2. An extended class of GPPs and some preliminaries

The GPP was introduced as an extended model of the NHPP, and it is characterized in [Reference Cha8]. In this section, we consider an extended class of GPPs. We start our discussion with the definition of the GPP, which is based on the concept of the stochastic intensity. For a regular counting process $\{N(t),\ t \geq 0\}$ , the stochastic intensity is defined by the following ([Reference Aven and Jensen3, Reference Aven and Jensen4, Reference Finkelstein13]):

(1) \begin{eqnarray} \lambda_{t} \equiv \lim_{\Delta t\rightarrow 0} \frac{\mathbb{P}\left(N(t,t+\Delta t)=1|{\mathcal{H}}_{t-}\right)}{\Delta t} = \lim_{\Delta t\rightarrow 0} \frac{\mathbb{E}\left[N(t, t+\Delta t)|{\mathcal{H}}_{t-}\right]}{\Delta t},\end{eqnarray}

where ${\mathcal{H}}_{t-} \equiv \{N(u),\ 0\leq u<t\}$ is the history of the process in [0, t) and $N(t_1,t_2)$ , $t_1<t_2$ , stands for the number of events in $[t_1 , t_2)$ . See p. 1150 of [Reference Cha8] for the interpretation of the stochastic intensity in (1) and several examples of $\lambda_{t}$ . Note that, in a regular counting process, the joint distribution for the event occurrences in a time interval can be stated using the stochastic intensity in (1), which is essential in deriving various properties to be obtained in this paper (e.g., the distributions of number(s) of events, the characterization of the process as the mixture of other processes, the superposition of the processes, the characterization of the marginal processes in a multivariate process). Thus, the stochastic intensity plays a key role in this paper and is a measure of great importance.

As mentioned on p. 1150 of [Reference Cha8], the GPP with $\alpha=0$ and $\beta=1$ reduces to the NHPP having intensity function $\lambda(t)$ . In defining the GPP, except for this specific case with $\alpha=0$ , the parameter $\alpha$ is restricted to a positive real number, and because of this, the susceptibility to the event occurrence in the next time interval increases as the number of events occurring in the previous time interval increases. This is a kind of positive dependence property in the increments. In this paper, we consider an extended class of GPPs when the parameter $\alpha$ is a negative real value and, accordingly, the corresponding counting process model would exhibit a kind of negative dependence in the increments. However, as a counting process is not properly defined if the stochastic intensity can take negative value(s), we define the extended class of GPPs as follows, by setting the parameter $\beta = \alpha l_0$ with $l_0$ a fixed positive integer, not depending on t.

For convenience, the EGPP with the set of parameters $(\lambda(t),\alpha,l_0)$ will be denoted by EGPP $(\lambda(t),\alpha,l_0)$ , and the function $\lambda(t)$ will be called the ‘base intensity function’. According to Definition 2, when $N(t-\!)<l_0$ , the stochastic intensity of EGPP $(\lambda(t),\alpha,l_0)$ satisfies $\lambda_t>0$ (if the base intensity function $\lambda(t)$ is positive at t), whereas, when $N(t-\!)=l_0$ , it is given by $\lambda_t=0$ , which implies that additional events cannot occur anymore. Thus, the EGPP in Definition 2 has $\lambda_t \geq 0$ , $t \geq0$ , and the corresponding counting process is well defined. Obviously, in EGPP $(\lambda(t),\alpha,l_0)$ , $\lambda_t=0,\ t>T_{l_0}$ , where $T_{l_0}$ is the occurrence time of the $l_0$ th event, and no event occurs after the occurrence of the $l_0$ th event.

Remark 1. In Definition 2, one could consider a further extended model by replacing (ii) with

\begin{eqnarray*}\lambda_t=\max\{(\!-\alpha N(t-\!)+\beta)\lambda(t), 0\},\end{eqnarray*}

where $\beta$ is an arbitrary positive real value, not necessarily a multiple of $\alpha$ . For this model, to derive stochastic properties, basically the same procedures as those which will be applied to the model in Definition 2 can be used. However, the results cannot be stated in explicit forms; for this reason, the discussion in this paper will just be focused on the model in Definition 2.

As the GPP has been extended to a multivariate version in [Reference Cha and Giorgio9], the EGPP in Definition 2 can be generalized to a multivariate process. For consistency and convenience, we employ similar notation as that of [Reference Cha and Giorgio9]. Let $\{{\mathbf{N}}(t),\ t\geq0\}$ , where ${\mathbf{N}}(t)=(N_1(t),N_2(t), \cdots, N_m(t))$ , be a multivariate counting process, and let $\{M(t),\ t\geq0\}$ , where $M(t)=N_1(t)+N_2(t)+\cdots+N_m(t)$ , be the corresponding pooled process. A multivariate process is said to be regular if the pooled process is regular, whereas it is called a marginally regular process if its marginal processes, considered as univariate point processes, are all regular (see also [Reference Cha and Giorgio9]). In this paper, the multivariate process $\{{\mathbf{N}}(t),\ t\geq0\}$ of interest is assumed to be a regular process.

In the following discussion, for convenience, the marginal point processes are denoted by $\{N_i(t),\ t\geq0\}$ , and they will be called the type i point processes, $i=1,2,\cdots,m$ . In addition, the events occurring in the type i point process $\{N_i(t),\ t\geq0\}$ will be called type i events. Denote by ${\mathcal{H}}^{(P)}_{t-} \equiv \{M(u),\ 0\leq u<t\}$ the history of the pooled process in [0,t). Define $M(t-\!)$ as the total number of events in [0,t) and $T^{(P)}_{i}$ as the time from 0 until the arrival of the ith event in the pooled process $\{M(t),\ t\geq0\}$ . Similarly, define the marginal histories of the marginal processes ${\mathcal{H}}^{(i)}_{t-} \equiv \{N_i(u),\ 0\leq u<t\}$ , $i=1,2,\cdots,m$ , and the sequential arrival points of the events $0\leq T_{i1}\leq T_{i2} \leq \cdots \leq T_{iN_i(t-\!)}<t$ in [0,t), $i=1,2,\cdots,m$ , where $N_i(t-\!)$ is the total number of events of the type i point process in [0,t), $i=1,2,\cdots,m$ .

