Some Time-changed fractional Poisson processes

TL;DR

This paper investigates two types of fractional Poisson processes time-changed by Lévy subordinators, analyzing their distributional properties, dependence structures, and providing simulations to illustrate their behaviors.

Contribution

It introduces and studies TCFPP-I and TCFPP-II, new fractional Poisson processes with time changes, detailing their properties, distributions, and differential equations, which were not previously explored.

Findings

TCFPP-I exhibits long-range dependence under certain conditions.

TCFPP-II is characterized as a renewal process with a specific waiting time distribution.

Sample path simulations demonstrate the processes' behaviors.

Abstract

In this paper, we study the fractional Poisson process (FPP) time-changed by an independent L\'evy subordinator and the inverse of the L\'evy subordinator, which we call TCFPP-I and TCFPP-II, respectively. Various distributional properties of these processes are established. We show that, under certain conditions, the TCFPP-I has the long-range dependence property and also its law of iterated logarithm is proved. It is shown that the TCFPP-II is a renewal process and its waiting time distribution is identified. Its bivariate distributions and also the governing difference-differential equation are derived. Some specific examples for both the processes are discussed. Finally, we present the simulations of the sample paths of these processes.

Figures

Equations

$$N_{\beta }(t,\lambda )=N(E_{\beta }(t),\lambda ),\text{leavevmode}t\ge 0,\text{leavevmode}0<\beta <1,$$

$$E_{\beta }(t)=\inf \{r\ge 0:D_{\beta }(r)>t\},\text{leavevmode}\text{leavevmode}\text{leavevmode}t\ge 0,$$

$$\mathbb{E}[e^{-s{D}_f(t)}]=e^{-tf(s)},$$

$$f(s)=bs+{\int }_0^{\infty }(1-e^{-sx})\nu (dx),\text{leavevmode}b\ge 0,$$

$${\int }_0^{\infty }(x\wedge 1)\nu (dx)<\infty .$$

$$\{Q_{\beta }^f(t,\lambda )\}=\{N_{\beta }(D_f(t),\lambda )\},\text{leavevmode}t\ge 0,$$

$$E_f(t)=\inf \{r\ge 0:D_f(r)>t\},\text{leavevmode}\text{leavevmode}\text{leavevmode}t\ge 0,$$

$$\{W_{\beta }^f(t,\lambda )\}=\{N_{\beta }(E_f(t),\lambda )\},\text{leavevmode}t\ge 0.$$

$$L_{\alpha }(z)=\sum _{k=0}^{\infty }\frac{z^k}{\Gamma (\alpha k+1)},\alpha ,z\in \mathrm{I}\mathrm{C}\text{ and }\Re (\alpha )>0.$$

$$L_{\alpha ,\beta }^{\gamma }(z)=\sum _{k=0}^{\infty }\frac{\Gamma (\gamma +k)}{\Gamma (\gamma )\Gamma (\alpha k+\beta )}\frac{z^k}{k!},\alpha ,\beta ,\gamma ,z\in \mathrm{I}\mathrm{C}\text{ and }\Re (\alpha ),\Re (\beta ),\Re (\gamma )>0.$$

$${\partial }_t^{\beta }{p}_{{}_{\beta }}(n|t,\lambda )$$

$${\partial }_t^{\beta }{p}_{{}_{\beta }}(0|t,\lambda )$$

$${\partial }_t^{\beta }f(t)=\{\begin{array}{ll}\text{hfill}\frac{1}{\Gamma (1-\beta )}{\int }_0^t\frac{f^{\prime }(s)}{(t-s{)}^{\beta }}ds,\text{hfill}& 0<\beta <1,\text{vspace}\ast 0.3cm\\ \frac{d}{dt}f(t),\beta =1.& \end{array}$$

$$p_{{}_{\beta }}(n|t,\lambda )=\frac{(\lambda t^{\beta }{)}^n}{n!}\sum _{k=0}^{\infty }\frac{(n+k)!}{k!}\frac{(-\lambda t^{\beta }{)}^k}{\Gamma (\beta (k+n)+1)}\text{leavevmode}.$$

$$\mathbb{E}[N_{\beta }(t,\lambda )]$$

$$\text{Cov}[N_{\beta }(s,\lambda ),N_{\beta }(t,\lambda )]$$

$$\{Q_{\beta }^f(t,\lambda )\}=\{N_{\beta }(D_f(t),\lambda )\},\text{leavevmode}t\ge 0,$$

$$\mathbb{P}[Q_{\beta }^f(t)=n]=\frac{{\lambda }^n}{n!}\sum _{k=0}^{\infty }\frac{(n+k)!}{k!}\frac{(-\lambda {)}^k}{\Gamma (\beta (k+n)+1)}\mathbb{E}[D_f^{\beta (n+k)}(t)],\text{leavevmode}n\ge 0.$$

$$\mathbb{P}[Q_{\beta }^f(t)=n]$$

$$={\int }_0^{\infty }\frac{(\lambda y^{\beta }{)}^n}{n!}\sum _{k=0}^{\infty }\frac{(n+k)!}{k!}\frac{(-\lambda y^{\beta }{)}^k}{\Gamma (\beta (k+n)+1)}{g}_f(y,t)dy$$

$$=\frac{{\lambda }^n}{n!}\sum _{k=0}^{\infty }\frac{(n+k)!}{k!}\frac{(-\lambda {)}^k}{\Gamma (\beta (k+n)+1)}\mathbb{E}[D_f^{\beta (n+k)}(t)].\text{qed}$$

$$\sum _{n=0}^{\infty }{\delta }_{\beta }^f(n|t,\lambda )$$

$$=\sum _{n=0}^{\infty }\frac{{\lambda }^n}{n!}\sum _{r=n}^{\infty }\frac{r!}{(r-n)!}\frac{(-\lambda {)}^{r-n}}{\Gamma (\beta r+1)}\mathbb{E}[D_f^{\beta r}(t)]\text{leavevmode}\text{leavevmode}\text{leavevmode}(\text{taking }r=k+n)$$

$$=\sum _{r=0}^{\infty }\sum _{n=0}^r\frac{{\lambda }^n}{n!}\frac{r!}{(r-n)!}\frac{(-\lambda {)}^{r-n}}{\Gamma (\beta r+1)}\mathbb{E}[D_f^{\beta r}(t)]$$

$$=\sum _{r=0}^{\infty }\frac{{\lambda }^r\mathbb{E}[D_f^{\beta r}(t)]}{\Gamma (\beta r+1)}\sum _{n=0}^r\left(\genfrac{}{}{0ex}{}{r}{n}\right)(-1{)}^{r-n}=\sum _{r=0}^{\infty }\frac{{\lambda }^r\mathbb{E}[D_f^{\beta r}(t)]}{\Gamma (\beta r+1)}(1-1{)}^r$$

