0,\end{equation*}

where the STFPP $\{N^{\alpha,\beta}(t)\}_{t\ge0}$ is independent of the gamma subordinator. For $\alpha=1$ it reduces to the time-fractional negative binomial process (see [Reference Vellaisamy and Maheshwari36]), and for $\beta=1$ it reduces to the space-fractional negative binomial process. The PGF of the time-fractional negative binomial process is (see [Reference Kataria and Khandakar22])

(23) \begin{equation}\mathbb{E}(u^{\mathcal{N}^{1}_{\beta}(t)})=\sum_{k=0}^{\infty}\frac{(\!-\lambda)^{k}(1-u)^{k}}{\Gamma(k\beta+1)}\frac{\Gamma (bt+k\beta)}{a^{k\beta}\Gamma(bt)},\quad |u|\le 1.\end{equation}

Its state probability is given by (see [Reference Vellaisamy and Maheshwari36, Section 4.1])

(24) \begin{equation}\mathbb{P}\{\mathcal{N}^{1}_{\beta}(t)=n\}=\sum_{l=n}^{\infty}(\!-1)^{l+n}\binom{l}{n}\frac{\Gamma(l\beta+bt)}{\Gamma(bt)\Gamma(\beta l+1)}\left(\frac{\lambda}{a^{\beta}}\right)^{l},\quad n\ge0.\end{equation}

2.5. Long-range dependence property

Let $s\gt0$ be fixed. The process $\{X(t)\}_{t\ge0}$ is said to exhibit the LRD property if its correlation function has the following asymptotic behaviour (see [Reference D’Ovidio and Nane15, Reference Maheshwari and Vellaisamy27]):

\begin{equation*}\operatorname{Corr}(X(s),X(t))\sim c(s)t^{-\theta}\quad \text{as}\ t\rightarrow\infty\end{equation*}

for some $c(s)\gt0$ and $\theta\in(0,1)$ . If $\theta\in(1,2)$ , then it is said to possesses the short-range dependence property.

3.1. The NGCP time-changed by the inverse stable subordinator

For $0\lt\beta\lt1$ , let $\{\mathcal{M}_{\beta}(t)\}_{t\ge0}$ be a time-changed NGCP defined as follows:

\begin{equation*}\mathcal{M}_{\beta}(t)\,:\!=\mathcal{M}(Y_{\beta}(t)),\quad t\ge0,\end{equation*}

where $\{\mathcal{M}(t)\}_{t\ge0}$ is independent of the inverse stable subordinator $\{Y_{\beta}(t)\}_{t\ge0}$ . We call it the NGFCP. Its increment process is defined as

\begin{equation*}I_{\beta}(t,v)\,:\!=\mathcal{M}(Y_{\beta}(t)+v)-\mathcal{M}(v),\quad t\ge0,\end{equation*}

whose marginal distributions, $q^{\beta}_{n}(t,v)=\mathbb{P}\{I_{\beta}(t,v)=n\}$ for $n\ge0$ , are given by

(33) \begin{align}q^{\beta}_{n}(t,v)&=\int_{0}^{\infty}q_{n}(u,v)h_{\beta}(t,u)\,\mathrm{d}u\\&=\int_{0}^{\infty}\sum_{\Omega(k,n)}\biggl(\prod_{j=1}^{k}\frac{(\Lambda_{j}(v,u+v))^{x_{j}}}{x_{j}!}\biggr){\mathrm{e}}^{-\sum_{j=1}^{k}\Lambda_{j}(v,u+v)}h_{\beta}(t,u)\,\mathrm{d}u.\nonumber\end{align}

Here, $h_{\beta}(t,u)$ is the density of $\{Y_{\beta}(t)\}_{t\ge0}$ . So the marginal distributions of the NGFCP are given by

(34) \begin{align}q^{\beta}_{n}(t)=q^{\beta}_{n}(t,0)&=\int_{0}^{\infty}q_{n}(u,0)h_{\beta}(t,u)\mathrm{d}u\nonumber\\&=\int_{0}^{\infty}\sum_{\Omega(k,n)}\biggl(\prod_{j=1}^{k}\frac{(\Lambda_{j}(u))^{x_{j}}}{x_{j}!}\biggr){\mathrm{e}}^{-\sum_{j=1}^{k}\Lambda_{j}(u)}h_{\beta}(t,u)\,\mathrm{d}u,\end{align}

where we have used $\Lambda_{j}(0,u)=\Lambda_{j}(u)$ .

Let $\delta_{n,0}$ denote the Kronecker delta function.

Theorem 1. The PMF of the increment process of the NGFCP satisfies the following system of fractional differential–integral equations:

(35) \begin{equation}\frac{\mathrm{d}^{\beta}}{\mathrm{d}t^{\beta}}q^{\beta}_{n}(t,v)=\int_{0}^{\infty}\!\biggl(-\sum_{j=1}^{k}\lambda_{j}(u+v)q_{n}(u,v)+\sum_{j=1}^{n\wedge k}\lambda_{j}(u+v)q_{n-j}(u,v)\biggr)h_{\beta}(t,u)\,\mathrm{d}u,\quad n\ge 0,\end{equation}

with initial condition $q^{\beta}_{n}(0,v)=\delta_{n,0}$ .

Proof. The characteristic function of $I_{\beta}(t,v)$ can be written as

(36) \begin{equation}\hat{q}_\xi^\beta(t,v)=\int_{0}^{\infty}\hat{q}_{\xi}(u,v)h_{\beta}(t,u)\,\mathrm{d}u,\end{equation}

where $\hat{q}_{\xi}(u,v)$ is the characteristic function of I(t, v) given in (32).

Let $\tilde{h}_{\beta}(s,u)$ be the Laplace transform of the density of $\{Y_{\beta}(t)\}_{t\ge0}$ as given in (18). On taking the Laplace transform of (36), we get

\begin{align*}\tilde{\hat{q}}_\xi^\beta(s,v)&=\int_{0}^{\infty}\hat{q}_{\xi}(u,v)\tilde{h}_{\beta}(s,u)\,\mathrm{d}u\\&=s^{\beta-1}\int_{0}^{\infty}\exp\biggl(\sum_{j=1}^{k}\Lambda_{j}(v, u+v)({\mathrm{e}}^{\omega \xi j}-1)\biggr){\mathrm{e}}^{-us^{\beta}}\,\mathrm{d}u\quad (\text{using}\ ({18})\ \text{and} \ ({32}))\\&=s^{\beta-1}\bigg\{\bigg[-\frac{{\mathrm{e}}^{-us^{\beta}}}{s^{\beta}}\exp\biggl(\sum_{j=1}^{k}\Lambda_{j}(v, u+v)({\mathrm{e}}^{\omega \xi j}-1)\biggr)\bigg]_{0}^{\infty}\\&\qquad\quad +\frac{1}{s^{\beta}}\int_{0}^{\infty}\sum_{j=1}^{k}\lambda_{j}(u+v)({\mathrm{e}}^{\omega \xi j}-1)\exp\biggl(\sum_{j=1}^{k}\Lambda_{j}(v, u+v)({\mathrm{e}}^{\omega \xi j}-1)\biggr){\mathrm{e}}^{-us^{\beta}}\,\mathrm{d}u\bigg\}.\end{align*}

Thus,

\begin{equation*}s^{\beta}\tilde{\hat{q}}_\xi^\beta(s,v)-s^{\beta-1}=\!\int_{0}^{\infty}\!\!\sum_{j=1}^{k}\lambda_{j}(u+v)({\mathrm{e}}^{\omega \xi j}-1)\exp\biggl(\sum_{j=1}^{k}\Lambda_{j}(v,u+v)({\mathrm{e}}^{\omega \xi j}-1)\biggr)s^{\beta-1}{\mathrm{e}}^{-us^{\beta}}\mathrm{d}u.\end{equation*}

On taking the inverse Laplace transform and using (17) and (18), we get

\begin{align*}\frac{\mathrm{d}^{\beta}}{\mathrm{d}t^{\beta}}\hat{q}_\xi^\beta(t,v)&=\int_{0}^{\infty}\sum_{j=1}^{k}\lambda_{j}(u+v)({\mathrm{e}}^{\omega \xi j}-1)\exp\biggl(\sum_{j=1}^{k}\Lambda_{j}(v,u+v)({\mathrm{e}}^{\omega \xi j}-1)\biggr)h_{\beta}(t,u)\,\mathrm{d}u\\&=\int_{0}^{\infty}\sum_{j=1}^{k}\lambda_{j}(u+v)({\mathrm{e}}^{\omega \xi j}-1)\hat{p}_\xi(u,v)h_{\beta}(t,u)\,\mathrm{d}u,\end{align*}

where we have used (32) in the last step. Finally, using the inversion formula for the characteristic function gives the required result.

On substituting $v=0$ in Theorem 1, we get the system of differential equations that governs the PMF of the NGFCP.

Corollary 1. The PMF $q^{\beta}_{n}(t)$ of the NGFCP satisfies

(37) \begin{equation}\frac{\mathrm{d}^{\beta}}{\mathrm{d}t^{\beta}}q^{\beta}_{n}(t)=\int_{0}^{\infty}\biggl(-\sum_{j=1}^{k}\lambda_{j}(u)q_{n}(u)+\sum_{j=1}^{n\wedge k}\lambda_{j}(u)q_{n-j}(u)\biggr)h_{\beta}(t,u)\,\mathrm{d}u,\quad n\ge 0,\end{equation}

with initial condition $q^{\beta}_{n}(0)=\delta_{n,0}$ .

Remark 4. On choosing constant rates, that is, $\lambda_{j}(t)=\lambda_{j}$ for $1\le j \le k$ , in (37) we get

\begin{align*}\frac{\mathrm{d}^{\beta}}{\mathrm{d}t^{\beta}}p^{\beta}(n,t)&=\int_{0}^{\infty}\biggl(-\sum_{j=1}^{k}\lambda_{j}p(n,u)+\sum_{j=1}^{n\wedge k}\lambda_{j}p(n-j,u)\biggr)h_{\beta}(t,u)\,\mathrm{d}u,\end{align*}

which reduces to (2). Similarly, by taking constant rates in (34), we get

\begin{equation*}p^{\beta}(n,t)=\int_{0}^{\infty}\sum_{\Omega(k,n)}\biggl(\prod_{j=1}^{k}\frac{(\lambda_{j}u)^{x_{j}}}{x_{j}!}\biggr){\mathrm{e}}^{-\Lambda u}h_{\beta}(t,u)\,\mathrm{d}u,\end{equation*}

which coincides with the PMF of the GFCP.