A multivariate counting process can be efficiently defined through the stochastic intensity approach (see [Reference Cha and Giorgio9, Reference Cox and Lewis10]). As mentioned in [Reference Cha and Giorgio9], a regular multivariate process can be specified by the following set of intensity functions (see also [Reference Cox and Lewis10]):

\begin{eqnarray*} &&\lambda^{(i)}_{t} \equiv \lim_{\Delta t\rightarrow 0} \frac{\mathbb{P}\left(N_i(t,t+\Delta t)=1|{\mathcal{H}}^{(1)}_{t-};\ {\mathcal{H}}^{(2)}_{t-};\cdots;{\mathcal{H}}^{(m)}_{t-}\right)}{\Delta t}, i=1,2,\cdots,m,\end{eqnarray*}

where $N_i(t_1,t_2)$ , $t_1<t_2$ , represents the number of events of the type i point process in $[t_1 , t_2)$ , $i=1,2\cdots,m$ .

Now the multivariate extended generalized Pólya process is defined based on the intensity functions.

The main properties of the MEGPP will be derived in the next section. For this, as preliminaries, we need some basic results on the univariate process EGPP, which will be discussed in the remaining part of this section. Before deriving the basic results on the EGPP, we will state an important property of the EGPP. To this end, we recall the definition of the ‘restarting property’ of a stochastic process.

Several examples of counting process models which possess this restarting property are given on p. 1151 of [Reference Cha8]. To state our results, let us now define some necessary notation. Denote by $0\leq T_1\leq T_2 \leq \cdots$ the sequential arrival points in $\{N(t),\ t\geq0\}$ . We also define $N^{(u)}(t)\equiv N(u+t)-N(u)$ , $u,t>0$ , and $\Lambda(t)\equiv \int_{0}^{t} \lambda(s)ds$ . From the definition of the EGPP in Definition 2, the following proposition clearly holds.

Now we will state the results on the distribution of the number(s) of events in the EGPP. The results in Theorem 1 and their proofs will form an important base for the derivations of the stochastic properties of the MEGPP. Throughout this paper, for convenience, we will use the following notation:

\begin{eqnarray*}\left(\begin{array}{c}{n }\\ \\[-7pt] {n_1,n_2,\cdots,n_k}\end{array}\right) \equiv \frac{n!}{n_1!n_2! \cdots n_k!},\end{eqnarray*}

where $n=\sum_{i=1}^{k}n_i$ , and

\begin{eqnarray*}n^{\underline{i}} \equiv n(n-1)(n-2)\cdots(n-i+1).\end{eqnarray*}

Theorem 1. Let $\{N(t),\ t \geq0\}$ be EGPP $(\lambda(t),\alpha,l_0)$ . For $t>0$ and $0\equiv u_0<u_1<u_2<\cdots<u_r$ , the following properties hold:

\begin{eqnarray*} &&(i)\ \mathbb{P}(N(t)=n)= \left(\begin{array}{c}{l_0 }\\ \\[-7pt] {n}\end{array}\right) \left( 1-\exp\{-\alpha \Lambda(t)\}\right)^n\left( \exp\{-\alpha\Lambda(t)\} \right)^{l_0-n}, \ n=0,1,\cdots,l_0. \\[4pt]&&(ii)\ \mathbb{P}(N(u_2)-N(u_1)=n) \\ &&=\left(\begin{array}{c}{l_0}\\ \\[-7pt] {n}\end{array}\right) \left( \exp\{-\alpha\Lambda(u_1)\}-\exp\{-\alpha\Lambda(u_2)\}\right)^{n} \\&&\,\,\quad\qquad \times\left( 1-\exp\{-\alpha \Lambda(u_1)\}+\exp\{-\alpha\Lambda(u_2)\}\right)^{l_0-n},\ n=0,1,\cdots,l_0. \\[4pt]&&(iii)\ \mathbb{P}(N(u_{i})-N(u_{i-1})=n_i, \ i=1,2,\cdots,r) \\ &&=\left(\begin{array}{c}{l_0 }\\ \\[-7pt] {n_1,n_2,\cdots,n_r,l_0-\sum_{k=1}^{r}n_k }\end{array}\right) \\&&\ \times\Bigg[\prod_{i=1}^{r}\Big( \exp\{-\alpha \Lambda(u_{i-1})\}-\exp\{-\alpha \Lambda(u_{i})\}\Big)^{n_i}\Bigg] \Big( \exp\{-\alpha \Lambda(u_r)\}\Big)^{l_0-\sum_{k=1}^{r}n_k},\\&&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad 0\leq n_1+n_2+\cdots+n_r \leq l_0.\end{eqnarray*}

\begin{eqnarray*}\\[-35pt]&&(iv)\ \mathbb{P}(N(u_2)-N(u_1)=n_2|N(u_1)=n_1)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \\&&=\left(\begin{array}{c}{l_0-n_1 }\\ \\[-7pt] {n_2}\end{array}\right) \left( 1-\exp\{-\alpha (\Lambda(u_2)-\Lambda(u_1))\}\right)^{n_2} \\&&\qquad\qquad\qquad\quad\, \times \left( \exp\{-\alpha(\Lambda(u_2)-\Lambda(u_1))\} \right)^{l_0-n_1-n_2}, \ n_2=0,1,\cdots,l_0-n_1.\end{eqnarray*}

A counting process is said to possess independent increments if the numbers of events that occur in disjoint time intervals are independent (see, e.g., Section 1.9, p. 42, of [Reference Ross23]). Clearly, it can be easily seen from Definition 2 that the EGPP does not have the independent increments property. Based on Theorem 1, the dependence property of the EGPP is now formally discussed. The following proposition states that the increments in any two disjoint time intervals have negative covariance and correlation coefficient.