$$=\frac{\mathbb{E}[1]}{\Gamma (1)}=1.$$

$$f(x|\alpha ,pt)=\frac{{\alpha }^{pt}}{\Gamma (pt)}{e}^{-\alpha x}{x}^{pt-1},\text{leavevmode}x>0,$$

$$\mathbb{E}[e^{-sY(t)}]={\left(1+\frac{s}{\alpha }\right)}^{-pt}=\exp \left(-pt\log \left(1+s/\alpha \right)\right).$$

$$\{Q_{\beta }^{(1)}(t)\}=\{N_{\beta }(Y(t))\},\text{leavevmode}t\ge 0.$$

$$\mathbb{E}[Y^{\rho }(t)]=\frac{\Gamma (pt+\rho )}{{\alpha }^{\rho }\Gamma (pt)},\text{leavevmode}\text{leavevmode}\rho >0.$$

$$\mathbb{P}[Q^{(1)}_{\beta}(t)=n]=\bigg{(}\frac{\lambda}{\alpha^{\beta}}\bigg{)}^{n}\frac{1}{n!\Gamma(pt)}\sum\limits_{k=0}^{\infty}\frac{(n+k)!}{k!}\frac{\Gamma{((n+k)\beta+pt)}}{\Gamma{(\beta(n+k)+1)}}\bigg{(}\frac{-\lambda}{\alpha^{\beta}}\bigg{)}^{k},\leavevmode\nobreak\ n\geq 0,$$

$$\mathbb{E}[e^{-s{D}_{\alpha }^{\mu }(t)}]=e^{-t\left((\mu +s{)}^{\alpha }-{\mu }^{\alpha }\right)}.$$

$$g_{\mu }(x,t)=e^{-\mu x+{\mu }^{\beta }t}g(x,t),\text{leavevmode}x>0,$$

$$\{Q_{\beta }^{(2)}(t)\}=\{N_{\beta }(D_{\alpha }^{\mu }(t))\},\text{leavevmode}t\ge 0.$$

Full text

1. Introduction

Recently, there has been a considerable interest in studying the fractional Poisson process (FPP) $\{N_{\beta}(t,\lambda)\}_{t\geq 0}$. The early development of the FPP is due [32, 16, 21]. Later, a rich growth of the literature is contributed by [24, 25, 5, 6]. It is proved in [25] that the FPP can be seen as the subordination of the Poisson process by an independent inverse $\beta$-stable subordinator, that is,

[TABLE]

where $\{N(t,\lambda)\}_{t\geq 0}$ is the Poisson process with rate $\lambda>0$ and $\{E_{\beta}(t)\}_{t\geq 0}$ is the inverse $\beta$-stable subordinator. The relation between the inverse $\beta$-stable subordinator and the $\beta$-stable subordinator $\{D_{\beta}(t)\}_{t\geq 0}$ is

[TABLE]

where the Laplace transform (LT) of the $\beta$-stable subordinator is given by $\mathbb{E}[e^{-sD_{\beta}(t)}]=e^{-ts^{\beta}}$, $s>0$. [18] studied the time-changed Poisson process subordinated with the inverse Gaussian, the first-exit time of the inverse Gaussian, the stable and the tempered stable subordinator. [28] studied the Poisson process subordinated by an independent $\beta$-stable subordinator $\{D_{\beta}(t)\}_{t\geq 0}$, called the space fractional Poisson process. In [30], studied the Poisson process subordinated with independent Lévy subordinator and [36] studied the FPP subordinated with an independent gamma subordinator to obtain the fractional negative binomial process (FNBP). Observe that the Lévy subordinator covers most of the special subordinators (see [3, Theorem 1.3.15]) considered in the literature.

The goal of the present work is to study the FPP $\{N_{\beta}(t,\lambda)\}_{t\geq 0}$ time-changed by an independent Lévy subordinator (hereafter referred to as the subordinator) $\{D_{f}(t)\}_{t\geq 0}$ with LT (see [3, Section 1.3.2])

[TABLE]

where

[TABLE]

is the Bernstein function. Here $b$ is the drift coefficient and $\nu$ is a non-negative Lévy measure on positive half-line such that

[TABLE]

The assumption $\nu(0,\infty)=\infty$ guarantees that the sample paths of $D_{f}(t)$ are almost surely $(a.s.)$ strictly increasing. [30] studied the process $\{N(D_{f}(t),\lambda)\}_{t\geq 0},$ where $\{D_{f}(t)\}_{t\geq 0}$ is the subordinator with drift coefficient $b=0$. We investigate the process

[TABLE]

where the time variable $t$ is replaced by an independent subordinator $\{D_{f}(t)\}_{t\geq 0}$ and call the time-changed fractional Poisson process version one (TCFPP-I). The probability mass function (pmf) of TCFPP-I $\{Q_{\beta}^{f}(t,\lambda)\}_{t\geq 0}$ is obtained and its mean and covariance functions are computed. We discuss the asymptotic behavior of the covariance function for large $t$. Using these results, we prove the long-range dependence (LRD) property for the TCFPP-I process, under certain conditions. The law of iterated logarithm for the TCFPP-I is also proved.

The first-exit time of $\{D_{f}(t)\}_{t\geq 0}$ is defined as

[TABLE]

which is the right-continuous inverse of the subordinator $\{D_{f}(t)\}_{t\geq 0}$. We consider also the time-changed fractional Poisson process version two (TCFPP-II) defined by

[TABLE]

The pmf, mean and covariance functions for the TCFPP-II are derived. We also discuss the asymptotic behavior of the mean and variance functions of the TCFPP-II. The bivariate distribution and the difference-differential equation governing the pmf of the TCFPP-II are derived. Lastly, we present the simulations for some special TCFPP-I and TCFPP-II processes.

The paper is organized as follows. In Section 2, we present some preliminary definitions and results. The TCFPP-I and the TCFPP-II processes are investigated in detail in Section 3 and 4, respectively. In Section 5, we present the simulations for some specific TCFPP-I and TCFPP-II processes.

2. Preliminaries

In this section, we present some preliminary results which are required later in the paper.