On taking $\lambda_{j}(t)=\lambda(t)$ and $\lambda_{j}(t)=(1-\rho)\rho^{j-1}\lambda(t)/(1-\rho^{k})$ for $1\le j\le k$ in Theorem 1, we get the systems of differential–integral equations that govern the PMF of the corresponding increment process of the non-homogeneous fractional PPoK and the non-homogeneous fractional PAPoK, respectively (see [Reference Kadankova, Leonenko and Scalas18, Theorems 3.3 and 4.4]).

Remark 5. For $\beta=1$ , the Laplace transform of the inverse stable subordinator reduces to ${\mathrm{e}}^{-us}$ , whose inverse Laplace transform is $\delta(t-u)$ , the Dirac delta distribution. Hence,

\begin{equation*}\int_{0}^{\infty}q_{n}(u,v)\delta(t-u)\,\mathrm{d}u=q_{n}(t,v).\end{equation*}

From (33) it follows that $q_{n}^{1}(t,v)=q_{n}(t,v)$ , and the system of differential–integral equations in (35) reduces to

\begin{align*}\frac{\mathrm{d}}{\mathrm{d}t}q_{n}(t,v)&=\int_{0}^{\infty}\biggl(-\sum_{j=1}^{k}\lambda_{j}(u+v)q_{n}(u,v)+\sum_{j=1}^{n\wedge k}\lambda_{j}(u+v)q_{n-j}(u,v)\biggr)\delta(t-u)\,\mathrm{d}u\\&=-\sum_{j=1}^{k}\lambda_{j}(t+v)q_{n}(t,v)+\sum_{j=1}^{n\wedge k}\lambda_{j}(t+v)q_{n-j}(t,v).\end{align*}

The mean, variance, and covariance of the NGCP are given by $\mathbb{E}(\mathcal{M}(t))=\sum_{j=1}^{k}j\Lambda_{j}(t)$ , $\operatorname{Var}(\mathcal{M}(t))=\sum_{j=1}^{k}j^{2}\Lambda_{j}(t)$ , and $\operatorname{Cov}(\mathcal{M}(s),\mathcal{M}(t))=\sum_{j=1}^{k}j^{2}\Lambda_{j}(s\wedge t)$ , respectively. As the NGCP has independent increments, it follows that the process $\{\mathcal{M}(t)-\sum_{j=1}^{k}j\Lambda_{j}(t)\}_{t\geq0}$ is a martingale with respect to a natural filtration $\mathscr{F}_{t}=\sigma(\mathcal{M}(s), 0\lt s\le t)$ .

Next, we discuss the arrival times of the NGCP. Let $\mathscr{T}_{n}=\min\{t\ge 0\,:\,\mathcal{M}(t)=n\}$ be the arrival time of the nth NGCP event. Then

\begin{equation*}F_{\mathscr{T}_{n}}(t)=\mathbb{P}\{\mathscr{T}_{n}\le t\}=\mathbb{P}\{\mathcal{M}(t)\ge n\}=1-\sum_{x=0}^{n-1}\sum_{\Omega(k,x)}\biggl(\prod_{j=1}^{k}\frac{(\Lambda_{j}(t))^{x_{j}}}{x_{j}!}\biggr){\mathrm{e}}^{-\sum_{j=1}^{k}\Lambda_{j}(t)}.\end{equation*}

We note that $F_{\mathscr{T}_{n}}(t)$ gives a distribution function if and only if the $\Lambda_{j}(t)$ satisfy the conditions given in Remark 5 of [Reference Leonenko, Scalas and Trinh26].

Similarly, let $\mathscr{T}_{n}^{\,\beta}=\min\{t\ge 0\,:\,\mathcal{M}_{\beta}(t)=n\}$ be the arrival time of the nth NGFCP event. On using (34), its distribution can be written as

\begin{align*}F_{\mathscr{T}_{n}^{\,\beta}}(t)&=\sum_{x=n}^{\infty}\int_{0}^{\infty}\sum_{\Omega(k,x)}\biggl(\prod_{j=1}^{k}\frac{(\Lambda_{j}(u))^{x_{j}}}{x_{j}!}\biggr){\mathrm{e}}^{-\sum_{j=1}^{k}\Lambda_{j}(u)}h_{\beta}(t,u)\,\mathrm{d}u\\&=\int_{0}^{\infty}\biggl(1-\sum_{x=0}^{n-1}\sum_{\Omega(k,x)}\biggl(\prod_{j=1}^{k}\frac{(\Lambda_{j}(u))^{x_{j}}}{x_{j}!}\biggr){\mathrm{e}}^{-\sum_{j=1}^{k}\Lambda_{j}(u)}\biggr)h_{\beta}(t,u)\,\mathrm{d}u\\&=\int_{0}^{\infty}F_{\mathscr{T}_{n}}(u)h_{\beta}(t,u)\,\mathrm{d}u.\end{align*}

3.2. The NGCP time-changed by the inverse subordinator

Let $\{Y_{f}(t)\}_{t\ge0}$ be the inverse subordinator defined in (11) such that $c_{1}=c_{2}=0$ in its associated Bernštein function given in (10). We consider a time-changed process defined as follows:

\begin{equation*}\mathcal{M}_{f}(t)\,:\!=\mathcal{M}(Y_{f}(t)), \quad t\ge0,\end{equation*}

where the NGCP $\{\mathcal{M}(t)\}_{t\ge0}$ is independent of $\{Y_{f}(t)\}_{t\ge0}$ .

For $f(s)=s^{\beta}$ with $0\lt\beta\lt1$ , the process $\{\mathcal{M}_{f}(t)\}_{t\ge0}$ reduces to the NGFCP. For $k=1$ , it reduces to the non-homogeneous Poisson process time-changed by the inverse subordinator that was introduced and studied by Buchak and Sakhno [Reference Buchak and Sakhno9].

Let $v\ge 0$ . The increment process of $\{\mathcal{M}_{f}(t)\}_{t\ge0}$ is defined as

\begin{equation*}I_{f}(t,v)=I(Y_{f}(t),v)=\mathcal{M}\left(Y_{f}(t)+v\right)-\mathcal{M}(v),\quad t\ge0.\end{equation*}

Its marginal distribution $q_n^f(t,v)=\mathbb{P}\{I_{f}(t,v)=n\}$ for $n\ge0$ is given by

\begin{align*}q_n^f(t,v)&=\int_{0}^{\infty}q_n(u,v)h_{f}(t,u)\,\mathrm{d}u\\&=\int_{0}^{\infty}\sum_{\Omega(k,n)}\biggl(\prod_{j=1}^{k}\frac{(\Lambda_{j}(v,u+v))^{x_{j}}}{x_{j}!}\biggr){\mathrm{e}}^{-\sum_{j=1}^{k}\Lambda_{j}(v,u+v)}h_{f}(t,u)\,\mathrm{d}u,\end{align*}

where $h_{f}(t,u)$ is the density of $\{Y_{f}(t)\}_{t\ge0}$ , which exists under Condition I. So the marginal distribution of $\{\mathcal{M}_{f}(t)\}_{t\ge0}$ , i.e. $q_n^f(t)=q_n^f(t,0)=\mathbb{P}\{\mathcal{M}_{f}(t)=n\}$ , is given by

\begin{align*}q_n^f(t)&=\int_{0}^{\infty}q_n(u)h_{f}(t,u)\,\mathrm{d}u\\&=\int_{0}^{\infty}\sum_{\Omega(k,n)}\biggl(\prod_{j=1}^{k}\frac{(\Lambda_{j}(u))^{x_{j}}}{x_{j}!}\biggr){\mathrm{e}}^{-\sum_{j=1}^{k}\Lambda_{j}(u)}h_{f}(t,u)\,\mathrm{d}u.\end{align*}

Theorem 2. Let $^f\mathcal{D}_t$ be the generalized Caputo derivative with respect to the Bernštein function f defined in (13), and suppose that Condition I holds. Then the PMF of $\{I_{f}(t,v)\}_{t\ge0}$ satisfies the following system of differential–integral equations:

\begin{equation*}{^f}\mathcal{D}_tq_n^f(t,v)=\int_{0}^{\infty}\biggl(-\sum_{j=1}^{k}\lambda_{j}(u+v)q_n(u,v)+\sum_{j=1}^{n\wedge k}\lambda_{j}(u+v)q_{n-j}(u,v)\biggr)h_{f}(t,u)\,\mathrm{d}u,\end{equation*}

with initial condition $q^{f}_{n}(0,v)=\delta_{n,0}$ .

Proof. The proof follows similar lines to that of Theorem 1. We present it here for the sake of completeness.

The characteristic function of $I_{f}(t,v)$ can be written as

(38) \begin{equation}\hat{q}_\xi^f(t,v)=\int_{0}^{\infty}\hat{q}_{\xi}(u,v)h_{f}(t,u)\,\mathrm{d}u.\end{equation}

On taking the Laplace transform in (38) and using (12) and (32), we get

\begin{align*}\tilde{\hat{q}}_\xi^f(s,v)&=\frac{f(s)}{s}\int_{0}^{\infty}\exp\biggl(\sum_{j=1}^{k}\Lambda_{j}(v, u+v)({\mathrm{e}}^{\omega \xi j}-1)\biggr){\mathrm{e}}^{-uf(s)}\,\mathrm{d}u\\&=\frac{f(s)}{s}\Bigg\{\biggl[-\frac{{\mathrm{e}}^{-uf(s)}}{f(s)}\exp\biggl(\sum_{j=1}^{k}\Lambda_{j}(v, u+v)({\mathrm{e}}^{\omega \xi j}-1)\biggr)\biggr]_{0}^{\infty}\\&+\frac{1}{f(s)}\int_{0}^{\infty}\sum_{j=1}^{k}\lambda_{j}(u+v)({\mathrm{e}}^{\omega \xi j}-1)\exp\biggl(\sum_{j=1}^{k}\Lambda_{j}(v,u+v)({\mathrm{e}}^{\omega\xi j}-1)\biggr){\mathrm{e}}^{-uf(s)}\mathrm{d}u\Bigg\}.\end{align*}

Thus,

\begin{equation*}f(s)\tilde{\hat{q}}_\xi^f(s,v)-\frac{f(s)}{s}=\!\int_{0}^{\infty}\!\sum_{j=1}^{k}\lambda_{j}(u+v)({\mathrm{e}}^{\omega \xi j}-1)\exp\biggl(\sum_{j=1}^{k}\Lambda_{j}(v,u+v)({\mathrm{e}}^{\omega \xi j}-1)\biggr)\frac{f(s)}{s}{\mathrm{e}}^{-uf(s)}\mathrm{d}u.\end{equation*}

On taking the inverse Laplace transform and using (12) and (15), we get

\begin{align*}^f\mathcal{D}_t\hat{q}_\xi^f(t,v)&=\int_{0}^{\infty}\sum_{j=1}^{k}\lambda_{j}(u+v)({\mathrm{e}}^{\omega \xi j}-1)\exp\biggl(\sum_{j=1}^{k}\Lambda_{j}(v,u+v)({\mathrm{e}}^{\omega \xi j}-1)\biggr)h_{f}(t,u)\,\mathrm{d}u\\&=\int_{0}^{\infty}\sum_{j=1}^{k}\lambda_{j}(u+v)({\mathrm{e}}^{\omega \xi j}-1)\hat{q}_\xi(u,v)h_{f}(t,u)\,\mathrm{d}u,\end{align*}

where we have used (32) in the last step. Now, the desired result follows on using the inversion formula for the characteristic function.