Proposition 2. For $0 \leq u_1 <u_2 \leq u_3<u_4$ ,

\begin{eqnarray*}&&Cov(N(u_2)-N(u_1), N(u_4)-N(u_3))\\[3pt] &&=-l_0\prod_{i=2,4}\Big( \exp\{-\alpha \Lambda(u_{i-1})\}-\exp\{-\alpha \Lambda(u_{i})\}\Big)<0,\end{eqnarray*}

and

\begin{eqnarray*}&&Corr(N(u_2)-N(u_1), N(u_4)-N(u_3))\\[3pt] &&=-\sqrt {\frac {\prod_{i=2,4}\Big( \exp\{-\alpha \Lambda(u_{i-1})\}-\exp\{-\alpha \Lambda(u_{i})\}\Big)}{\prod_{i=2,4}\Big( 1-\exp\{-\alpha \Lambda(u_{i-1})\}+\exp\{-\alpha \Lambda(u_{i})\}\Big)}}<0.\end{eqnarray*}

Proof. First assume that $u_1>0$ . It can be shown from Theorem 1(iii) that

\begin{eqnarray*}&&\mathbb{P}(N(u_{i})-N(u_{i-1})=n_i,\ i=2,4) \\[3pt] &&=\frac{l_0!}{n_2!n_4!(l_0-n_2-n_4)!}\Bigg[\prod_{i=2,4}\Big( \exp\{-\alpha \Lambda(u_{i-1})\}-\exp\{-\alpha \Lambda(u_{i})\}\Big)^{n_i}\Bigg] \\[3pt] &&\quad \times\Big(1-\sum_{i=2,4} \Big( \exp\{-\alpha \Lambda(u_{i-1})\}-\exp\{-\alpha \Lambda(u_{i})\}\Big)\Big)^{l_0-n_2-n_4},\end{eqnarray*}

where $0\leq n_2+n_4 \leq l_0$ . Then, by the properties of the multinomial distribution (see, e.g., [Reference Johnson, Kots and Balakrishnan14], p. 34), the results can be directly obtained. In the case when $u_1=0$ , the results can be shown in the same way.

As mentioned in Proposition 1, for the EGPP, the restarting property holds conditionally as in the case of the GPP. On the other hand, for the GPP, the future process from an arbitrary time point u is ‘unconditionally’ the GPP as well (see Theorem 2 in [Reference Cha8], pp. 1152–1153). The importance of this unconditional restarting property is explained in [Reference Cha8], on pp. 1152–1153. Now, for the EGPP, we will check whether the future process from an arbitrary time point u is still unconditionally the EGPP or not. Employing similar notation as in [Reference Cha8] for consistency, for a fixed $u>0$ , define $T^{(u)}_{i}$ as the time from 0 to the arrival of the event that occurred in $(u,\infty)$ at the ith time, $u \leq T^{(u)}_{1}\leq T^{(u)}_{2} \leq \cdots$ . Denote by $N^{(u)}(t_1,t_2)$ , $t_1<t_2$ , the number of events in $[u+t_1,u+t_2)$ and by ${\mathcal{H}}_{[u,u+t)}$ the history of the process in $[u,u+t)$ . We now need to derive the stochastic intensity $\lambda_{t}^{u}$ of the future process $\{N^{(u)}(t),\ t\geq0\}$ , which is defined by

\begin{align*} \lambda_{t}^{u}\,{ \equiv} \lim_{\Delta t \rightarrow 0}\frac{\mathbb{P}\left(N^{(u)}(t,t+\Delta t)=1|{\mathcal{H}}_{[u,u+t)}\right)}{\Delta t} =\lim_{\Delta t \rightarrow 0} \frac{\mathbb{E}\left[N^{(u)}(t, t+\Delta t)|{\mathcal{H}}_{[u,u+t)}\right]}{\Delta t}. \end{align*}

Note that ${\mathcal{H}}_{[u,u+t)}$ can completely be defined by the number of events and the sequential arrival times during the interval $[u,u+t)$ . The following theorem implies that, ‘unconditionally’, the future process of the EGPP is also the EGPP.

Theorem 2. Let $\{N(t),\ t \geq0\}$ be EGPP $(\lambda(t),\alpha,l_0)$ . The stochastic intensity $\lambda_{t}^{u}$ is given by

\begin{eqnarray*}\lambda_{t}^{u}=\Big(\!-\alpha[N((u+t)-\!)-N(u-\!)]+\alpha l_0\Big)\frac{\lambda(u+t)\exp\{-\alpha\Lambda(u+t)\}}{1-\exp\{-\alpha\Lambda(u)\}+\exp\{-\alpha\Lambda(u+t)\}},\end{eqnarray*}

and $\{N^{(u)}(t),\ t\geq0\}$ is unconditionally EGPP with the set of parameters

\begin{eqnarray*}\left(\frac{\lambda(u+t)\exp\{-\alpha\Lambda(u+t)\}}{1-\exp\{-\alpha\Lambda(u)\}+\exp\{-\alpha\Lambda(u+t)\}},\alpha, l_0 \right).\end{eqnarray*}

Now the conditional distribution of the arrival times in the EGPP will be obtained. As a more general result, we consider that of the arrival times in an ‘arbitrary time interval’ (u, v], $0\leq u<v$ :

\begin{eqnarray*} \Big(T^{(u)}_{1}, T^{(u)}_{2},\cdots,T^{(u)}_{(N(v)-N(u))}|N(v)-N(u)=n\Big). \end{eqnarray*}

Theorem 3. Let $\{N(t),\ t \geq0\}$ be EGPP $(\lambda(t),\alpha,l_0)$ . The conditional joint probability density function of

\begin{eqnarray*} \Big(T^{(u)}_{1}, T^{(u)}_{2},\cdots,T^{(u)}_{(N(v)-N(u))}|N(v)-N(u)=n\Big), \qquad 0\leq u<v, \end{eqnarray*}

is given by

\begin{eqnarray*} &&n!\prod_{i=1}^{n} \left(\frac{\alpha\lambda\Big(t^{(u)}_{i}\Big)\exp\Big\{-\alpha\Lambda\Big(t^{(u)}_{i}\Big)\Big\}}{\exp\{-\alpha \Lambda(u)\}-\exp\{-\alpha\Lambda(v)\}}\right),\end{eqnarray*}

where $u< t^{(u)}_{1}\leq t^{(u)}_{2}\leq \cdots \leq t^{(u)}_{n} \leq v$ .