The Mittag-Leffler function $L_{\alpha}(z)$ is defined as (see [27])

[TABLE]

The generalized Mittag-Leffler function $L_{\alpha,\beta}^{\gamma}(z)$ is defined as (see [31])

[TABLE]

Let $0<\beta\leq 1$. The fractional Poisson process (FPP) $\{N_{\beta}(t,\lambda)\}_{t\geq 0}$, which is a generalization of the Poisson process $\{N(t,\lambda)\}_{t\geq 0}$, is defined to be a stochastic process for which $p_{{}_{\beta}}(n|t,\lambda)=\mathbb{P}[N_{\beta}(t,\lambda)=n]$ satisfies (see [21, 24, 25])

[TABLE]

$\text{with }p_{{}_{\beta}}(n|0,\lambda)=1\text{ if }n=0\text{ and is zero if }n\geq 1.$ Here, $\partial^{\beta}_{t}$ denotes the Caputo-fractional derivative defined by

[TABLE]

The pmf $p_{{}_{\beta}}(n|t,\lambda)$ of the FPP is given by (see [21, 25])

[TABLE]

The mean, variance and covariance functions (see [21, 22]) of the FPP are given by

[TABLE]

where $q=\lambda/\Gamma(1+\beta)$, $R=\frac{\lambda^{2}}{\beta}\left(\frac{1}{\Gamma(2\beta)}-\frac{1}{\beta\Gamma^{2}(\beta)}\right)>0$, $d=\beta q^{2}B(\beta,1+\beta)$, and $B(a,b;x)=\int_{0}^{x}t^{a-1}(1-t)^{b-1}dt,\leavevmode\nobreak\ 0<x<1$, is the incomplete beta function.

3. Time-changed fractional Poisson process-I

In this section, we consider the FPP time-changed by a subordinator $\{D_{f}(t)\}_{t\geq 0}$, defined in (1.2), for which the moments $\mathbb{E}[D_{f}^{\rho}(t)]<\infty$ for all $\rho>0$. Note $\{D_{f}(t)\}_{t\geq 0}$ is an increasing process with $D_{f}(0)=0\leavevmode\nobreak\ a.s.$

Definition 3.1** **(TCFPP-I).

The time-changed fractional Poisson process version one (TCFPP-I) is defined as

[TABLE]

where $\{N_{\beta}(t,\lambda)\}_{t\geq 0}$ is the FPP and is independent of the subordinator $\{D_{f}(t)\}_{t\geq 0}$.

We suppress the parameter $\lambda$, unless the context requires, associated with the processes $\{N_{\beta}(t,\lambda)\}_{t\geq 0}$ and $\{Q_{\beta}^{f}(t,\lambda)\}_{t\geq 0}$.

Theorem 3.1**.**

The one-dimensional distributions of the TCFPP-I is given by

[TABLE]

Proof.

Let $g_{f}(x,t)$ be the pdf of $D_{f}(t)$. Then, from (2.4),

[TABLE]

Remark 3.1**.**

It can be seen that the *pmf * $\delta_{\beta}^{f}(n|t,\lambda)$ satisfies the normalizing condition $\sum_{n=0}^{\infty}\delta_{\beta}^{f}(n|t,\lambda)=1$. We have

[TABLE]

We next present some examples of the TCFPP-I processes.

Example 3.1** (Fractional negative binomial process).**

Let $\{Y(t)\}_{t\geq 0}$ be the gamma subordinator, where $Y(t)\sim G(\alpha,pt)$, the gamma distribution with density

[TABLE]

where both $\alpha$ and $p$ are positive. It is known that (see [3, p. 54])

[TABLE]

The fractional negative binomial process (FNBP), introduced and studied in detail in [36], is defined by time-changing the FPP by an independent gamma subordinator, that is,

[TABLE]

It is known that (see [36, eq. (4.4)])

[TABLE]

From (3.1), the pmf of $Q^{(1)}_{\beta}(t)$ is

[TABLE]

which coincides with the pmf of the FNBP obtained in [36]. Also, it holds that

$\sum_{n=0}^{\infty}\mathbb{P}[Q^{(1)}_{\beta}(t)=n]=1.$

Example 3.2** (FPP subordinated by tempered $\alpha$-stable subordinator).**

Let $\{D_{\alpha}^{\mu}(t)\}_{t\geq 0},\mu>0,0<\alpha<1$ be the tempered $\alpha$-stable subordinator with LT

[TABLE]

The pdf of the tempered $\alpha$-stable subordinator is given by (see [2, eq. (2.2)])

[TABLE]

where $g(x,t)$ is the pdf of the $\alpha$-stable subordinator $\{D_{\alpha}(t)\}_{t\geq 0}$. The FPP time-changed by an independent tempered $\alpha$-stable subordinator is defined as

[TABLE]

In this case, the pmf (3.1) reduces to

[TABLE]

It is easy to show that $\sum_{n=0}^{\infty}\mathbb{P}[Q_{\beta}^{(2)}(t)=n]=1.$

Example 3.3** (FPP subordinated by inverse Gaussian subordinator).**

Let $\{G(t)\}_{t\geq 0}$ be the inverse Gaussian subordinator with LT (see [3, Example 1.3.21])

[TABLE]

The FPP time-changed by an independent inverse Gaussian subordinator is defined as

[TABLE]

It is known that (see [15, 18]) the moments of $\{G(t)\}_{t\geq 0}$ are given by

[TABLE]

where $K_{\nu}(z),z>0$ is the modified Bessel function of third kind with index $\nu\in\mathbb{R}$. We can substitute the moments of $\{G(t)\}_{t\geq 0}$ in (3.1) to obtain the pmf of $\{Q^{(3)}_{\beta}(t)\}_{t\geq 0}$. Moreover, it can be shown that $\sum_{n=0}^{\infty}\mathbb{P}[Q^{(3)}_{\beta}(t)=n]=1.$

We next obtain the mean, the variance and the covariance functions of the TCFPP-I.

Theorem 3.2**.**

Let $0<s\leq t<\infty,$ $q=\lambda/\Gamma(1+\beta)$ and $d=\beta q^{2}B(\beta,1+\beta)$. The distributional properties of the TCFPP-I $\{Q_{\beta}^{f}(t,\lambda)\}_{t\geq 0}$ are as follows:

(i) $\leavevmode\nobreak\ \leavevmode\nobreak\ \mathbb{E}[Q^{f}_{\beta}(t)]=q\mathbb{E}[D_{f}^{\beta}(t)]$,

(ii)$\leavevmode\nobreak\ \leavevmode\nobreak\ \text{Var}[Q^{f}_{\beta}(t)]=q\mathbb{E}[D_{f}^{\beta}(t)]\left(1-q\mathbb{E}[D_{f}^{\beta}(t)]\right)+2d\mathbb{E}[D_{f}^{2\beta}(t)]$,

[TABLE]

Proof.