On substituting $v=0$ in Theorem 2, we get the system of differential–integral equations that governs the PMF of $\{\mathcal{M}_{f}(t)\}_{t\ge0}$ .

Corollary 2. Assume that Condition I holds. Then the PMF of $\{\mathcal{M}_{f}(t)\}_{t\ge0}$ satisfies the following system of differential–integral equations:

(39) \begin{equation}^f\mathcal{D}_tq_n^f(t)=\int_{0}^{\infty}\biggl(-\sum_{j=1}^{k}\lambda_{j}(u)q_n(u)+\sum_{j=1}^{n\wedge k}\lambda_{j}(u)q_{n-j}(u)\biggr)h_{f}(t,u)\,\mathrm{d}u,\quad n\ge0,\end{equation}

with initial condition $q^{f}_{n}(0)=\delta_{n,0}$ .

On choosing constant rates, that is, $\lambda_{j}(t)=\lambda_{j}$ for $1\le j \le k$ , in (39) we get

(40) \begin{equation}^f\mathcal{D}_tp_n^f(t)=-\Lambda p_n^f(t)+\sum_{j=1}^{n\wedge k}\lambda_{j}p_n^f(t),\end{equation}

where $p_n^f(t)=\mathbb{P}\{M(Y_f(t))=n\}$ are the state probabilities of $\{M(Y_{f}(t))\}_{t\ge0}$ , a process studied in [Reference Kataria and Khandakar20].

Remark 6. Alternatively, the system of differential equations (40) can be obtained as follows. Note that

(41) \begin{equation}p_n^f(t)=\int_{0}^{\infty}p(n,u)h_{f}(t,u)\,\mathrm{d}u, \quad n\ge0.\end{equation}

On taking the generalized Riemann–Liouville derivative $^f\mathscr{D}_t$ and using the results

\begin{equation*}{^f}\mathscr{D}_th_{f}(t,u)=-\frac{\partial}{\partial u}h_{f}(t,u), \quad h_{f}(t,0)=\nu(t),\quad h_{f}(0,u)=\delta(u)\end{equation*}

(see [Reference Toaldo34, Theorem 4.1]) in (41), we get

(42) \begin{align}^f\mathscr{D}_tp_n^f(t)&=-\int_{0}^{\infty}p(n,u)\frac{\partial}{\partial u}h_{f}(t,u)\,\mathrm{d}u\nonumber\\&=-\left[p(n,u)h_{f}(t,u)\right]_{u=0}^{\infty}+\int_{0}^{\infty}h_{f}(t,u)\frac{\mathrm{d}}{\mathrm{d}u}p(n,u)\,\mathrm{d}u\nonumber\\&=p(n,0)h_{f}(t,0)+\int_{0}^{\infty}h_{f}(t,u)\biggl(-\Lambda p(n,u)+\sum_{j=1}^{n\wedge k}\lambda_{j}p(n-j,u)\biggr)\,\mathrm{d}u\nonumber\\&=p(n,0)\nu(t)-\Lambda p_n^f(t)+\sum_{j=1}^{n\wedge k}\lambda_{j}p_{n-j}^f(t),\end{align}

where in the penultimate step we have used (2) with $\beta=1$ . Also,

\begin{equation*}p_n^f(0)=\int_{0}^{\infty}p(n,u)h_{f}(0,u)\,\mathrm{d}u=\int_{0}^{\infty}p(n,u)\delta(u)\,\mathrm{d}u=p(n,0).\end{equation*}

From (14) we get

\begin{equation*}^f\mathcal{D}_tp_n^f(t)={^f\mathscr{D}}_tp_n^f(t)-\nu(t)p_n^f(0),\end{equation*}

which on using in (42) gives the required result.

4.1. Generalized space-fractional counting process

The multistable subordinator $\{H(t)\}_{t\ge0}$ reduces to the stable subordinator $\{D_{\beta}(t)\}_{t\ge0}$ when the stability index is a constant, that is, $\beta(t)=\beta\in (0,1)$ for all $t\ge0$ . In this case, the GSMCP reduces to the following time-changed process:

(46) \begin{equation}Y(t)=M(D_{\beta}(t)), \quad t\ge0,\end{equation}

where the GCP is independent of the stable subordinator. We call the process $\{Y(t)\}_{t\ge0}$ the generalized space-fractional counting process (GSFCP).

When $k=1$ , the GSFCP reduces to the SFPP (see [Reference Orsingher and Polito28]). For $\lambda_{j}=\lambda$ $(j=1,2,\ldots,k)$ , it reduces to the space-fractional PPoK (see [Reference Gupta and Kumar17]). For $0\le \rho\lt1$ and $\lambda_{j}=\lambda(1-\rho)\rho^{j-1}/(1-\rho^{k})$ $(j=1,2,\ldots,k)$ , it reduces to the space-fractional version of the PAPoK.

Let $l_{\beta}(t,\cdot)$ be the density of the stable subordinator and G(u, t) the PGF of the GCP given in (8). From (46), the PGF of the GSFCP can be obtained as

(47) \begin{align}G_{\beta}(u,t)&=\int_{0}^{\infty}G(u,x)l_{\beta}(t,x)\,\mathrm{d}x\nonumber\\&=\int_{0}^{\infty}\exp\biggl(-\sum_{j=1}^{k}\lambda_{j}(1-u^{j})x\biggr)l_{\beta}(t,x)\,\mathrm{d}x\nonumber\\&=\exp\biggl(-t\Bigl(\sum_{j=1}^{k}\lambda_{j}(1-u^{j})\Bigr)^{\beta}\biggr).\end{align}

The PGF of the GSFCP satisfies the differential equation

\begin{equation*}\frac{\partial}{\partial t}G_{\beta}(u,t)=-\biggl(\sum_{j=1}^{k}\lambda_{j}(1-u^{j})\biggr)^{\beta}G_{\beta}(u,t), \quad G_{\beta}(u,0)=1.\end{equation*}

Proposition 3. Let $X_{i}$ , $i\ge1$ be i.i.d. uniform random variables in [0, 1]. Then

\begin{equation*}G_{\beta}(u,t)=\mathbb{P}\bigg\{\min_{0\le i\le N(t)}X_{i}^{1/\beta}\ge 1-\frac{1}{\Lambda}\sum_{j=1}^{k}\lambda_{j}u^{j}\bigg\},\quad 0\lt u\lt1,\end{equation*}

where $\{N(t)\}_{t\ge0}$ is a homogeneous Poisson process with rate $\Lambda^{\beta}$ such that $\min_{0\le i \le N(t)} X_{i}^{1/\beta}= 1$ when $N(t)=0$ .

Proof. Observe that

\begin{align*}\mathbb{P}\bigg\{\min_{0\le i\le N(t)}X_{i}^{1/\beta}\ge 1-\frac{1}{\Lambda}\sum_{j=1}^{k}\lambda_{j}u^{j}\bigg\}&=\sum_{n=0}^{\infty}\mathbb{P}\bigg\{\min_{0\le i\le n} X_{i}^{1/\beta}\ge 1-\frac{1}{\Lambda}\sum_{j=1}^{k}\lambda_{j}u^{j}\bigg\}\frac{(\Lambda^{\beta}t)^{n}}{n!}{\mathrm{e}}^{-t\Lambda^{\beta}}\\&={\mathrm{e}}^{-t\Lambda^{\beta}}\sum_{n=0}^{\infty}\biggl(\mathbb{P}\Big\{X_{1}^{1/\beta}\ge 1-\frac{1}{\Lambda}\sum_{j=1}^{k}\lambda_{j}u^{j}\Big\}\biggr)^{n}\frac{(\Lambda^{\beta}t)^{n}}{n!}\\&={\mathrm{e}}^{-t\Lambda^{\beta}}\sum_{n=0}^{\infty}\biggl(1-\Big(1-\frac{1}{\Lambda}\sum_{j=1}^{k}\lambda_{j}u^{j}\Big)^{\beta}\biggr)^{n}\frac{(\Lambda^{\beta}t)^{n}}{n!}\\&=\exp\biggl(-t\Bigl(\sum_{j=1}^{k}\lambda_{j}(1-u^{j})\Bigr)^{\beta}\biggr),\end{align*}

which coincides with (47). This completes the proof.

Note that the distribution of jumps of the TCGCP-I $\{M_{f}(t)\}_{t\ge0}$ is given by (see [Reference Kataria and Khandakar20])

(48) \begin{equation}\mathbb{P}\{M_{f}(h)=n\}=\begin{cases}1-hf(\Lambda)+o(h), & n=0,\\\displaystyle-h\sum_{ \Omega(k,n)}f^{(z_{k})}(\Lambda)\prod_{j=1}^{k}\frac{(\!-\lambda_{j})^{x_{j}}}{x_{j}!}+o(h), & n\ge1.\end{cases}\end{equation}

Thus, from (48) and Remark 9, the jump distribution of the GSFCP is given by

\begin{equation*}\mathbb{P}\{Y(h)=n\}=\begin{cases}1-h\Lambda^{\beta}+o(h), & n=0,\\\displaystyle- h\sum_{ \Omega(k,n)}(\beta)_{z_{k}}\Lambda^{\beta-z_{k}}\prod_{j=1}^{k}\frac{(\!-\lambda_{j})^{x_{j}}}{x_{j}!}+o(h), & n\ge1,\end{cases}\end{equation*}

where $(\beta)_{z_{k}}=\beta(\beta-1)\cdots(\beta-z_k+1)$ .