3. Main properties of the MEGPP

The following proposition provides some basic properties of the MEGPP, to help with structural understanding of the process. The results in Proposition 3 play crucial roles in the analysis of the MEGPP. For convenience and consistency, we define the necessary notation as follows: $N^{(u)}_{i}(t)\equiv N_i(u+t)-N_i(u)$ , $\Lambda_i(t) \equiv \int_{0}^{t}\lambda_i(s)ds$ , $i=1,2,\cdots,m$ , $\lambda(t)\equiv \lambda_1(t)+\lambda_2(t)+\cdots+\lambda_m(t)$ , $\Lambda(t)=\int_{0}^{t}\lambda(s)ds=\Lambda_1(t)+\Lambda_2(t)+\cdots+\Lambda_m(t)$ , and $p_i(t)\equiv \lambda_i(t)/\lambda(t)$ , $i=1,2,\cdots,m$ . We also use the concept of thinning of a univariate process, from Definition 4 of [Reference Cha and Giorgio9].

Now we will derive the main stochastic properties of the MEGPP. First of all, the joint distribution of the numbers of events is of prime interest to us.

Proof. First, let us prove Property (i). We have

\begin{eqnarray*}&&\mathbb{P}(N_i(t)=n_i,\ i=1,2,\cdots,m)\\&&=\mathbb{P}\Bigg(N_i(t)=n_i,\ i=1,2,\cdots,m|M(t)=\sum_{i=1}^{m}n_i\Bigg)\mathbb{P}\Bigg(M(t)=\sum_{i=1}^{m}n_i\Bigg),\end{eqnarray*}

where, by Proposition 3(i) and Theorem 1,

\begin{eqnarray*}\mathbb{P}\Bigg(M(t)=\sum_{i=1}^{m}n_i\Bigg)=\left(\begin{array}{c}{l_0 }\\ \\[-7pt] {\sum_{i=1}^{m}n_i}\end{array}\right) \left( 1-\exp\{-\alpha \Lambda(t)\}\right)^{\sum_{i=1}^{m}n_i}\left( \exp\{-\alpha\Lambda(t)\} \right)^{l_0-\sum_{i=1}^{m}n_i}.\end{eqnarray*}

From Theorem 3, given $M(t)=\sum_{i=1}^{m}n_i$ , the sequential arrival times

\begin{equation*}0\le T^{(P)}_{1}\le T^{(P)}_{2}\le \cdots \le T^{(P)}_{\sum_{i=1}^{m}n_i}\leq t\end{equation*}

have the following joint probability density function:

\begin{eqnarray*}&&\Bigg(\sum_{i=1}^{m}n_i\Bigg)!\prod_{i=1}^{\sum_{i=1}^{m}n_i} \Bigg(\frac{\alpha\lambda(t_{i})\exp\{-\alpha\Lambda(t_{i})\}}{1-\exp\{-\alpha\Lambda(t)\}}\Bigg).\end{eqnarray*}

Thus they have the same distribution as the order statistics corresponding to $\sum_{i=1}^{m}n_i$ independent random variables identically distributed according to the probability density function

\begin{eqnarray*} \Bigg(\frac{\alpha\lambda(x)\exp\{-\alpha\Lambda(x)\}}{1-\exp\{-\alpha \Lambda(t)\}}\Bigg), \qquad 0<x\leq t. \end{eqnarray*}

Thus, if an event has occurred in (0,t], the probability that it will be a type i event is given by

\begin{eqnarray*}\psi_i(t)\equiv \Bigg(\frac{\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx}{1-\exp\{-\alpha \Lambda(t)\}}\Bigg), \qquad i=1,2,\cdots,m.\end{eqnarray*}

Thus,

\begin{eqnarray*}\mathbb{P}\Bigg(N_i(t)=n_i,\ i=1,2,\cdots,m|M(t)=\sum_{i=1}^{m}n_i\Bigg)=\left(\begin{array}{c}{\sum_{i=1}^{m}n_i }\\ \\[-7pt] {n_1,n_2,\cdots,n_m}\end{array}\right) \prod_{i=1}^{m} \Big(\psi_i(t)\Big)^{n_i}.\end{eqnarray*}

Therefore,

\begin{align*}\mathbb{P}(N_i(t)=n_i,\ i=1,2,\cdots,m)&=\left(\begin{array}{c}{l_0 }\\ \\[-7pt] {n_1,n_2,\cdots,n_m,l_0-\sum_{i=1}^{m}n_i }\end{array}\right) \\[3pt]&\quad \times \prod_{i=1}^{m}\Bigg(\!\alpha\!\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx\!\Bigg)^{n_i}\Big(\!\exp\{-\alpha\Lambda(t)\}\!\Big)^{l_0-\sum_{i=1}^{m}n_i}.\end{align*}

By using Proposition 3(iv) and the above Property (i), Property (ii) can be directly obtained.

Now we show Property (iii). By Proposition 3(iii), given $N_i(u_{1})-N_i(u_{0})=n_{i1},\ i=1,2,\cdots,m$ , we have that $\{{\mathbf{N}}^{(u_1)}(t),\ t\geq0\}$ is MEGPP $(\lambda_1(u_1+t), \lambda_2(u_1+t),\cdots, \lambda_m(u_1+t),\alpha,l_0-\sum_{i=1}^{m}n_{i1})$ . Then, relying on the above Property (i),

\begin{eqnarray*}&&\mathbb{P}(N_i(u_{2})-N_i(u_{1})=n_{i2},\ i=1,2,\cdots,m|N_i(u_{1})-N_i(u_{0})=n_{i1},\ i=1,2,\cdots,m)\\[3pt]&&=\left(\begin{array}{c}{l_0-\sum_{i=1}^{m}n_{i1} }\\ \\[-7pt] {n_{12},n_{22},\cdots, n_{m2}, l_0-\sum_{i=1}^{m}\sum_{j=1}^{2}n_{ij}}\end{array}\right) \\[3pt] &&\qquad \times \prod_{i=1}^{m} \Bigg(\alpha\int_{u_1}^{u_2}\lambda_i(x)\exp\{-\alpha[\Lambda(x)-\Lambda(u_1)]\}dx\Bigg)^{n_{i2}}\\[3pt] &&\qquad\qquad\qquad\qquad\quad\quad \times \Big(\exp\{-\alpha[\Lambda(u_2)-\Lambda(u_1)]\}\Big)^{l_0-\sum_{i=1}^{m}\sum_{j=1}^{2}n_{ij}}.\end{eqnarray*}