Using (2.5), we get

[TABLE]

which proves Part (i). From (2.6) and (2.5),

[TABLE]

which leads to

[TABLE]

By (3.3) and (3.4), Part (iii) follows. Part (ii) follows from Part (iii) by putting $s=t$. ∎

Index of dispersion. The index of dispersion for a counting process $\{X(t)\}_{t\geq 0}$ is defined by (see [12, p. 72])

[TABLE]

The stochastic process $\{X(t)\}_{t\geq 0}$ is said to be overdispersed if $I(t)>1$ for all $t\geq 0$ (see [7, 36]). Since the mean of the TCFPP-I $\{Q_{\beta}^{f}(t)\}_{t\geq 0}$ is nonnegative, it suffices to show that Var$[Q_{\beta}^{f}(t)]-\mathbb{E}[Q_{\beta}^{f}(t)]>0$. From Theorem 3.2, we have that

[TABLE]

where $R=\frac{\lambda^{2}}{\beta}\left(\frac{1}{\Gamma(2\beta)}-\frac{1}{\beta\Gamma^{2}(\beta)}\right)>0$ and $\lambda>0$ for all $\beta\in(0,1)$ (see [7, Section 3.1]). Hence, the TCFPP-I exhibits overdispersion.

We next derive the asymptotic expansion for the covariance function of the TCFPP-I process. The following result generalizes Lemma 4.1 of [23] to the subordinator $\{D_{f}(t)\}_{t\geq 0}$. First, recall the following definition.

Definition 3.2**.**

Let $f(x)$ and $g(x)$ be positive functions. We say that $f(x)$ is asymptotically equal to $g(x)$, written as $f(x)\sim g(x)$, as $x\to\infty$, if

[TABLE]

Theorem 3.3**.**

Let $0<\beta<1$, $0<s\leq t$ and $s$ be fixed. Let $\{D_{f}(t)\}_{t\geq 0}$ be the subordinator with $\mathbb{E}[e^{-sD_{f}(t)}]=e^{-tf(s)}$, where $f(s)$ is the Bernstein function defined in (1.3). If $\mathbb{E}[D_{f}^{\beta}(t)]\rightarrow\infty$ as $t\rightarrow\infty$, then

(i) the asymptotic expansion of ${\mathbb{E}}\left[D_{f}^{\beta}(s)D_{f}^{\beta}(t)\right]$ is

[TABLE]

(ii) the asymptotic expansion of $\beta\mathbb{E}\left[D_{f}^{2\beta}(t)B\left(\beta,1+\beta;D_{f}(s)/D_{f}(t)\right)\right]$ is

[TABLE]

Proof.

(i) Since the subordinator $\{D_{f}(t)\}_{t\geq 0}$ has stationary and independent increments, it suffices to show that

[TABLE]

Also, $\{D_{f}(t)\}_{t\geq 0}$ is an increasing process with $D_{f}(0)=0\leavevmode\nobreak\ a.s.$ so that

[TABLE]

Now consider

[TABLE]

From (3.5) and (3.6), we have that

[TABLE]

Taking the limit as $t$ tends to infinity in the above equation and using the fact that $\{D_{f}(t)\}_{t\geq 0}$ has independent increments, we get

[TABLE]

which proves Part (i).

To prove Part (ii), it suffices to show that, in view of Part (i),

[TABLE]

Note first that

[TABLE]

which leads to

[TABLE]

Hence,

[TABLE]

On the other hand,

[TABLE]

This leads to

[TABLE]

using Part (i). By (3.7) and (3.8), we have that

[TABLE]

This completes the proof of Part (ii). ∎

Remark 3.2**.**

The assumption, in Theorem 3.3, that $\mathbb{E}[D_{f}^{\beta}(t)]\rightarrow\infty$ as $t\rightarrow\infty$, is satisfied for the following subordinators.

(a) Gamma subordinator: It is known (see [36]) that $\mathbb{E}[Y^{\beta}(t)]=\frac{\Gamma(pt+\beta)}{\alpha^{\beta}\Gamma(pt)}\sim\left(\frac{pt}{\alpha}\right)^{\beta}$, which implies $\mathbb{E}[Y^{\beta}(t)]\to\infty$ as $t\to\infty$.

(b) Inverse Gaussian subordinator: The moments of inverse Gaussian subordinator $\{G(t)\}_{t\geq 0}$ are given in (3.2). The asymptotic expansion of $K_{\nu}(z)$, for large $z$, is (see [15, eq. (A.9)])

[TABLE]

where $\mu=4\nu^{2}$. For $0<\beta<1$, the asymptotic expansion of (3.2), as $t\to\infty$, is

[TABLE]

which implies that $\mathbb{E}[G^{\beta}(t)]\to\infty$ as $t\to\infty.$

We next show that the TCFPP-I, under certain conditions on the subordinator $\{D_{f}(t)\}_{t\geq 0}$, possesses the LRD property. There are various definitions in the literature for the LRD property of a stochastic process. We now present the definition (see [13, 23]) that will be used in this paper.

Definition 3.3**.**

Let $0<s<t$ and $s$ be fixed. Assume a stochastic process $\{X(t)\}_{t\geq 0}$ has the correlation function Corr$[X(s),X(t)]$ that satisfies

[TABLE]

for large $t$, $d>0$, $c_{1}(s)>0$ and $c_{2}(s)>0$. That is,

[TABLE]

for some $c(s)>0$ and $d>0.$ We say $\{X(t)\}_{t\geq 0}$ has the long-range dependence (LRD) property if $d\in(0,1)$ and has the short-range dependence (SRD) property if $d\in(1,2)$.

Note (3.10) implies that Corr$[X(s),X(t)]$ behaves like $t^{-d}$, for large $t$.

Theorem 3.4**.**

Let $\{D_{f}(t)\}_{t\geq 0}$ be such that $\mathbb{E}[D_{f}^{\beta}(t)]\sim k_{1}t^{\rho}$ and $\mathbb{E}[D_{f}^{2\beta}(t)]\sim k_{2}t^{2\rho},$ for some $0<\rho<1,$ and positive constants $k_{1}$ and $k_{2}$ with $k_{2}\geq k_{1}^{2}$. Then the TCFPP-I $\{Q^{f}_{\beta}(t)\}_{t\geq 0}$ has the LRD property.

Proof.