The Lévy measure and the distribution of first passage times of the GSFCP follow from (44) and (45), respectively.

Remark 10. In view of (4), it can be observed that the GSFCP is equal in distribution to a space-fractional compound Poisson process, that is,

\begin{equation*}Y(t)\;\stackrel{\rm d}{=}\;\sum_{j=1}^{N(D_{\beta}(t))}X_{j},\end{equation*}

where $\{N(D_{\beta}(t))\}_{t\ge0}$ is the SFPP with intensity $\Lambda$ and is independent of the sequence of i.i.d. random variables $\{X_{j}\}_{j\ge 1}$ with distribution (3).

The state probabilities of the GSFCP can be obtained by putting $f(s)=s^{\beta}$ in Remark 9 of [Reference Kataria and Khandakar20] as follows:

(49) \begin{align}\mathbb{P}\{Y(t)=n\}&=\sum_{\Omega(k,n)}\biggl(\prod_{j=1}^{k}\frac{\lambda_{j}^{x_{j}}}{x_{j}!}\biggr)\frac{(\!-1)^{z_{k}}}{\Lambda^{z_{k}}}\frac{\mathrm{d}^{z_{k}}}{\mathrm{d}v^{z_{k}}}{\mathrm{e}}^{-t\Lambda^{\beta} v^{\beta}}\Big|_{v=1}\nonumber\\&=\sum_{\Omega(k,n)}\biggl(\prod_{j=1}^{k}\frac{\lambda_{j}^{x_{j}}}{x_{j}!}\biggr)\frac{(\!-1)^{z_{k}}}{\Lambda^{z_{k}}}\sum_{r=0}^{\infty}\frac{(\!-\Lambda^{\beta}t)^{r}}{r!}\frac{\Gamma(\beta r+1)}{\Gamma(\beta r+1-z_{k})}\nonumber\\&=\sum_{\Omega(k,n)}\biggl(\prod_{j=1}^{k}\frac{\lambda_{j}^{x_{j}}}{x_{j}!}\biggr)\frac{(\!-1)^{z_{k}}}{\Lambda^{z_{k}}}\:{}_1\psi_1\left[\left.\begin{array}{l}(1,\beta)\\(1-z_{k},\beta)\end{array}\right|-\Lambda^\beta t \right],\quad n\ge0,\end{align}

where ${}_1\psi_1(\!\cdot\!)$ is the generalized Wright function (see [Reference Kilbas, Srivastava and Trujillo24, Equation (1.11.14)]).

Further, for $k=1$ and $\Lambda=\lambda$ in (49), we get the PMF of the SFPP (see [Reference Orsingher and Polito28, Equation (2.15)]).

5.1. First passage times of the TCGCP-I

Here, we study the first passage times of the TCGCP-I, defined as

\begin{equation*}T_{f}^{n}\,:\!=\inf\{s\ge0\,:\,M_{f}(s)\ge n\}.\end{equation*}

The distribution of $T_{f}^{n}$ can be written as

\begin{equation*}\mathbb{P}\{T_{f}^{n}\lt s\}=\mathbb{P}\{M_{f}(s)\ge n\}=\sum_{m=n}^{\infty}\int_{0}^{\infty}\sum_{\Omega(k,m)}\biggl(\prod_{j=1}^{k}\frac{(\lambda_{j}x)^{x_{j}}}{x_{j}!}\biggr){\mathrm{e}}^{-\Lambda x}\mathbb{P}\{D_{f}(s)\in \mathrm{d}x\},\end{equation*}

so its density function $\mathscr{P}_{f}(n,s)=\mathbb{P}\{T_{f}^{n}\in \mathrm{d}s\}/\mathrm{d}s$ can be obtained as

(57) \begin{align}\mathscr{P}_{f}(n,s)&=\frac{\mathrm{d}}{\mathrm{d}s}\sum_{m=n}^{\infty}\int_{0}^{\infty}\sum_{\Omega(k,m)}\biggl(\prod_{j=1}^{k}\frac{(\lambda_{j}x)^{x_{j}}}{x_{j}!}\biggr){\mathrm{e}}^{-\Lambda x}\mathbb{P}\{D_{f}(s)\in \mathrm{d}x\}\\&=\frac{\mathrm{d}}{\mathrm{d}s}\int_{0}^{\infty}\biggl(1-\sum_{l=0}^{n-1}\sum_{\Omega(k,l)}\biggl(\prod_{j=1}^{k}\frac{(\lambda_{j}x)^{x_{j}}}{x_{j}!}\biggr){\mathrm{e}}^{-\Lambda x}\biggr)\mathbb{P}\{D_{f}(s)\in \mathrm{d}x\}\nonumber\\&=-\frac{\mathrm{d}}{\mathrm{d}s}\sum_{l=0}^{n-1}\sum_{\Omega(k,l)}\biggl(\prod_{j=1}^{k}\frac{(\!-\lambda_{j})^{x_{j}}}{x_{j}!}\biggr)\int_{0}^{\infty}\frac{\mathrm{d}^{z_{k}}}{\mathrm{d}\Lambda^{z_{k}}}{\mathrm{e}}^{-\Lambda x}\mathbb{P}\{D_{f}(s)\in \mathrm{d}x\}\nonumber\end{align}

(58) \begin{align}=-\frac{\mathrm{d}}{\mathrm{d}s}\sum_{l=0}^{n-1}\sum_{\Omega(k,l)}\biggl(\prod_{j=1}^{k}\frac{(\!-\lambda_{j})^{x_{j}}}{x_{j}!}\biggr)\frac{\mathrm{d}^{z_{k}}}{\mathrm{d}\Lambda^{z_{k}}}{\mathrm{e}}^{-sf(\Lambda)}.\end{align}

Equivalently,

\begin{equation*}\mathbb{P}\{T_{f}^{n}\in \mathrm{d}s\}=\mathbb{P}\{T_{f}^{n-1}\in \mathrm{d}s\}-\sum_{\Omega(k,n-1)}\biggl(\prod_{j=1}^{k}\frac{(\!-\lambda_{j})^{x_{j}}}{x_{j}!}\biggr)\frac{\mathrm{d}}{\mathrm{d}s}\frac{\mathrm{d}^{z_{k}}}{\mathrm{d}\Lambda^{z_{k}}}{\mathrm{e}}^{-sf(\Lambda)}\mathrm{d}s,\end{equation*}

which gives a recursive relation between the distributions of $T_{f}^{n}$ .

Remark 14. From (58), we have

\begin{equation*}\mathbb{P}\{T_{f}^{1}\in \mathrm{d}s\}=f(\Lambda){\mathrm{e}}^{-sf(\Lambda)}\,\mathrm{d}s,\end{equation*}

that is, the waiting time of the first event for the TCGCP-I is exponential.

Next, we derive the system of differential equations that governs the distribution of $T_{f}^{n}$ . The following result will be used.

Proposition 6. Let $\mathcal{O}_{f}(u,t)=\sum_{n=1}^{\infty}u^{n}\mathscr{P}_{f}(n,t)$ for $|u|\lt1$ . Then

\begin{equation*}\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{O}_{f}(u,t)=-f\biggl(\sum_{j=1}^{k}(1-u^{j})\lambda_{j}\biggr)\mathcal{O}_{f}(u,t).\end{equation*}

Proof. On using (57), we get

\begin{align*}\mathcal{O}_{f}(u,t)&=\frac{\mathrm{d}}{\mathrm{d}t}\sum_{m=1}^{\infty}\sum_{n=1}^{m}u^{n}\int_{0}^{\infty}\sum_{\Omega(k,m)}\biggl(\prod_{j=1}^{k}\frac{(\lambda_{j}x)^{x_{j}}}{x_{j}!}\biggr){\mathrm{e}}^{-\Lambda x}\mathbb{P}\{D_{f}(t)\in \mathrm{d}x\}\\&=\frac{\mathrm{d}}{\mathrm{d}t}\frac{u}{u-1}\sum_{m=1}^{\infty}(u^{m}-1)\int_{0}^{\infty}\sum_{\Omega(k,m)}\biggl(\prod_{j=1}^{k}\frac{(\lambda_{j}x)^{x_{j}}}{x_{j}!}\biggr){\mathrm{e}}^{-\Lambda x}\mathbb{P}\{D_{f}(t)\in \mathrm{d}x\}\\&=\frac{\mathrm{d}}{\mathrm{d}t}\frac{u}{u-1}\int_{0}^{\infty}\Big({\mathrm{e}}^{\sum_{j=1}^{k}(u^{j}-1)\lambda_{j}x}-1\Big)\mathbb{P}\{D_{f}(t)\in \mathrm{d}x\}\\&=\frac{\mathrm{d}}{\mathrm{d}t}\frac{u}{u-1}{\mathrm{e}}^{-tf\bigl(\sum_{j=1}^{k}(1-u^{j})\lambda_{j}\bigr)}\\&=\frac{u}{1-u}f\biggl(\sum_{j=1}^{k}(1-u^{j})\lambda_{j}\biggr){\mathrm{e}}^{-tf\bigl(\sum_{j=1}^{k}(1-u^{j})\lambda_{j}\bigr)}.\end{align*}

The result follows on taking the derivative with respect to t.