Thus,

\begin{eqnarray*}&&\mathbb{P}(N_i(u_{j})-N_i(u_{j-1})=n_{ij},\ i=1,2,\cdots,m, j=1,2) \\[3pt]&&=\prod_{k=1}^{2} \Bigg[ \left(\begin{array}{c}{ l_0-\sum_{j=1}^{k-1}\Big(\sum_{i=1}^{m}n_{ij}\Big) }\\ \\[-7pt] {n_{1k}, n_{2k}, \cdots, n_{mk}, l_0-\sum_{j=1}^{k}\Big(\sum_{i=1}^{m}n_{ij}\Big) }\end{array}\right) \\[3pt] &&\quad\quad\quad \times \prod_{i=1}^{m} \Bigg(\alpha\int_{u_{k-1}}^{u_{k}}\lambda_i(x)\exp\{-\alpha[\Lambda(x)-\Lambda(u_{k-1})]\}dx\Bigg)^{n_{ik}} \\[3pt]&&\qquad\qquad\quad \times \Big(\exp\{-\alpha[\Lambda(u_k)-\Lambda(u_{k-1})]\}\Big)^{l_0-\sum_{i=1}^{m}\sum_{j=1}^{k}n_{ij}}\Bigg],\end{eqnarray*}

where $\sum_{j=1}^{k-1} ({\cdot})\equiv 0$ when $k=1$ . By applying a similar procedure recursively, we obtain the desired result.

Note that the results in Theorems 1–4 hold whether $\Lambda(\infty)\equiv \int_{0}^{\infty}\lambda(s)ds=\infty$ or $\Lambda(\infty)<\infty$ . In the EGPP, if $\Lambda(\infty)<\infty$ , then $N(\infty)\equiv \lim_{t \rightarrow \infty} N(t)<l_0$ with positive probability. Thus, from the pure-death-process point of view stated in Remark 2, the process may end incompletely with positive probability. In this sense, if $\Lambda(\infty)<\infty$ , then the corresponding EGPP (MEGPP) will be said to be incomplete, whereas an EGPP (MEGPP) with $\Lambda(\infty)=\infty$ will be called a complete EGPP (MEGPP). In the following theorem, it will be shown that an incomplete EGPP (MEGPP) can be characterized as a mixture of complete EGPPs (MEGPPs). Throughout this paper, $X\sim{\mbox{Bin}}(n,p)$ signifies that the random variable X follows the binomial distribution with parameters (n,p). In the following discussion, MEGPP $(\lambda_1(t),\lambda_2(t), \cdots, \lambda_m(t),\alpha,l_0)$ with $l_0=0$ represents a counting process whose counting random variables are given by 0 with probability 1 for all $t\geq 0$ .

Theorem 5. Let $\{{\mathbf{N}}(t),\ t\geq0\}$ be MEGPP $(\lambda_1(t),\lambda_2(t), \cdots, \lambda_m(t),\alpha,l_0)$ with $\Lambda(\infty)<\infty$ . Let L be a binomial random variable: $L\sim {\mbox{Bin}} (l_0, 1-\exp\{-\alpha \Lambda(\infty)\})$ . For $t>0$ , let

\begin{eqnarray*}\phi_i(t) \equiv \frac{\lambda_i(t)\exp\{-\alpha \Lambda(t)\}}{\exp\{-\alpha \Lambda(t)\}-\exp\{-\alpha \Lambda(\infty)\}}, \qquad i=1,2,\cdots,m.\end{eqnarray*}

If $\{{\mathbf{N}}^*(t),\ t\geq0\}$ satisfies $\{{\mathbf{N}}^*(t),\ t\geq0\}|L \sim {\mbox{MEGPP}} (\phi_1(t), \phi_2(t),\cdots, \phi_m(t), \alpha, L)$ , then it is stochastically equivalent to $\{{\mathbf{N}}(t),\ t\geq0\}$ .

From Proposition 3(i), the properties of the EGPP can be directly obtained from those of the MEGPP, as the pooled process of MEGPP $(\lambda_1(t), \lambda_2(t), \cdots, \lambda_m(t),\alpha,l_0)$ is EGPP $(\lambda(t),\alpha,l_0)$ . Thus, from Theorem 5, we have the following corollary.

Corollary 1. Let $\{N(t),\ t \geq0\}$ be EGPP $(\lambda(t),\alpha,l_0)$ with $\Lambda(\infty)<\infty$ . Let L be a binomial random variable: $L\sim {\mbox{Bin}} (l_0, 1-\exp\{-\alpha \Lambda(\infty)\})$ . For $t>0$ , let

\begin{eqnarray*}\phi(t) \equiv \frac{\lambda(t)\exp\{-\alpha \Lambda(t)\}}{\exp\{-\alpha \Lambda(t)\}-\exp\{-\alpha \Lambda(\infty)\}}, \qquad t\geq 0.\end{eqnarray*}

If $\{N^*(t),\ t\geq0\}$ satisfies $\{N^*(t),\ t\geq0\}|L \sim {\mbox{EGPP}} (\phi(t), \alpha, L)$ , then it is stochastically equivalent to $\{N(t),\ t\geq0\}$ .

Note that, in Corollary 1, if $\Lambda(\infty)<\infty$ , then

\begin{eqnarray*}\frac{\lambda(t)\exp\{-\alpha \Lambda(t)\}}{\exp\{-\alpha \Lambda(t)\}-\exp\{-\alpha \Lambda(\infty)\}} >\lambda(t), \qquad \forall t >0.\end{eqnarray*}

Thus, roughly reinterpreting Corollary 1, an incomplete EGPP with the base intensity function $\lambda(t)$ can be characterized as a mixture of complete EGPPs with a larger base intensity function. A natural question in a more general sense could be the following: Is the mixture representation unique? Or can an incomplete EGPP be also characterized as the mixture of another group of EGPPs with a larger base intensity function? The following theorem provides the answer to this question.