Consider the last term of $\text{Cov}[Q^{f}_{\beta}(s),Q_{\beta}^{f}(t)]$ given in Theorem 3.2 (iii), namely,

[TABLE]

Using Theorem 3.3 (ii), we get for large $t$,

[TABLE]

Using (3.11), Theorem 3.2 (iii) and $\mathbb{E}[D_{f}^{\beta}(t)]\sim k_{1}t^{\rho}$, we get,

[TABLE]

since $t^{\rho}-(t-s)^{\rho}\sim\rho st^{\rho-1}$ for large $t$, and $\rho>0.$

Similarly, from Theorem 3.2 (ii) and $\mathbb{E}[D_{f}^{2\beta}(t)]\sim k_{2}t^{2\rho}$, we have that

[TABLE]

where $d_{1}=\frac{\lambda^{2}}{\beta}\left(\frac{k_{2}}{\Gamma(2\beta)}-\frac{k_{1}^{2}}{\beta\Gamma^{2}(\beta)}\right)$. Thus, from (3.12) and (3.13), we have for large $t>s$,

[TABLE]

which decays like the power law $t^{-\rho},\leavevmode\nobreak\ 0<\rho<1$. Hence, the TCFPP-I exhibits the LRD property.∎

Remark 3.3**.**

From Remark 3.2, it can be seen that moments of the gamma subordinator has the asymptotic expansion $\mathbb{E}[Y^{\beta}(t)]\sim(p/\alpha)^{\beta}t^{\beta}$ and $\mathbb{E}[Y^{2\beta}(t)]\sim\left(p/\alpha\right)^{2\beta}t^{2\beta}$. Therefore, the FNBP $\{Q^{(1)}_{\beta}(t)\}_{t\geq 0}$ exhibits the LRD property. Similarly, for the inverse Gaussian subordinator $\{G(t)\}_{t\geq 0}$, we have the asymptotic expansion $\mathbb{E}[G^{\beta}(t)]\sim\left(\delta/\gamma\right)^{\beta}t^{\beta}$ and $\mathbb{E}[G^{2\beta}(t)]\sim\left(\delta/\gamma\right)^{2\beta}t^{2\beta}$. Hence, $\{Q^{(3)}_{\beta}(t)\}_{t\geq 0}$ also has the LRD property.

Definition 3.4**.**

We call a function $l:(0,\infty)\rightarrow(0,\infty)$ regularly varying at 0+ with index $\alpha\in\mathbb{R}$ (see [8]) if

[TABLE]

We first reproduce the following law of iterated logarithm (LIL) for the subordinator from [8, Chapter III, Theorem 14].

Lemma 3.1**.**

Let $X(t)$ be a subordinator with $\mathbb{E}[e^{-sX(t)}]=e^{-tf(s)}$, where $f(s)$ is regularly varying at $0+$ with index $\alpha\in(0,1)$. Let $h$ be the inverse function of $f$ and

[TABLE]

Then

[TABLE]

We next prove the LIL for the TCFPP-I.

Theorem 3.5** (Law of iterated logarithm).**

Let the Laplace exponent $f(s)$ of the subordinator $\{D_{f}(t)\}_{t\geq 0}$ be regularly varying at 0+ with index $\alpha\in(0,1)$. Then, for $0<\beta<1$,

[TABLE]

where

[TABLE]

and $E_{\beta}(t)$ is the inverse $\beta$-stable subordinator.

Proof.

Since $E_{\beta}(t)\stackrel{{\scriptstyle d}}{{=}}t^{\beta}E_{\beta}(1)$ (see [26, Proposition 3.1]), we have

[TABLE]

The law of large numbers for the Poisson process implies

[TABLE]

Note that $D_{f}(t)\to\infty$, $a.s.$ as $t\to\infty$ (see [8, page 73]). Consider now,

[TABLE]

where the last step follows from (3.14). ∎

When $\beta=1$, the LIL for the time-changed Poisson process $\{Q_{1}^{f}(t)\}\stackrel{{\scriptstyle d}}{{=}}\{N_{1}(D_{f}(t))\},\leavevmode\nobreak\ t\geq 0,$ (discussed in [30]) can be proved in a similar way and is stated below.

Corollary 3.1**.**

Let the Laplace exponent $f(s)$ of the subordinator $\{D_{f}(t)\}_{t\geq 0}$ be regularly varying at 0+ with index $\alpha\in(0,1)$. Then

[TABLE]

where

[TABLE]

Example 3.4**.**

The space fractional Poisson process, introduced in [28], defined by time changing the Poisson process by an independent $\alpha$-stable subordinator, that is,

[TABLE]

where $\{D_{\alpha}(t)\}_{t\geq 0}$ is the $\alpha$-stable subordinator with LT $\mathbb{E}[e^{-sD_{\alpha}(t)}]=e^{-ts^{\alpha}}$. Here, the corresponding Bernstein function $f(s)=s^{\alpha}$ is regularly varying with index $\alpha\in(0,1)$. Therefore, by Corollary 3.1, we have the LIL for the space fractional Poisson process with

[TABLE]

4. Time-changed fractional Poisson process-II

The first-exit time of the subordinator $\{D_{f}(t)\}_{t\geq 0}$ is its right-continuous inverse, defined by

[TABLE]

and is called an inverse subordinator (see [8]). Note that for any $\rho>0$, $\mathbb{E}[E_{f}^{\rho}(t)]<\infty$ (see [1, Section 2.1]). We now consider the FPP time-changed by an inverse subordinator.

Definition 4.1** **(TCFPP-II).

The time-changed fractional Poisson process version two (TCFPP-II) is defined as

[TABLE]

where $\{N_{\beta}(t)\}_{t\geq 0}$ is the FPP and is independent of the inverse subordinator $\{E_{f}(t)\}_{t\geq 0}$.

We now present some results and distributional properties of the TCFPP-II. The proofs of some of them are shortened or omitted to avoid repetition from the previous section.

The one-dimensional distributions of the TCFPP-II can be written as

[TABLE]

which follows from (2.4).

Theorem 4.1**.**

Let $0<s\leq t$. Then

(i) $\leavevmode\nobreak\ \leavevmode\nobreak\ \mathbb{E}[W^{f}_{\beta}(t)]=q\mathbb{E}[E_{f}^{\beta}(t)]$,

(ii)$\leavevmode\nobreak\ \leavevmode\nobreak\ \text{Var}[W^{f}_{\beta}(t)]=q\mathbb{E}[E_{f}^{\beta}(t)]\left(1-q\mathbb{E}[E_{f}^{\beta}(t)]\right)+2d\mathbb{E}[E_{f}^{2\beta}(t)]$,

[TABLE]

Proof.

The proof is similar to the proof of Theorem 3.2 and hence is omitted. ∎

We next discuss the asymptotic behavior of moments of the TCFPP-II. The mean and variance functions contain the term of the form $\mathbb{E}[E^{\beta}_{f}(t)]$. Therefore, we study the asymptotic behavior of $\mathbb{E}[E^{\beta}_{f}(t)]$. It will be studied using the Tauberian theorem (see [34, Theorem 4.1] and [8, p. 10]), which we reproduce here. Recall that a function $\ell(t)$, $t>0$, is slowly varying at [math] (respectively $\infty$) if for all $c>0$, $\lim(\ell(ct)/\ell(t))=1$, as $t\rightarrow 0$ (respectively $t\rightarrow\infty$).