Theorem 3. The density function of the first passage times of the TCGCP-I solves the following system of differential equations:

\begin{equation*}\frac{\mathrm{d}}{\mathrm{d}t}\mathscr{P}_{f}(n,t)=-f\biggl(\Lambda\Big(I-\frac{1}{\Lambda}\sum_{j=1}^{n\wedge k}\lambda_{j}B^{j}\Big)\biggr)\mathscr{P}_{f}(n,t),\quad n\ge1.\end{equation*}

Proof. On using (52) for $\mathscr{P}_{f}(n,t)$ , we get

\begin{align*}\sum_{n=1}^{\infty}&u^{n}f\biggl(\Lambda\Big(I-\frac{1}{\Lambda}\sum_{j=1}^{n\wedge k}\lambda_{j}B^{j}\Big)\biggr)\mathscr{P}_{f}(n,t)\\&=f(\Lambda)\mathcal{O}_{f}(u,t)-\sum_{n=1}^{\infty}u^{n}\sum_{m=1}^{n}\sum_{ \Omega(k,m)}\biggl(\prod_{j=1}^{k}\frac{\lambda_{j}^{x_{j}}}{x_{j}!}\biggr)\mathscr{P}_{f}(n-m,t)\int_{0}^{\infty}s^{z_{k}}{\mathrm{e}}^{-\Lambda s}\overline{\nu}(\mathrm{d}s)\\&=f(\Lambda)\mathcal{O}_{f}(u,t)-\sum_{m=1}^{\infty}\sum_{n=m}^{\infty}u^{n}\sum_{ \Omega(k,m)}\biggl(\prod_{j=1}^{k}\frac{\lambda_{j}^{x_{j}}}{x_{j}!}\biggr)\mathscr{P}_{f}(n-m,t)\int_{0}^{\infty}s^{z_{k}}{\mathrm{e}}^{-\Lambda s}\overline{\nu}(\mathrm{d}s)\\&=f(\Lambda)\mathcal{O}_{f}(u,t)-\mathcal{O}_{f}(u,t)\sum_{m=1}^{\infty}u^{m}\sum_{\Omega(k,m)}\biggl(\prod_{j=1}^{k}\frac{\lambda_{j}^{x_{j}}}{x_{j}!}\biggr)\int_{0}^{\infty}s^{z_{k}}{\mathrm{e}}^{-\Lambda s}\overline{\nu}(\mathrm{d}s)\\&=f(\Lambda)\mathcal{O}_{f}(u,t)-\mathcal{O}_{f}(u,t)\int_{0}^{\infty}{\mathrm{e}}^{-\Lambda s}\Big({\mathrm{e}}^{\sum_{j=1}^{k}u^{j}\lambda_{j}s}-1\Big)\overline{\nu}(\mathrm{d}s)\\&=f\biggl(\sum_{j=1}^{k}(1-u^{j})\lambda_{j}\biggr)\mathcal{O}_{f}(u,t).\end{align*}

The result follows on applying Proposition 6.

Let $\mathcal{T}_{f}^{n}$ be the first-hitting times of the TCGCP-I, that is,

\begin{equation*}\mathcal{T}_{f}^{n}\,:\!= \inf \{s\ge0\,:\,M_{f}(s)=n\},\quad n\ge1.\end{equation*}

As the TCGCP-I is a Lévy process, it has independent increments, so the hitting-time distribution for state n of the TCGCP-I has the form

\begin{align*}\mathbb{P}\bigl\{\mathcal{T}_{f}^{n}\in \mathrm{d}s\bigr\}&=\mathbb{P}\bigg\{\bigcup_{l=0}^{n-1}\{M_{f}(s)=l, M_{f}[s,s+\mathrm{d}s)=n-l)\}\bigg\}\\&=\sum_{l=0}^{n-1}\sum_{\Omega(k,l)}\biggl(\prod_{j=1}^{k}\frac{\lambda_{j}^{x_{j}}}{x_{j}!}\biggr)\frac{(\!-1)^{z_{k}}}{\Lambda^{z_{k}}}\frac{\mathrm{d}^{z_{k}}}{\mathrm{d}v^{z_{k}}}{\mathrm{e}}^{-sf(\Lambda v)}\Big|_{v=1}\!\!\cdot(\!-\mathrm{d}s)\!\!\sum_{\Omega(k,n-l)}\!f^{(z_{k})}(\Lambda)\prod_{j=1}^{k}\frac{(\!-\lambda_{j})^{x_{j}}}{x_{j}!},\end{align*}

where we have used Remark 9 of [Reference Kataria and Khandakar20] and (48) in the last step. Therefore,

(59) \begin{equation}\mathbb{P}\bigl\{\mathcal{T}_{f}^{n}\lt\infty\bigr\}=\sum_{l=0}^{n-1}\sum_{\Omega(k,l)}\biggl(\prod_{j=1}^{k}\frac{\lambda_{j}^{x_{j}}}{x_{j}!}\biggr)\frac{(\!-1)^{z_{k}+1}}{\Lambda^{z_{k}}}\frac{\mathrm{d}^{z_{k}}}{\mathrm{d}v^{z_{k}}}\frac{1}{f(\Lambda v)}\bigg|_{v=1}\sum_{ \Omega(k,n-l)}f^{(z_{k})}(\Lambda)\prod_{j=1}^{k}\frac{(\!-\lambda_{j})^{x_{j}}}{x_{j}!}.\end{equation}

On taking $k=1$ and $\Lambda=\lambda$ in (59), we get

\begin{equation*}\mathbb{P}\bigl\{\mathcal{T}_{f}^{n}\lt\infty\bigr\}\big|_{k=1}=\sum_{l=0}^{n-1}\frac{(\!-1)^{n+1}}{l!(n-l)!}\frac{\mathrm{d}^{l}}{\mathrm{d}v^{l}}\frac{1}{f(\lambda v)}\bigg|_{v=1} \lambda^{n-l}f^{(n-l)}(\lambda),\end{equation*}

which agrees with Equation (2.7) of [Reference Garra, Orsingher and Scavino16].

5.2. An application to risk theory

Recently, several authors have introduced and studied risk processes in which the TFPP is used to model the number of claims received by a insurance company (see [Reference Beghin and Macci5, Reference Constantinescu, Ramirez and Zhu10]). If the claims arrive in groups of size less than or equal to a fixed number k, then the GCP can be used to model the arrivals of these claims (see [Reference Kataria and Khandakar21]). In real life there are situations in which a large number of claims arrive in an infinitesimal time interval. It is important to note that the TCGCP-I allows arbitrary arrivals in a short span of time.

Consider the following risk model with the TCGCP-I as the counting process:

(60) \begin{equation}X_{f}(t)= ct-\sum_{j=1}^{M_{f}(t)}Z_j, \quad t\geq 0,\end{equation}

where $c\gt0$ denotes the constant premium rate and $ \{Z_j\}_{j\ge1}$ is a sequence of positive i.i.d. random variables with distribution F. Here, the $Z_j$ represent the claim sizes, which are independent of $\{M_{f}(t)\}_{t\ge0}$ . The expected value of $\{X_{f}(t)\}_{t\ge0}$ is given by

\begin{equation*}\mathbb{E}(X_{f}(t))=ct-\mu \mathbb{E}(M_f(t)),\end{equation*}

where $\mu=\mathbb{E}(Z_j)$ . The relative safety loading factor $\rho$ for (60) is given by

\begin{equation*}\rho = \frac{ \mathbb{E}(X_{f}(t))}{\mathbb{E}\big(\sum_{j=1}^{M_f(t)} Z_j\big)} = \frac{ct}{\mu \mathbb{E}(M_f(t))}-1.\end{equation*}

Let $U(t)= u+X_{f}(t)$ for $t\ge0$ be the surplus process, where $u\ge0$ is the initial capital, let $\tau$ be the time to ruin of an insurance company, i.e. $ \tau=\inf\{t\gt0\,:\, U(t)\lt0\}$ with $\inf \phi=\infty$ , and let $\psi(u)=\mathbb{P}_{u}\{\tau\lt\infty\}$ be the probability of ruin. Let

(61) \begin{equation}K(u,y)= \mathbb{P}_{u}\{\tau\lt\infty, D\leq y \},\quad y\ge0\end{equation}

be the joint probability that the ruin occurs in finite time and the deficit, i.e. $D=|U(\tau)|$ , at the time of ruin is not more than y. Also, let $u'=u+ch$ and let $F^{*j}$ be the distribution of $Z_{1}+Z_{2}+\cdots+Z_{j}$ for all $j\ge 1$ . Now, using (48) and (61), we get

\begin{align*}K(u,y)&= (1-hf(\Lambda))K(u',y)+o(h) -h\sum_{n=1}^{\infty}\sum_{\Omega(k,n)}f^{(z_{k})}(\Lambda)\biggl(\prod_{j=1}^{k}\frac{(\!-\lambda_{j})^{x_{j}}}{x_{j}!}\biggr)\\&\quad\times\biggl(\int_0^{u'}K(u'-x,y)\,\mathrm{d}F^{*n}(x)+F^{*n}(u'+y)-F^{*n}(u')\biggr).\end{align*}

After rearranging the terms, we have

\begin{align*}\frac{K(u',y)-K(u,y)}{ch}&=\frac{f(\Lambda)}{c} K(u',y)+\frac{o(h)}{h}-\frac{1}{c}\sum_{n=1}^{\infty}\sum_{\Omega(k,n)}f^{(z_{k})}(\Lambda)\biggl(\prod_{j=1}^{k}\frac{(\!-\lambda_{j})^{x_{j}}}{x_{j}!}\biggr)\\&\quad\times \biggl(\int_0^{u'}K(u'-x,y)\,\mathrm{d}F^{*n}(x)+F^{*n}(u'+y)-F^{*n}(u')\biggr).\end{align*}

On taking $ h\rightarrow 0$ , we get

(62) \begin{align}\frac{\partial K(u,y)}{\partial u}& = \frac{f(\Lambda)}{c}K(u,y)-\frac{1}{c} \sum_{n=1}^{\infty} \sum_{\Omega(k,n)}f^{(z_{k})}(\Lambda)\biggl(\prod_{j=1}^{k}\frac{(\!-\lambda_{j})^{x_{j}}}{x_{j}!}\biggr)\nonumber\\&\quad\times\biggl(\int_0^{u}K(u-x,y)\,\mathrm{d}F^{*n}(x) + F^{*n}(u+y)-F^{*n}(u)\biggr).\end{align}

Note that

(63) \begin{equation}\sum_{n=1}^{\infty}\sum_{\Omega(k,n)}f^{(z_{k})}(\Lambda)\prod_{j=1}^{k}\frac{(\!-\lambda_{j})^{x_{j}}}{x_{j}!}=\sum_{r=0}^{\infty}\frac{(\!-\Lambda)^r}{r!}f^{(r)}(\Lambda)-f(\Lambda)=f(0)-f(\Lambda)=-f(\Lambda),\end{equation}

as $f(0)=0$ . Now, let

(64) \begin{equation}W(x)=-\frac{1}{f(\Lambda)}\sum_{n=1}^{\infty} \sum_{\Omega(k,n)}f^{(z_{k})}(\Lambda)\biggl(\prod_{j=1}^{k}\frac{(\!-\lambda_{j})^{x_{j}}}{x_{j}!}\biggr)F^{*n}(x)\end{equation}

be the mixer distribution of the aggregated claims. Thus, (62) reduces to

(65) \begin{equation}\frac{\partial K(u,y)}{\partial u} = \frac{f(\Lambda)}{c}\left( K(u,y)+W(u)-W(u+y)-\int_0^{u}K(u-x,y)\,\mathrm{d}W(x) \right).\end{equation}