Theorem 6. Assume that $\Lambda(\infty)<\infty$ . Let $\{{\mathbf{N}}(t),\ t\geq0\}$ be a regular multivariate counting process and let L be a binomial random variable: $L\sim {\mbox{Bin}} (l_0, p)$ , where $l_0 \geq 1$ and $1-\exp\{-\alpha\Lambda(\infty)\}<p<1$ . For $t>0$ , let

\begin{eqnarray*}\phi_i(t) \equiv \frac{\lambda_i(t)\exp\{-\alpha \Lambda(t)\}}{\exp\{-\alpha \Lambda(t)\}-(1-p)}, \qquad i=1,2,\cdots,m.\end{eqnarray*}

If $\{{\mathbf{N}}(t),\ t\geq0\}$ satisfies $\{{\mathbf{N}}(t),\ t\geq0\}|L \sim {\mbox{MEGPP}} (\phi_1(t),\phi_2(t), \cdots, \phi_m(t), \alpha, L)$ , then it is MEGPP $(\lambda_1(t),\lambda_2(t),\cdots, \lambda_m(t), \alpha,l_0)$ .

Proof. Similarly to the proof of Theorem 5, it is sufficient to show the property for the univariate process. By the condition that $1-\exp\{-\alpha\Lambda(\infty)\}<p<1$ , it holds that $\phi(t)\equiv \sum_{i=1}^{m} \phi_i(t)\geq 0$ for all $t \geq 0$ . Then, following similar procedures to those described in the proof of Theorem 5, it can be shown that the stochastic intensity $\lambda_t$ of the univariate process $\{N(t),\ t \geq0\}$ is given by

\begin{eqnarray*}\lambda_t=(\!-\alpha N(t-\!)+\alpha l_0)\lambda(t),\end{eqnarray*}

which implies that $\{N(t),\ t \geq0\}$ is EGPP $(\lambda(t), \alpha, l_0)$ .

From Theorem 6, we can obtain the following corollary. In the remaining part of this paper, other properties of the EGPP can also be derived from those of the MEGPP in similar ways, and we do not state them.

Corollary 2. Assume that $\Lambda(\infty)<\infty$ . Let $\{N(t),\ t\geq0\}$ be a regular counting process and let L be a binomial random variable: $L\sim {\mbox{Bin}} (l_0, p)$ , where $l_0 \geq 1$ and $1-\exp\{-\alpha\Lambda(\infty)\}<p<1$ . For $t>0$ , let

\begin{eqnarray*}\phi(t) \equiv \frac{\lambda(t)\exp\{-\alpha \Lambda(t)\}}{\exp\{-\alpha \Lambda(t)\}-(1-p)}.\end{eqnarray*}

If $\{N(t),\ t\geq0\}$ satisfies $\{N(t),\ t\geq0\}|L \sim {\mbox{EGPP}} (\phi(t),\alpha, L)$ , then it is EGPP $(\lambda(t),\alpha,l_0)$ .

Note that, in Corollary 2, $\Phi(\infty)\equiv \int_{0}^{\infty}\phi(s)ds<\infty$ . Thus, in Corollary 2, an incomplete EGPP has been characterized as the mixture of another group of incomplete EGPPs with a larger base intensity function. Corollary 2 (Theorem 6) improves understanding of the ‘hierarchical structure’ of an incomplete EGPP (MEGPP). By Corollary 2, if $\Lambda(\infty)<\infty$ , then EGPP $(\lambda(t),\alpha,l_0)$ can be represented as the mixture of the baseline counting process

(2) \begin{eqnarray} \mbox{EGPP}\bigg(\frac{\lambda(t)\exp\{-\alpha \Lambda(t)\}}{\exp\{-\alpha \Lambda(t)\}-(1-p_1)},\alpha,l_1\bigg), \quad \mbox{where}\ p_1> 1-\exp\{-\alpha\Lambda(\infty)\},\end{eqnarray}

with the mixing distribution $L_1 \sim {\mbox{Bin}} (l_0, p_1)$ . Again by Corollary 2, the baseline counting process in (2) for each $l_1=1,\cdots,l_0$ can also be represented as the mixture of the baseline counting process

\begin{eqnarray*}\mbox{EGPP}\bigg(\frac{\lambda(t)\exp\{-\alpha \Lambda(t)\}}{\exp\{-\alpha \Lambda(t)\}-(1-p_1p_2)},\alpha,l_2\bigg), \quad \mbox{where}\ p_1p_2> 1-\exp\{-\alpha\Lambda(\infty)\},\end{eqnarray*}

with the mixing distribution $L_2 \sim{\mbox{Bin}} (l_1, p_2)$ . This separation procedure can be performed repeatedly. Note that, at each separation stage, the third parameter of the baseline EGPP can be $1,2,\cdots,l_0$ . Thus, if $p_1,p_2,\cdots$ are appropriately determined, it can be seen that any incomplete EGPP is separable into the corresponding baseline counting process and the mixing distribution for an infinite number of times. For example, choose $p^*$ and $n_0$ so that $1-\exp\{-\alpha\Lambda(\infty)\}<p^*<1$ and $p^*+1/(1+n_0)<1$ . Furthermore, determine $p_i,$ $i=1,2,\cdots$ , as follows:

\begin{eqnarray*}p_1 = p^*+\frac{1}{1+n_0},\qquad p_i=\frac{p^*+1/(i+n_0)}{p^*+1/((i-1)+n_0)}, \qquad i=2,3,\cdots.\end{eqnarray*}

Then $\prod_{i=1}^{m}p_i =p^*+1/(m+n_0)> 1-\exp\{-\alpha\Lambda(\infty)\}$ , for $m=1,2,\cdots$ , and the separation procedure can be performed for an infinite number of times. In this case, the baseline counting process at the mth stage is given by

\begin{eqnarray*}\mbox{EGPP}\bigg(\frac{\lambda(t)\exp\{-\alpha \Lambda(t)\}}{\exp\{-\alpha \Lambda(t)\}-\big(1-\prod_{i=1}^{m}p_i\big)},\alpha,l_m\bigg),\end{eqnarray*}

and the base intensity function becomes larger and larger as the number of iterations increases; the ‘upper bound function’ for the base intensity function is given by

\begin{eqnarray*}\phi^*(t) \equiv \frac{\lambda(t)\exp\{-\alpha \Lambda(t)\}}{\exp\{-\alpha \Lambda(t)\}-\exp\{-\alpha \Lambda(\infty)\}}, \qquad t\geq 0,\end{eqnarray*}

which is the base intensity function in Corollary 1. In point process theory, a point process is said to be infinitely divisible if, for every k, it can be represented as the superposition of k independent, identically distributed point process components (see, e.g., [Reference Daley and Vere-Jones11], p. 255). In a similar sense, the property stated in Theorem 6 (Corollary 2) can be called the infinitely separable property of the incomplete MEGPP (EGPP).