Theorem 4.2**.**

(Tauberian theorem) Let $\ell:(0,\infty)\rightarrow(0,\infty)$ be a slowly varying function at [math] (respectively $\infty$) and let $\rho\geq 0$. Then for a function $U:(0,\infty)\rightarrow(0,\infty)$, the following are equivalent:

(i) $U(x)\sim x^{\rho}\ell(x)/\Gamma(1+\rho),\quad x\rightarrow 0$ (respectively $x\rightarrow\infty$).

(ii) $\widetilde{U}(s)\sim s^{-\rho-1}\ell(1/s),\quad s\rightarrow\infty$ (respectively $s\rightarrow 0$),

where $\widetilde{U}(s)$ is the LT of $U(x)$.

Let $M_{p}(t)={\mathbb{E}}[E_{f}^{p}(t)],\leavevmode\nobreak\ p>0$. The LT of the $p$-th moment of $E_{f}(t)$ is given by (see [19])

[TABLE]

where $f(s)$ is the Bernstein function associated with $\{D_{f}(t)\}_{t\geq 0}$. The asymptotic moments can be specifically computed for special cases, which also serves examples of the TCFPP-II processes.

Example 4.1** (FPP subordinated with inverse gamma subordinator).**

We study the FPP time-changed by the inverse of the gamma subordinator $\{Y(t)\}_{t\geq 0}$, with corresponding Bernstein function $f(s)=p\log(1+s/\alpha)$. The right-continuous inverse of the gamma subordinator $\{Y(t)\}_{t\geq 0}$ is defined as

[TABLE]

We study the asymptotic behavior of the mean of $\{W_{\beta}^{(1)}(t)\}_{t\geq 0}=\{N_{\beta}(E_{Y}(t))\}_{t\geq 0}$, that is, the function $\mathbb{E}[W_{\beta}^{(1)}(t)]=q\mathbb{E}[E_{Y}^{\beta}(t)]$. The LT of $\mathbb{E}[E^{\beta}_{Y}(t)]$ is given by

[TABLE]

Note that $p\log(1+s/\alpha)\sim ps/\alpha,$ as $s\to 0$. Now using Theorem 4.2, we get (see also [20, Proposition 4.1])

[TABLE]

The asymptotic behavior of variance function of $\{W^{(1)}_{\beta}(t)\}$ can also be computed using above expression.

Example 4.2** (FPP subordinated with the inverse tempered $\alpha$-stable subordinator).**

Consider the FPP subordinated with the inverse tempered $\alpha$-stable subordinator $\{E^{\mu}_{\alpha}(t)\}_{t\geq 0}$. The inverse tempered $\alpha$-stable subordinator is introduced by [19] and they studied its asymptotic behavior of moments. The $p$-th moment of $E^{\mu}_{\alpha}(t)$ satisfies (see [19, Proposition 3.1])

[TABLE]

Therefore, we have that

[TABLE]

Example 4.3** (FPP subordinated with inverse of the inverse Gaussian subordinator).**

The right-continuous inverse of the inverse Gaussian subordinator $\{G(t)\}_{t\geq 0}$, with corresponding Bernstein function $f(s)=\delta\left(\sqrt{2s+\gamma^{2}}-\gamma\right)$, denoted by $\{E_{G}(t)\}_{t\geq 0}$, defined as (see [35])

[TABLE]

Hence from (4.2)

[TABLE]

Using above result and Theorem 4.2, we get

[TABLE]

We finally get the asymptotic moments as

[TABLE]

We next show that the TCFPP-II $\{W_{\beta}^{f}(t)\}_{t\geq 0}$ is a renewal process. We begin with the following lemma.

Let $\{D_{f}(t)\}_{t\geq 0}$ be a subordinator with the associated Bernstein function $f(s)$. Let $\{E_{f}(t)\}_{t\geq 0}$ be the right-continuous inverse of $\{D_{f}(t)\}_{t\geq 0}$. We call, rather loosely, $E_{f}(t)$ the inverse subordinator corresponding to $f(s)$.

Lemma 4.1**.**

Let $\{E_{f_{1}}(t)\}_{t\geq 0}$ and $\{E_{f_{2}}(t)\}_{t\geq 0}$ be two independent inverse subordinators corresponding to Bernstein functions $f_{1}(s)$ and $f_{2}(s)$, respectively. Then

[TABLE]

where $(f_{1}\circ f_{2})(s)=f_{1}(f_{2}(s)).$

Proof.

Consider two independent subordinators $\{D_{f_{1}}(t)\}_{t\geq 0}$ and $\{D_{f_{2}}(t)\}_{t\geq 0}$ with

[TABLE]

where $f_{1}(s)$ and $f_{2}(s)$ are the associated Bernstein functions. We claim that

[TABLE]

where $\circ$ denotes the composition of functions. To see this, let us compute the LT of the left-hand side

[TABLE]

Since $f(s)=(f_{1}\circ f_{2})(s)$ is again a Bernstein function (see [33, Remark 5.28 (ii)]) and $\{D_{f_{2}}(D_{f_{1}}(t))\}_{t\geq 0}$ is a Lévy process (see [3, Theorem 1.3.25]), it follows that $\{D_{f}(t)\}_{t\geq 0}$ is a subordinator with associated Bernstein function $f(s)=(f_{1}\circ f_{2})(s)$.

Consider next have the inverse subordinators defined by

[TABLE]

Then the process

[TABLE]

which completes the proof. ∎

Corollary 4.1**.**

Let $\{E_{\beta}(t)\}_{t\geq 0}$ be inverse $\beta$-stable subordinator corresponding to $f_{1}(s)=s^{\beta}$, and $\{E_{f}(t)\}_{t\geq 0}$ be an inverse subordinator corresponding to $f_{2}(s)=f(s)$. Then from (4.4),

[TABLE]

where $\phi(s)=(f(s))^{\beta}$.

Remark 4.1**.**

One can further generalize the TCFPP-I process $\{Q_{\beta}^{f}(t)\}_{t\geq 0}$ and TCFPP-II process $\{W_{\beta}^{f}(t)\}_{t\geq 0}$, by subordinating it again with a subordinator and an inverse subordinator, respectively. As it clearly shown in (4.5) and (4.4), the subordination of subordinator and inverse subordinator yields again a subordinator and an inverse subordinator, respectively. Hence, further subordination leads again to the processes of type TCFPP-I $\{Q_{\beta}^{f}(t)\}_{t\geq 0}$ and TCFPP-II $\{W_{\beta}^{f}(t)\}_{t\geq 0}$. This is also valid for $n$-iterated subordination.