As $\lim_{y\rightarrow \infty} K(u,y)=\psi(u)$ , by letting $y\to \infty$ in (65) we get the following governing equation for the probability of ruin:

\begin{equation*}\frac{\mathrm{d}}{\mathrm{d} u}\psi(u)= \frac{f(\Lambda)}{c}\left(\psi(u)+W(u)-1-\int_0^{u}\psi(u-x)\,\mathrm{d}W(x) \right).\end{equation*}

Theorem 4. The joint probability of the time to ruin and the deficit at the time of ruin with zero initial capital is given by

(66) \begin{equation}K(0,y)= \frac{f(\Lambda)}{c} \int_0^{y}\left(1-W(u)\right)\mathrm{d}u.\end{equation}

Proof. On integrating (65) with respect to u on $ (0,\infty)$ , we get

\begin{align*}K(\infty,y)-K(0,y)&=\frac{f(\Lambda)}{c}\left(\int_0^{\infty}K(u,y)\,\mathrm{d}u+\int_0^{\infty}\left(W(u)-W(u+y)\right)\mathrm{d}u\right)\\&\quad -\frac{f(\Lambda)}{c}\int_0^{\infty}\int_0^{u}K(u-x,y)\,\mathrm{d}W(x)\,\mathrm{d}u.\end{align*}

Hence, using $K(\infty,y)=0$ and the change of variable yields

\begin{equation*}K(0,y)=\frac{f(\Lambda)}{c}\int_0^{\infty}\left(W(u+y)-W(u)\right)\mathrm{d}u=\frac{f(\Lambda)}{c}\int_0^{y}\left(1-W(u)\right)\mathrm{d}u.\end{equation*}

This completes the proof.

Remark 16. Suppose that the claim sizes $Z_j$ follow the exponential distribution with mean $\mu\gt0$ , that is, $F(x)=1-{\mathrm{e}}^{-x/\mu}$ for $x\ge 0$ . Then

\begin{equation*}F^{*n}(x)=1-\sum_{m=0}^{n-1}\frac{(x/\mu)^{m}}{m!}{\mathrm{e}}^{-x/\mu}\end{equation*}

is an Erlang distribution function. Hence, from (64) we get

(67) \begin{align}W(x)&=-\frac{1}{f(\Lambda)}\sum_{n=1}^{\infty}\sum_{\Omega(k,n)}f^{(z_{k})}(\Lambda)\biggl(\prod_{j=1}^{k}\frac{(\!-\lambda_{j})^{x_{j}}}{x_{j}!}\biggr)\biggl(1-\sum_{m=0}^{n-1}\frac{(x/\mu)^{m}}{m!}{\mathrm{e}}^{-x/\mu}\biggr)\nonumber\\&=1+\frac{1}{f(\Lambda)}\sum_{n=1}^{\infty}\sum_{\Omega(k,n)}f^{(z_{k})}(\Lambda)\biggl(\prod_{j=1}^{k}\frac{(\!-\lambda_{j})^{x_{j}}}{x_{j}!}\biggr)\sum_{m=0}^{n-1}\frac{(x/\mu)^{m}}{m!}{\mathrm{e}}^{-x/\mu},\end{align}

where in the last step we have used (63). Thus, on substituting (67) into (66), we obtain the value of K(0, y) in the following form:

\begin{equation*}K(0,y)=\frac{1}{c}\sum_{n=1}^{\infty} \sum_{\Omega(k,n)}f^{(z_{k})}(\Lambda)\biggl(\prod_{j=1}^{k}\frac{\lambda_{j}^{x_{j}}}{x_{j}!}\biggr)(\!-1)^{z_{k}+1}\sum_{m=0}^{n-1}\frac{\mu}{m!}\gamma(m+1, y/\mu),\end{equation*}

where $\gamma(s,x)=\displaystyle\int_{0}^{x}t^{s-1}{\mathrm{e}}^{-t}\,\mathrm{d}t$ for $s\gt0$ is the lower incomplete gamma function.

5.3. The GFCP time-changed by the Lévy subordinator

Kataria and Khandakar [Reference Kataria and Khandakar20] introduced the following time-changed process, the TCGFCP-I:

(68) \begin{equation}M_{f}^{\beta}(t)\,:\!= M^{\beta}(D_{f}(t)), \quad t\ge0,\end{equation}

where the GFCP $\{M^{\beta}(t)\}_{t\ge0}$ is independent of the Lévy subordinator $\{D_{f}(t)\}_{t\ge0}$ , whose moments are finite.

First, we give an additional result for $\{M_{f}^{\beta}(t)\}_{t\ge0}$ .

Proof. Using the self-similarity property of the inverse stable subordinator, we get

\begin{equation*}M_{f}^{\beta}(t)=M(Y_{\beta}(D_{f}(t)))\;\stackrel{\rm d}{=}\;M\bigl((D_{f}(t))^{\beta}Y_{\beta}(1)\bigr).\end{equation*}

Because $D_{f}(t)\to \infty$ as $t\to \infty$ almost surely, we have

\begin{align*}\lim\limits_{t\to\infty}\frac{M_{f}^{\beta}(t)}{t^{\beta}}&\;\stackrel{\rm d}{=}\;Y_{\beta}(1)\lim\limits_{t\to\infty}\frac{M\left((D_{f}(t))^{\beta}Y_{\beta}(1)\right)}{(D_{f}(t))^{\beta}Y_{\beta}(1)}\left(\frac{D_{f}(t)}{t}\right)^{\beta}\\&\;\stackrel{\rm d}{=}\;\sum_{j=1}^{k}j\lambda_{j}Y_{\beta}(1)\lim\limits_{t\to\infty}\left(\frac{D_{f}(t)}{t}\right)^{\beta}\quad(\text{using}{9}))\\&\;\stackrel{\rm d}{=}\;\sum_{j=1}^{k}j\lambda_{j}Y_{\beta}(1)\left(\mathbb{E}(D_{f}(1))\right)^{\beta}\!,\end{align*}

where in the last step we have used the strong law of large numbers for a Lévy subordinator (see [Reference Bertoin8, p. 92]). Thus, the result follows as $Y_{\beta}(1)$ is not infinitely divisible (see [Reference Vellaisamy and Kumar35]).

In (68), we take the Bernštein function $f_1$ to obtain the GFCP time-changed by an independent gamma subordinator, that is,

\begin{equation*}M_{f_{1}}^{\beta}(t)\,:\!= M^{\beta}(Z(t)), \quad t\ge0.\end{equation*}

The mean, variance, covariance, and LRD property of $\{M_{f_{1}}^{\beta}(t)\}_{t\ge0}$ follow from Section 4.1 of [Reference Kataria and Khandakar20]. It can be shown that the increment process $\{M_{f_{1}}^{\beta}(t+h)-M_{f_{1}}^{\beta}(t)\}_{t\ge0}$ , for $h\gt0$ fixed, exhibits the short-range dependence property. The proof follows similar lines to that of Theorem 4 of [Reference Maheshwari and Vellaisamy27].

For $k=1$ , the process $\{M_{f_{1}}^{\beta}(t)\}_{t\ge0}$ reduces to the time-fractional negative binomial process $\{\mathcal{N}^{1}_{\beta}(t)\}_{t\ge0}$ with intensity $\lambda$ (see [Reference Vellaisamy and Maheshwari36]). For notational convenience, we write $Q^{\beta}(t,\lambda)=\mathcal{N}^{1}_{\beta}(t)$ for $t\ge0$ . Using (23), its mth factorial moment is given by

(69) \begin{equation}\mathbb{E}\!\left(Q^{\beta}(t,\lambda)(Q^{\beta}(t,\lambda)-1)\cdots(Q^{\beta}(t,\lambda)-m+1)\right)=\frac{\lambda^{m}m!\Gamma (bt+m\beta)}{\Gamma(m\beta+1)a^{m\beta}\Gamma(bt)},\quad m\ge 1.\end{equation}

On taking $\beta=1$ in (69), we get the mth factorial moment of the negative binomial process (see [Reference Orsingher and Toaldo29, Remark 4.7]).

The next result gives the conditional distribution of $X_{(k)}^{Q^{\beta}(t,\lambda)}$ , a kth order statistic from the sample of size $Q^{\beta}(t,\lambda)$ , where $k\in \{1,2,\ldots, Q^{\beta}(t,\lambda)\}$ .

Theorem 5. Let $\{X_{i}\}_{i\ge1}$ be a sequence of i.i.d. random variables with distribution function F. Then

\begin{equation*}\mathbb{P}\bigl\{X_{(k)}^{Q^{\beta}(t,\lambda)}\lt z\,\big|\,Q^{\beta}(t,\lambda)\ge k\bigr\}=\frac{\mathbb{P}\{Q^{\beta}(t,\lambda F(z))\ge k\}}{\mathbb{P}\{Q^{\beta}(t,\lambda)\ge k\}}.\end{equation*}

Proof. For $1\le k\le n$ , we will use the following standard result for the distribution of kth order statistics:

(70) \begin{equation}\mathbb{P}\{X_{(k)}^{n}\lt z\}=\sum_{j=k}^{n}\binom{n}{j}F^{j}(z)\left(1-F(z)\right)^{n-j}.\end{equation}

Now,

(71) \begin{align}\mathbb{P}\bigl\{X_{(k)}^{Q^{\beta}(t,\lambda)}\lt z\,\big|\,Q^{\beta}(t,\lambda)\ge k\bigr\}&=\frac{\sum_{n=k}^{\infty}\mathbb{P}\{X_{(k)}^{Q^{\beta}(t,\lambda)}\lt z,Q^{\beta}(t,\lambda)=n\}}{\mathbb{P}\{Q^{\beta}(t,\lambda)\ge k\}}\nonumber\\&=\frac{\sum_{n=k}^{\infty}\mathbb{P}\{X_{(k)}^{Q^{\beta}(t,\lambda)}\lt z\,|\,Q^{\beta}(t,\lambda)=n\}\,\mathbb{P}\{Q^{\beta}(t,\lambda)=n\}}{\mathbb{P}\{Q^{\beta}(t,\lambda)\ge k\}}.\end{align}