4. Further characterization and properties of the MEGPP

In this section, we will discuss the superposition of MEGPPs, the dependence property, the conditional distribution of arrival times, and the marginal processes in the MEGPP. The following theorem characterizes the MEGPP as a superposition of another group of MEGPPs.

Theorem 7. Let $\{{\tilde {\mathbf{N}}_i}(t),\ t \geq0\}$ , where ${\tilde {\mathbf{N}}_i}(t)=(\tilde {N}_{i1}(t),\tilde {N}_{i2}(t), \cdots, \tilde {N}_{im}(t))$ , be MEGPP with the set of parameters $(\lambda_1(t), \lambda_2(t),\cdots, \lambda_m(t), \alpha, l_i)$ , $i=1,2,\cdots,r$ . Suppose that $\{{\tilde {\mathbf{N}}_i}(t),\ t \geq0\}$ , $i=1,2,\cdots,r$ , are independent. Then $\{{\tilde {\mathbf{N}}}(t),\ t \geq0\}$ , where

\begin{equation*}{\tilde {\mathbf{N}}}(t)=\left(\sum_{i=1}^{r}\tilde {N}_{i1}(t),\sum_{i=1}^{r}\tilde {N}_{i2}(t), \cdots, \sum_{i=1}^{r}\tilde {N}_{im}(t)\right),\end{equation*}

is MEGPP $(\lambda_1(t),\lambda_2(t),\cdots, \lambda_m(t), \alpha, \sum_{i=1}^{r}l_i)$ .

Proof. Due to Proposition 3(ii), a MEGPP can be constructed by the thinning of an EGPP. Thus, for the proof of Theorem 7, it is sufficient to show the result for the univariate process. Thus, letting $\{{\tilde N_i}(t),\ t \geq0\}$ be independent EGPP $(\lambda(t), \alpha, l_i)$ , $i=1,2,\cdots,r$ , we will show that $\{{\tilde N}(t),\ t \geq0\}$ , where ${\tilde N}(t)\equiv \sum_{i=1}^{r}{\tilde N_i}(t)$ , is EGPP $\Big(\lambda(t), \alpha, \sum_{i=1}^{r}l_i\Big)$ . Denote by $\tilde{\lambda}_{t}$ and $\tilde{\lambda}^{(i)}_{t}$ the stochastic intensities of $\{{\tilde N}(t),\ t \geq0\}$ and $\{{\tilde N_i}(t),\ t \geq0\}$ , $i=1,2,\cdots,r$ , respectively, and by ${\tilde{\mathcal{H}}}_{t-}$ and ${\tilde{\mathcal{H}}}^{(i)}_{t-}$ , $i=1,2,\cdots,r$ , the corresponding histories. Then

\begin{eqnarray*}&&\sum_{i=1}^{r}\tilde{\lambda}^{(i)}_{t}=\lim_{\Delta t\rightarrow 0}\sum_{i=1}^{r} \frac{\mathbb{P}\left({\tilde{N_i}}(t,t+\Delta t)=1|{\tilde{\mathcal{H}}}^{(j)}_{t-},\ j=1,2,\cdots,r\right)}{\Delta t}\\&=&\lim_{\Delta t\rightarrow 0} \frac{\mathbb{P}\left(\bigcup_{i=1}^{r}\{{\tilde{N_i}}(t,t+\Delta t)=1\}|{\tilde{\mathcal{H}}}^{(j)}_{t-},\ j=1,2,\cdots,r\right)}{\Delta t}\\&=& \lim_{\Delta t\rightarrow 0} \frac{\mathbb{P}\left(\sum_{i=1}^{r}{\tilde{N_i}}(t,t+\Delta t)=1|{\tilde{\mathcal{H}}}^{(j)}_{t-},\ j=1,2,\cdots,r\right)}{\Delta t}\\&=& \lim_{\Delta t\rightarrow 0} \frac{\mathbb{P}\left({\tilde{N}}(t,t+\Delta t)=1|{\tilde{\mathcal{H}}}_{t-}\right)}{\Delta t}=\tilde{\lambda}_{t},\end{eqnarray*}

and $\tilde{\lambda}^{(i)}_{t}=(\!-\alpha {\tilde{N_i}}(t-\!)+\alpha l_i)\lambda(t)$ , $i=1,2,\cdots,r$ . Therefore,

\begin{align*}\tilde{\lambda}_{t}=\Bigg(\!-\alpha \sum_{i=1}^{r}{\tilde{N_i}}(t-\!)+\alpha \Bigg(\sum_{i=1}^{r}l_i\Bigg)\Bigg)\lambda(t)=\Bigg(\!-\alpha {\tilde{N}}(t-\!)+\alpha \Bigg(\sum_{i=1}^{r}l_i\Bigg)\Bigg)\lambda(t).\\[-45pt] \end{align*}

Now we will discuss the dependence property of the MEGPP. The following proposition states that there is a negative dependency between the increments in two different types of processes in any two (not necessarily disjoint) time intervals.

Proposition 4. Let $\{{\mathbf{N}}(t),\ t\geq0\}$ be MEGPP $(\lambda_1(t), \lambda_2(t),\cdots, \lambda_m(t), \alpha,l_0)$ . For any $t_{12}>t_{11}$ and $t_{22}>t_{21}$ , and for any $1\leq k_1 \ne k_2\leq m$ ,

\begin{eqnarray*}&&Cov(N_{k_1}(t_{12})-N_{k_1}(t_{11}), N_{k_2}(t_{22})-N_{k_2}(t_{21}))\\&&=-l_0\prod_{i=1,2}\Bigg( \alpha\int_{t_{i1}}^{t_{i2}}\lambda_{k_i}(x)\exp\{-\alpha\Lambda(x)\}dx\Bigg)<0,\end{eqnarray*}

and

\begin{eqnarray*}&&Corr(N_{k_1}(t_{12})-N_{k_1}(t_{11}), N_{k_2}(t_{22})-N_{k_2}(t_{21}))\\[3pt] &&=-\sqrt {\frac {\prod_{i=1,2}\Big( \alpha\int_{t_{i1}}^{t_{i2}}\lambda_{k_i}(x)\exp\{-\alpha\Lambda(x)\}dx\Big)}{\prod_{i=1,2}\Big(1- \alpha\int_{t_{i1}}^{t_{i2}}\lambda_{k_i}(x)\exp\{-\alpha\Lambda(x)\}dx\Big)}}<0.\end{eqnarray*}