Theorem 4.3**.**

The TCFPP-II $\{W_{\beta}^{f}(t)\}_{t\geq 0}$ is a renewal process with iid waiting times $\{J_{n}\}_{n\geq 1}$ with distribution

[TABLE]

where $E_{\phi}(t)$ is the inverse subordinator corresponding to $\phi(s)=(f(s))^{\beta}$.

Proof.

Using (1.1) and Corollary 4.1, we have

[TABLE]

where $\phi(s)=(f(s))^{\beta}.$ Therefore, the TCFPP-II $\{W_{\beta}^{f}(t)\}_{t\geq 0}$ is a Poisson process time-changed by an inverse subordinator $\{E_{\phi}(t)\}_{t\geq 0}$ corresponding to Bernstein function $\phi(s)=(f(s))^{\beta}.$ From [25, Theorem 4.1], we deduce that the time-changed Poisson process

[TABLE]

is a renewal process with iid waiting times $\{J_{n}\}_{n\geq 1}$ having the distribution (4.7). ∎

Remark 4.2**.**

By [25, Remark 5.4], the pmf $\eta_{\beta}^{f}(n|t,\lambda)$, given in (4.1), of the TCFPP-II $\{W_{\beta}^{f}(t)\}_{t\geq 0}$ satisfies

[TABLE]

in the mild sense, where $\phi(s)=\left(f(s)\right)^{\beta}$, $\psi_{\phi}(\cdot)$ is the Lévy measure associated to Bernstein function $\phi(s)$ and $H(x)=I(x\geq 0)$ is the Heaviside function.

We next present the bivariate distributions of the TCFPP-II, which generalizes a result by [29, Theorem 2.1]. Let $F(t)$ be the distribution function of the waiting time $J_{n}$ and $S_{n}=J_{1}+\ldots+J_{n}$ be the time of $n$th jump. Since $J_{n}$’s are iid, we have that $\mathbb{P}[S_{n}\leq t]=F^{\ast n}(t)$, where $F^{\ast n}(t)$ denotes the $n$-fold convolution of $F(t)$. For $n,k\geq 1$, define $\tau^{(k)}_{n}=S_{n+k}-S_{n}\stackrel{{\scriptstyle d}}{{=}}S_{k}$, where $\tau^{(k)}_{n}$ is the time elapsed between $n$-th and $(n+k)$-th jump. Clearly, $\mathbb{P}[\tau^{(1)}_{n}\in dt]=dF(t)$ and $\mathbb{P}[\tau^{(k)}_{n}\in dt]=dF^{\ast k}(t)$, for $k\geq 1$.

Theorem 4.4**.**

Let $0\leq s<t$ and $0\leq m\leq n$ be nonnegative integers. Let $\{E_{\phi}(t)\}_{t\geq 0}$ be the inverse subordinator corresponding to $\phi(s)=(f(s))^{\beta}.$ The TCFPP-II $\{W_{\beta}^{f}(t)\}_{t\geq 0}$ has the bivariate distributions given by,

[TABLE]

where $F(t)=1-\mathbb{E}[e^{-\lambda E_{\phi}(t)}]$ and $dF^{\ast n}(t)$ is the $n$-fold convolution of $dF(t),\leavevmode\nobreak\ n\geq 1$, with $dF^{*0}(t)=\delta_{0}(t)$, the Dirac delta function at zero.

Proof.

Case 1: When $n=m$, we have (see Figure 1)

[TABLE]

Case 2: When $n\geq m+1$, it follows that (see Figure 2)

[TABLE]

Since the waiting times between events are iid, we have that

[TABLE]

which completes the proof. ∎

Let us examine a special case of Theorem 4.4 for the FPP.

Remark 4.3**.**

It is known (see [25]) that the FPP $\{N_{\beta}(t)\}_{t\geq 0}=\{N(E_{\beta}(t))\}_{t\geq 0}$ is a renewal process whose inter-arrival times follow the Mittag-Leffler distribution, that is,

[TABLE]

where $L_{\beta}(z)$ is the Mittag-Leffler function defined in (2.1). Let us define $L_{\alpha,0}^{0}(-\lambda t^{\beta}):=t\delta_{0}(t)$, where $\delta_{0}(t)$ is the Dirac delta function at zero. This implies for $t\geq 0$ (see [29] and references therein),

[TABLE]

where $L_{\alpha,\beta}^{\gamma}(z)$ is the generalized Mittag-Leffler function defined in (2.2). The LT of the inverse $\beta$-stable subordinator is given by (see [9, eq. (16)])

[TABLE]

Using (4.8), (4.9) and Theorem 4.4, the bivariate distribution of the FPP, when $n=m\geq 0$, is

[TABLE]

For $n\geq m+1$,

[TABLE]

which coincides (2.9) of [29]. Indeed, it is shown in [29, eq. (2.6)] that

[TABLE]

When $\beta=1$, $L_{1,1}^{1}(x)=e^{x}$, and $L^{1}_{1,m}(x)=e^{x}/(m-1)!$ and

[TABLE]

the bivariate distribution of the Poisson process, as expected.

5. Simulation

In this section, we present simulated sample paths for some TCFPP-I and TCFPP-II processes. The sample paths for the FNBP, the FPP subordinated with tempered $\alpha$-stable subordinator (FPP-TSS) and the FPP subordinated with inverse Gaussian subordinator (FPP-IGN) are presented for a chosen set of parameters. The simulations of the corresponding TCFPP-II process of the FPP subordinated with inverse gamma subordinator (FPP-IG), the FPP subordinated with inverse tempered $\alpha$-stable subordinator (FPP-ITSS), and the FPP subordinated with inverse of inverse Gaussian subordinator (FPP-IIGN) are also given in this section. We first present the algorithm for simulation of the FPP.

Algorithm 1** **(Simulation of the FPP).

This algorithm (see [10]) gives the number of events $N_{\beta}(t),\leavevmode\nobreak\ 0<\beta<1$ of the FPP up to a fixed time $T$.

1. (a)

Fix the parameters $\lambda>0$ and $0<\beta<1$ for the FPP. 2. (b)

Set $n=0$ and $t=0.$ 3. (c)

Repeat while $t<T$

1. Generate three independent uniform random variables $U_{i}\sim U(0,1)$, $i=1,2,3$.
2. Compute (see [17])

[TABLE] 3. $t=t+dt$ and $n=n+1$. 4. (d)

Next $t$.