Using (24) and (70), the quantity in the numerator on the right-hand side of (71) can be evaluated as follows:

\begin{align*}\sum_{n=k}^{\infty}&\mathbb{P}\bigl\{X_{(k)}^{Q^{\beta}(t,\lambda)}\lt z\,\big|\,Q^{\beta}(t,\lambda)=n\bigr\}\mathbb{P}\{Q^{\beta}(t,\lambda)=n\}\\&=\sum_{n=k}^{\infty}\biggl(\sum_{j=k}^{n}\binom{n}{j}F^{j}(z)\left(1-F(z)\right)^{n-j}\biggr)\sum_{l=n}^{\infty}(\!-1)^{l+n}\binom{l}{n}\frac{\Gamma(l\beta+bt)}{\Gamma(bt)\Gamma(\beta l+1)}\left(\frac{\lambda}{a^{\beta}}\right)^{l}\\&=\sum_{l=k}^{\infty}\frac{(\!-1)^{l}\Gamma(l\beta+bt)}{\Gamma(bt)\Gamma(\beta l+1)}\left(\frac{\lambda}{a^{\beta}}\right)^{l}\sum_{n=k}^{l}\binom{l}{n}(\!-1)^{n}\sum_{j=k}^{n}\binom{n}{j}F^{j}(z)\left(1-F(z)\right)^{n-j}\\&=\sum_{l=k}^{\infty}\frac{(\!-1)^{l}\Gamma(l\beta+bt)}{\Gamma(bt)\Gamma(\beta l+1)}\left(\frac{\lambda}{a^{\beta}}\right)^{l}\sum_{j=k}^{l}F^{j}(z)\sum_{n=j}^{l}\binom{l}{n}\binom{n}{j}(\!-1)^{n}\left(1-F(z)\right)^{n-j}\\&=\sum_{l=k}^{\infty}\frac{(\!-1)^{l}\Gamma(l\beta+bt)}{\Gamma(bt)\Gamma(\beta l+1)}\left(\frac{\lambda}{a^{\beta}}\right)^{l}\sum_{j=k}^{l}\binom{l}{j}F^{j}(z)\sum_{n=0}^{l-j}\binom{l-j}{n}(\!-1)^{n+j}\left(1-F(z)\right)^{n}\\&=\sum_{l=k}^{\infty}\frac{(\!-1)^{l}\Gamma(l\beta+bt)}{\Gamma(bt)\Gamma(\beta l+1)}\left(\frac{\lambda}{a^{\beta}}\right)^{l}F^{l}(z)\sum_{j=k}^{l}\binom{l}{j}(\!-1)^{j}\\&=\sum_{j=k}^{\infty}\sum_{l=j}^{\infty}\binom{l}{j}(\!-1)^{l+j}\left(\frac{F(z)\lambda}{a^{\beta}}\right)^{l}\frac{\Gamma(l\beta+bt)}{\Gamma(bt)\Gamma(\beta l+1)}\\&=\sum_{j=k}^{\infty}\mathbb{P}\{Q^{\beta}(t,\lambda F(z))=j\}=\mathbb{P}\{Q^{\beta}(t,\lambda F(z))\ge k\}.\end{align*}

The proof follows upon substituting the above expression into (71).

Similarly, it can be shown that the space-fractional and space–time-fractional negative binomial processes exhibit the kth order statistic property as in Theorem 5. Also, this result holds for a more general process, namely the time-changed fractional Poisson process studied by Kataria and Khandakar [Reference Kataria and Khandakar22].

6.1. The GCP time-changed by a fractional gamma process

Beghin [Reference Beghin6] introduced a fractional gamma process $\{Z_{\alpha}(t)\}_{t\ge0}$ by time-changing the gamma subordinator $\{Z(t)\}_{t\ge0}$ with an independent inverse stable subordinator, that is,

\begin{equation*}Z_{\alpha}(t)\,:\!= Z(Y_{\alpha}(t)).\end{equation*}

For $x\ge0$ and $t\ge0$ , the following holds for its density function $g_{\alpha}(x,t)$ :

(77) \begin{equation}\left\{\begin{array}{l}\dfrac{\partial }{\partial x}g_{\alpha}(x,t)=-b(1-\mathscr{O}_{-1,t}^{\alpha})g_{\alpha}(x,t),\\g_{\alpha}(x,0)=0,\\\lim_{x\rightarrow +\infty }g_{\alpha}(x,t)=0,\end{array}\right.\end{equation}

where $\mathscr{O}_{-1,t}^{\alpha}$ is the fractional shift operator (see [Reference Beghin6, Theorem 5 and Equation (10)]).

For the Bernštein function $f_1$ given in (22), the process $\{\mathscr{M}_{f}^{\,\alpha}(t)\}_{t\ge0}$ reduces to

(78) \begin{equation}\mathscr{M}_{f_{1}}^{\,\alpha}(t)\,:\!= M(Z(Y_{\alpha}(t)))=M(Z_{\alpha}(t)), \quad t\ge0.\end{equation}

Proposition 8. The PMF $p_{f_{1}}^{\alpha}(n,t)=\mathbb{P}\{\mathscr{M}_{f_{1}}^{\,\alpha}(t)=n\}$ $(n\ge0)$ of $\{\mathscr{M}_{f_{1}}^{\,\alpha}(t)\}_{t\ge0}$ satisfies the fractional differential equations

(79) \begin{equation}\mathscr{O}_{-1,t}^{\alpha}p_{f_{1}}^{\alpha}(n,t)=(1+\Lambda/b)p_{f_{1}}^{\alpha}(n,t)-\frac{1}{b}\sum_{j=1}^{n\wedge k}\lambda_{j}p_{f_{1}}^{\alpha}(n-j,t),\end{equation}

with initial conditions

\begin{equation*}p_{f_{1}}^{\alpha}(n,0)=\begin{cases}1, &\; n=0,\\0, &\; n\ge1.\end{cases}\end{equation*}

Proof. From (78), we can write

(80) \begin{equation}p_{f_{1}}^{\alpha}(n,t)=\int_{0}^{\infty}p(n,x)g_{\alpha}(x,t)\,\mathrm{d}x.\end{equation}

On both sides of (80) we apply the fractional shift operator $\mathscr{O}_{-1,t}^{\alpha}$ and use (77) to obtain

\begin{align*}\mathscr{O}_{-1,t}^{\alpha}p_{f_{1}}^{\alpha}(n,t)&=\int_{0}^{\infty}p(n,x)\Big(g_{\alpha}(x,t)+\frac{1}{b}\frac{\partial }{\partial x}g_{\alpha}(x,t)\Big)\,\mathrm{d}x\\&=p_{f_{1}}^{\alpha}(n,t)+\frac{1}{b}\left[p(n,x)g_{\alpha}(x,t)\right]_{x=0}^{\infty}-\frac{1}{b}\int_{0}^{\infty}g_{\alpha}(x,t)\frac{\mathrm{d}}{\mathrm{d}x}p(n,x)\,\mathrm{d}x\\&=p_{f_{1}}^{\alpha}(n,t)-\frac{1}{b}\int_{0}^{\infty}g_{\alpha}(x,t)\biggl(-\Lambda p(n,x)+\sum_{j=1}^{n\wedge k}\lambda_{j}p(n-j,x)\biggr)\,\mathrm{d}x,\end{align*}

where in the last step we have used (2) with $\beta=1$ . This coincides with the desired result by considering (80).

6.2. Space–time-fractional version of the GCP

We take the Bernštein function $f_2$ in (72) to obtain the space–time-fractional version of the GCP, namely the generalized space–time-fractional counting process (GSTFCP) $\{\mathscr{M}_{f_{2}}^{\,\alpha}(t)\}_{t\ge0}$ , where

\begin{equation*}\mathscr{M}_{f_{2}}^{\,\alpha}(t)\,:\!= M(D_{\beta}(Y_{\alpha}(t))), \quad t\ge0.\end{equation*}

For $1\le j \le k$ , if $\lambda_{j}=\lambda$ or $\lambda_{j}=\lambda(1-\rho)\rho^{j-1}/(1-\rho^{k})$ for $0\le \rho\lt1$ , then the GSTFCP reduces to the space–time-fractional version of the PPoK or PAPoK, respectively.

From (74), the PGF of the GSTFCP is given by

(81) \begin{equation}\mathscr{G}_{f_{2}}^{\,\alpha}(u,t)=E_{\alpha,1}\biggl(-\Big(\sum_{j=1}^{k}(1-u^{j})\lambda_{j}\Big)^{\beta}t^{\alpha}\biggr).\end{equation}

For $k=1$ , it reduces to the PGF of the STFPP (see [Reference Orsingher and Polito28, Equation (2.28)]). It solves the differential equations

\begin{equation*}\frac{\mathrm{d}^{\alpha}}{\mathrm{d}t^{\alpha}}\mathscr{G}_{f_{2}}^{\,\alpha}(u,t)=-\biggl(\sum_{j=1}^{k}(1-u^{j})\lambda_{j}\biggr)^{\beta}\mathscr{G}_{f_{2}}^{\,\alpha}(u,t), \quad \mathscr{G}_{f_{2}}^{\,\alpha}(u,0)=1.\end{equation*}

Let $X_{i}$ for $i\ge1$ be i.i.d. uniform random variables in [0,Reference Applebaum1], and let $\{N_{\alpha}(t,\Lambda^{\beta})\}_{t\ge0}$ be the TFPP with intensity $\Lambda^{\beta}$ such that $\min_{0\le i \le N_{\alpha}(t,\Lambda^{\beta})} X_{i}^{1/\beta}= 1$ when $N_{\alpha}(t,\Lambda^{\beta})=0$ . Then for $0\lt u\lt1$ , the PGF of GSTFCP can be written as

(82) \begin{align}\mathscr{G}_{f_{2}}^{\,\alpha}(u,t)&=\sum_{m=0}^{\infty}\frac{(\!-t^{\alpha}\Lambda^{\beta})^{m}}{\Gamma(m\alpha+1)}\biggl(1-\frac{1}{\Lambda}\sum_{j=1}^{k}\lambda_{j}u^{j}\biggr)^{\beta m}\nonumber\\&=\sum_{m=0}^{\infty}\frac{(\!-t^{\alpha}\Lambda^{\beta})^{m}}{\Gamma(m\alpha+1)}\sum_{r=0}^{m}(\!-1)^{r}\binom{m}{r}\biggl(1-\Big(1-\frac{1}{\Lambda}\sum_{j=1}^{k}\lambda_{j}u^{j}\Big)^{\beta}\biggr)^{r}\nonumber\\&=\sum_{r=0}^{\infty}\biggl(1-\Big(1-\frac{1}{\Lambda}\sum_{j=1}^{k}\lambda_{j}u^{j}\Big)^{\beta}\biggr)^{r}\sum_{m=r}^{\infty}(\!-1)^{m-r}\binom{m}{r}\frac{(t^{\alpha}\Lambda^{\beta})^{m}}{\Gamma(m\alpha+1)}\nonumber\\&=\sum_{r=0}^{\infty}\biggl(\mathbb{P}\Big\{X_{r}^{1/\beta}\ge 1-\frac{1}{\Lambda}\sum_{j=1}^{k}\lambda_{j}u^{j}\Big\}\biggr)^{r}\mathbb{P}\{N_{\alpha}(t,\Lambda^{\beta})=r\}\nonumber\\&=\mathbb{P}\bigg\{\min_{0\le r\le N_{\alpha}(t,\Lambda^{\beta})} X_{r}^{1/\beta}\ge 1-\frac{1}{\Lambda}\sum_{j=1}^{k}\lambda_{j}u^{\,j}\bigg\}.\end{align}