Then $n$ denotes the number of events $N_{\beta}(t)$ occurred up to time $T$.

We next present the algorithms for the simulation of the gamma subordinator, the tempered $\alpha$-stable subordinator and the inverse Gaussian subordinator. The generated sample paths from these algorithms will then be used to simulate the inverse subordinator and the TCFPP-I.

Algorithm 2** **(Simulation for the gamma subordinator).

2. (a)

Fix the parameters $\alpha$ and $p$ for gamma subordinator. 3. (b)

Choose an interval $[0,T].$ Choose $n+1$ uniformly spaced time points $0=t_{0},t_{1},\ldots,t_{n}=T$ with $h=t_{2}-t_{1}.$ 4. (c)

Simulate $n$ independent gamma random variables $Q_{i}\sim G(\alpha,ph),1\leq i\leq n$, using GSS algorithm (see [4, p. 321]). 5. (d)

The discretized sample path of $Y(t)$ at $t_{i}$ is $Y(ih)=Y(t_{i})=\sum_{j=1}^{i}Q_{j},1\leq i\leq n$ with $Q_{0}=0.$

Algorithm 3** **(Simulation for the TSS).

2. (a)

Choose the parameters $\mu>0$ and $0<\alpha<1$. 3. (b)

Choose an interval $[0,T].$ Choose $n+1$ time points $0=t_{0},t_{1},\ldots,t_{n}=T.$ 4. (c)

Simulate $D_{\alpha}^{\mu(t_{i}-t_{i-1})^{1/\alpha}}(1)$ for $1\leq i\leq n$ from the Algorithm 3.2 of [14]. 5. (d)

Compute the increments

[TABLE]

with $D^{\mu}_{\alpha}(0)=0.$ 6. (e)

The discretized sample path of $D^{\mu}_{\alpha}(t)$ at $t_{i}$ is $D^{\mu}_{\alpha}(t_{i})=\sum_{j=1}^{i}\Delta D_{\alpha}^{\mu{(j)}},1\leq i\leq n.$

Algorithm 4** **(Simulation of the IGN subordinator).

The algorithm to generate the IGN random variables is given in [11, p. 183].

1. (a)

Choose an interval $[0,T].$ Choose $n+1$ uniformly spaced time points $0=t_{0},t_{1},\ldots,t_{n}=T$ with $h=t_{2}-t_{1}.$ 2. (b)

Since IGN subordinator $\{G(t)\}_{t\geq 0}$ has independent and stationary increments, $F_{i}\equiv G(t_{i})-G(t_{i-1})\stackrel{{\scriptstyle d}}{{=}}G(h)\sim$ IGN$(h,1)$ for $1\leq i\leq n$ and $h=T/n$. Now generate $n$ iid IGN variables $F_{i}$’s as follows (see [11, p. 183], therein substituted $\delta=1=\gamma$):

1. Generate a standard normal random variable $N$.
2. Assign $X=N^{2}$.
3. Assign $Y=h+\frac{X}{2}-\frac{1}{2}\sqrt{4hX+X^{2}}$.
4. Generate a uniform $[0,1]$ random variable $U$.
5. If $U\leq\frac{h}{h+Y}$, return $Y$; else return $\frac{h^{2}}{Y}$.
6. (c)

Assign $G(t_{0})=0$. The discretized sample path of $G(t)$ at $t_{i}$ is $G(t_{i})=\sum_{j=1}^{i}F_{j},1\leq i\leq n.$

Consider next the algorithm to simulate the inverse subordinator $\{E_{f}(t)\}_{t\geq 0}$. We first define $E_{f}^{\delta}(t)$ with the step length $\delta$ as (see [20])

[TABLE]

where $D_{f}(\delta n)$ is the value of the subordinator $D_{f}(t)$ evaluated at $\delta nδ$, which can be simulated by using the method presented above. Observe that trajectory of $E_{f}^{\delta}(t)$ has increments of length $\delta$ at random time instants governed by process $D_{f}(t)$ and therefore $E_{f}^{\delta}(t)$ is the approximation of operational time.

Algorithm 5** **(Simulation of the inverse subordinator).

2. (a)

Fix the parameters for the inverse subordinator, whichever under consideration. 3. (b)

Choose $n$ uniformly spaced time points $0=t_{1},t_{2},\ldots,t_{n}=T$ with $h=t_{2}-t_{1}.$ 4. (c)

Let $i=0$ and $t=0$. 5. (d)

Repeat while $t<T$

1. Generate an independent $D_{f}(t)$ random variables with $Q_{i}\sim D_{f}(h).$
2. Set $W(\lceil t/h\rceil+1):=h*i,\ldots,W(\lfloor(t+Q_{i})/h\rfloor+1):=h*i$.
3. $i=i+1,\leavevmode\nobreak\ t=t+Q_{i}.$
4. Next $t$.
5. (e)

The discretized sample path of $E_{f}(t)$ at $t_{i}$ is $W_{i},1\leq i\leq n$ with $W_{0}=0.$

Note that the simulations for the inverse of gamma subordinator, the inverse of tempered $\alpha$-stable subordinator and the inverse of inverse Gaussian subordinator can be done using the above algorithm by replacing the special case for the subordinator.

We next present a general algorithm to simulate the TCFPP-I, namely the FNBP, the FPP-TSS and the FPP-IGN processes. The same algorithm can be used to simulate the TCFPP-II, namely the FPP-IG, the FPP-ITSS and the FPP-IIGN processes.

Algorithm 6** **(Simulation of the TCFPP-I and the TCFPP-II).

2. (a)

Fix the parameters for the subordinator (inverse subordinator), under consideration. Choose the fractional index $\beta\leavevmode\nobreak\ (0<\beta<1)$ and rate parameter $\lambda>0$ for the FPP. 3. (b)

Fix the time $T$ for the time interval $[0,T]$ and choose $n+1$ uniformly spaced time points $0=t_{0},t_{1},\ldots,t_{n}=T$ with $h=t_{2}-t_{1}$. 4. (c)

Simulate the values $X(t_{i}),1\leq i\leq n,$ of the subordinator (inverse subordinator) at $t_{1},\ldots t_{n},$ using the algorithm for respective subordinator (inverse subordinator). 5. (d)

Using the values $X(t_{i}),1\leq i\leq n,$ generated in Step (c), as time points, compute the number of events of the FPP $N_{\beta}(X(t_{i})),1\leq i\leq n,$ using Algorithm 1.

Acknowledgments

A part of this work was done while the second author was visiting the Department of Statistics and Probability, Michigan State University, during Summer-2016.