The result in (82) is of particular interest as it connects the GSTFCP with the TFPP via uniform random variables. On using the generalized binomial theorem to expand the right-hand side of (81), we obtain the PMF of the GSTFCP in the following form:

(83) \begin{equation}\mathbb{P}\{\mathscr{M}_{f_{2}}^{\,\alpha}(t)=n\}=\sum_{ \Omega(k,n)}\biggl(\prod_{j=1}^{k}\frac{\lambda_{j}^{x_{j}}}{x_{j}!}\biggr)\frac{(\!-1)^{z_{k}}}{\Lambda^{z_{k}}}\sum_{m=0}^{\infty}\frac{(\!-\Lambda^{\beta}t^{\alpha})^{m}}{\Gamma(\alpha m+1)}\frac{\Gamma(\beta m+1)}{\Gamma(\beta m+1-z_{k})},\quad n\ge0.\end{equation}

The steps involved to obtain (83) are similar to those in the proof of (51).

Next, we obtain an alternate form of the PMF of the GSTFCP. The PMF (7) of the GCP can be rewritten as

\begin{equation*}p(n,t)=\sum_{\Omega(k,n)}\biggl(\prod_{j=1}^{k}\frac{\lambda_{j}^{x_{j}}}{x_{j}!}\biggr)(\!-\partial_{\Lambda})^{z_{k}}{\mathrm{e}}^{-\Lambda t}, \quad n\ge 0.\end{equation*}

Then, from (72), we have

(84) \begin{align}\mathbb{P}\{\mathscr{M}_{f_{2}}^{\,\alpha}(t)=n\}&=\int_{0}^{\infty}\sum_{\Omega(k,n)}\biggl(\prod_{j=1}^{k}\frac{\lambda_{j}^{x_{j}}}{x_{j}!}\biggr)(\!-\partial_{\Lambda})^{z_{k}}{\mathrm{e}}^{-\Lambda s}\mathbb{P}\{D_{\beta}(Y_{\alpha}(t))\in\mathrm{d}s\}\nonumber\\&=\sum_{\Omega(k,n)}\biggl(\prod_{j=1}^{k}\frac{\lambda_{j}^{x_{j}}}{x_{j}!}\biggr)(\!-\partial_{\Lambda})^{z_{k}}\mathbb{E}\!\left({\mathrm{e}}^{-\Lambda D_{\beta}(Y_{\alpha}(t))}\right)\end{align}

(85) \begin{align}=\sum_{\Omega(k,n)}\biggl(\prod_{j=1}^{k}\frac{\lambda_{j}^{x_{j}}}{x_{j}!}\biggr)(\!-\partial_{\Lambda})^{z_{k}}E_{\alpha,1}\!\left(-\Lambda^{\beta}t^{\alpha}\right),\end{align}

where in the last step we have used Lemma 3.1 of [Reference Beghin and D’Ovidio4]. Thus, the PGF of the GSTFCP can alternatively be obtained as follows:

\begin{align*}\mathscr{G}_{f_{2}}^{\,\alpha}(u,t)&=\sum_{n=0}^{\infty}u^{n}\sum_{\Omega(k,n)}\biggl(\prod_{j=1}^{k}\frac{\lambda_{j}^{x_{j}}}{x_{j}!}\biggr)(\!-\partial_{\Lambda})^{z_{k}}\mathbb{E}\left({\mathrm{e}}^{-\Lambda D_{\beta}(Y_{\alpha}(t))}\right)\quad (\text{using}\ ({84}))\\&=\sum_{r=0}^{\infty}\Big(\sum_{j=1}^{k}\lambda_{j}u^{j}\Big)^{r}\frac{(\!-\partial_{\Lambda})^{r}}{r!}\mathbb{E}\!\left({\mathrm{e}}^{-\Lambda D_{\beta}(Y_{\alpha}(t))}\right)\\&=\mathbb{E}\big({\mathrm{e}}^{-\sum_{j=1}^{k}(1-u^{j})\lambda_{j}D_{\beta}(Y_{\alpha}(t))}\big),\end{align*}

which reduces to (81) on using Lemma 3.1 of [Reference Beghin and D’Ovidio4].

Let $\{N^{\alpha,\beta}(t)\}_{t\ge0}$ denote the STFPP with intensity $\lambda$ . For notational convenience, let $N^{\alpha,\beta}(t, \lambda)=N^{\alpha,\beta}(t)$ . On taking $k=1$ and $\Lambda=\lambda_{1}=\lambda$ (say) in (83), we get

(86) \begin{equation}\mathbb{P}\{N^{\alpha,\beta}(t,\lambda)=n\}=\frac{(\!-1)^{n}}{n!}\sum_{m=0}^{\infty}\frac{(\!-\lambda^{\beta}t^{\alpha})^{m}\Gamma(\beta m+1)}{\Gamma(\beta m+1-n)\Gamma(\alpha m+1)},\end{equation}

which agrees with the PMF of the STFPP (see [Reference Orsingher and Polito28, Equation (2.29)]). Again taking $k=1$ and $\Lambda=\lambda$ in (85), we get an alternate version of the PMF of the STFPP (see [Reference Beghin and D’Ovidio4, Equation (3.19)]).

Theorem 7. Let $X_{(k)}^{N^{\alpha,\beta}(t,\lambda)}$ be a kth order statistic from a sample of size $N^{\alpha,\beta}(t,\lambda)$ , for $k\in \{1,2,\ldots, N^{\alpha,\beta}(t,\lambda)\}$ , and let $\{X_{i}\}_{i\ge1}$ be a sequence of i.i.d. random variables with distribution function F. Then

\begin{equation*}\mathbb{P}\bigl\{X_{(k)}^{N^{\alpha,\beta}(t,\lambda)}\lt z\,\big|\,N^{\alpha,\beta}(t,\lambda)\ge k\bigr\}=\frac{\mathbb{P}\{N^{\alpha,\beta}(t,\lambda F(z))\ge k\}}{\mathbb{P}\{N^{\alpha,\beta}(t,\lambda)\ge k\}}.\end{equation*}

Proof. On using the law of conditional probability, we get

(87) \begin{equation}\begin{split}&\mathbb{P}\big\{X_{(k)}^{N^{\alpha,\beta}(t,\lambda)}\lt z\,\big|\,N^{\alpha,\beta}(t,\lambda)\ge k\bigr\}\\&\quad=\frac{\sum_{n=k}^{\infty}\mathbb{P}\{X_{(k)}^{N^{\alpha,\beta}(t,\lambda)}\lt z\,|\,N^{\alpha,\beta}(t,\lambda)=n\}\,\mathbb{P}\{N^{\alpha,\beta}(t,\lambda)=n\}}{\mathbb{P}\{N^{\alpha,\beta}(t,\lambda)\ge k\}}.\end{split}\end{equation}

Using (70) and (86), the quantity in the numerator on the right-hand side of (87) can be evaluated as follows:

\begin{align*}\sum_{n=k}^{\infty}&\mathbb{P}\bigl\{X_{(k)}^{N^{\alpha,\beta}(t,\lambda)}\lt z\,\big|\,N^{\alpha,\beta}(t,\lambda)=n\bigr\}\,\mathbb{P}\{N^{\alpha,\beta}(t,\lambda)=n\}\\&=\sum_{n=k}^{\infty}\sum_{j=k}^{n}\binom{n}{j}F^{j}(z)\left(1-F(z)\right)^{n-j}\frac{(\!-1)^{n}}{n!}\sum_{m=0}^{\infty}\frac{(\!-\lambda^{\beta}t^{\alpha})^{m}\Gamma(\beta m+1)}{\Gamma(\beta m+1-n)\Gamma(\alpha m+1)}\\&=\sum_{j=k}^{\infty}\sum_{m=0}^{\infty}\sum_{n=j}^{\infty}\frac{F^{j}(z)}{(n-j)!j!}\left(1-F(z)\right)^{n-j}\frac{(\!-1)^{n}(\!-\lambda^{\beta}t^{\alpha})^{m}\Gamma(\beta m+1)}{\Gamma(\beta m+1-n)\Gamma(\alpha m+1)}\\&=\sum_{j=k}^{\infty}\frac{(\!-1)^{j}F^{j}(z)}{j!}\sum_{m=0}^{\infty}\frac{(\!-\lambda^{\beta}t^{\alpha})^{m}\Gamma(\beta m+1)}{\Gamma(\alpha m+1)}\sum_{n=0}^{\infty}\frac{(\!-1)^{n}\left(1-F(z)\right)^{n}}{n!\Gamma(\beta m+1-n-j)}\end{align*}

\begin{align*}&=\sum_{j=k}^{\infty}\frac{(\!-1)^{j}}{j!}\sum_{m=0}^{\infty}\frac{(\!-\lambda^{\beta}t^{\alpha})^{m}\Gamma(\beta m+1)}{\Gamma(\alpha m+1)}\frac{\left(F(z)\right)^{\beta m}}{\Gamma(\beta m+1-j)}\\&=\sum_{j=k}^{\infty}\frac{(\!-1)^{j}}{j!}\sum_{m=0}^{\infty}\frac{(\!-\lambda^{\beta}F^{\beta}(z)t^{\alpha})^{m}\Gamma(\beta m+1)}{\Gamma(\alpha m+1)\Gamma(\beta m+1-j)}\\&=\sum_{j=k}^{\infty}\mathbb{P}\{N^{\alpha,\beta}(t,\lambda F(z))=j\}=\mathbb{P}\{N^{\alpha,\beta}(t,\lambda F(z))\ge k\}.\end{align*}

The proof follows upon substituting the above expression into (87).