Proof. For any sequence of the elements in $(t_{11}, t_{12}, t_{21}, t_{22})$ satisfying $t_{12}>t_{11}$ and $t_{22}>t_{21}$ , from Theorem 4(iii), it can be shown that

\begin{eqnarray*}&&\mathbb{P}(N_{k_1}(t_{12})-N_{k_1}(t_{11})=n_1, N_{k_2}(t_{22})-N_{k_2}(t_{21})=n_2) \\[3pt] &&=\left(\begin{array}{c}{l_0 }\\ \\[-7pt] {n_1, n_2, l_0-n_1-n_2}\end{array}\right) \Bigg[\prod_{i=1,2}\Bigg( \alpha\int_{t_{i1}}^{t_{i2}}\lambda_{k_i}(x)\exp\{-\alpha\Lambda(x)\}dx\Bigg)^{n_i}\Bigg] \\[3pt]&&\quad \times\Bigg(1-\sum_{i=1,2}\Bigg( \alpha\int_{t_{i1}}^{t_{i2}}\lambda_{k_i}(x)\exp\{-\alpha\Lambda(x)\}dx\Bigg)\Bigg)^{l_0-n_1-n_2},\end{eqnarray*}

where $0\leq n_1+n_2 \leq l_0$ . Then, according to the properties of the multinomial distribution (see also [Reference Johnson, Kots and Balakrishnan14], p. 34), the results can be directly obtained.

We will now derive the conditional joint distribution of arrival times of events given that $N_i(t)=n_i$ , $i=1,2, \cdots,m$ .

Theorem 8. Let $\{{\mathbf{N}}(t),\ t\geq0\}$ be MEGPP $(\lambda_1(t), \lambda_2(t),\cdots, \lambda_m(t), \alpha,l_0)$ . The conditional joint arrival times distribution of

\begin{eqnarray*}(T_{i1},T_{i2},\cdots,T_{in_i}, i=1,2,\cdots,m |N_i(t)=n_i, i=1,2,\cdots,m)\end{eqnarray*}

is given by

\begin{eqnarray*}\prod_{i=1}^{m} \Bigg[ n_i! \prod_{j=1}^{n_i}\Bigg(\frac{\lambda_i(t_{ij})\exp\{-\alpha \Lambda(t_{ij})\}}{\int_{0}^{t}\lambda_i(x)\exp\{-\alpha \Lambda(x)\}dx}\Bigg)\Bigg],\end{eqnarray*}

$0<t_{i1}<\cdots<t_{in_i}\leq t$ , $i=1,2,\cdots,m$ .

Proof. Following a similar procedure to the proof of Theorem 1 and using Proposition 3(ii), it can be shown that the joint distribution of

\begin{eqnarray*}(T_{i1},T_{i2},\cdots,T_{in_i}, N_i(t)=n_i, i=1,2,\cdots,m)\end{eqnarray*}

is given by

(3) \begin{eqnarray} &&(l_0)^{\underline{\sum_{i=1}^{m}n_i}}\cdot \prod_{i=1}^{m} \left(\prod_{j=1}^{n_{i}} \alpha p_i(t_{ij})\lambda(t_{ij})\exp\{-\alpha\Lambda(t_{ij})\}\right) \cdot \exp\Bigg\{-\Bigg(l_0-\sum_{i=1}^{m}n_i\Bigg)\alpha\Lambda(t)\Bigg\} \nonumber \\&&=(l_0)^{\underline{\sum_{i=1}^{m}n_i}}\cdot \prod_{i=1}^{m} \left(\prod_{j=1}^{n_{i}} \alpha \lambda_i(t_{ij})\exp\{-\alpha\Lambda(t_{ij})\}\right) \cdot \Big(\exp\{-\alpha\Lambda(t)\}\Big)^{l_0-\sum_{i=1}^{m}n_i},\end{eqnarray}

for any sequence of elements $(t_{i1},\cdots,t_{in_i},i=1,2,\cdots,m)$ satisfying $0<t_{i1}<\cdots<t_{in_i}\leq t$ , $i=1,2,\cdots,m$ . On the other hand, from Theorem 4,

(4) \begin{eqnarray} &&\mathbb{P}(N_i(t)=n_i,\ i=1,2,\cdots,m)=\left(\begin{array}{c}{l_0 }\\ \\[-7pt] {n_1,n_2,\cdots,n_m,l_0-\sum_{i=1}^{m}n_i } \end{array}\right) \nonumber \\&& \qquad \times \prod_{i=1}^{m}\Bigg(\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx\Bigg)^{n_i}\Big(\exp\{-\alpha\Lambda(t)\}\Big)^{l_0-\sum_{i=1}^{m}n_i}.\end{eqnarray}

By combining (3) and (4), we have the desired result.

Now the marginal process of the MEGPP will be characterized.

Theorem 9. Let $\{{\mathbf{N}}(t),\ t\geq0\}$ be MEGPP $(\lambda_1(t), \lambda_2(t), \cdots,\lambda_m(t), \alpha,l_0)$ . The marginal process $\{N_i(t),\ t\geq 0\}$ is EGPP $({\tilde \lambda}_{i}(t), \alpha, l_0)$ , where

\begin{eqnarray*}{\tilde \lambda}_i(t)= \frac{\lambda_i(t)\exp\{-\alpha \Lambda(t)\}}{1-\alpha \int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx}, \qquad i=1,2,\cdots,m.\end{eqnarray*}

Proof. For the proof, the stochastic intensity functions of the marginal processes $\{N_i(t),\ t\geq0\}$ , $i=1,2, \cdots,m$ , of MEGPP $(\lambda_1(t), \lambda_2(t), \cdots,\lambda_m(t), \alpha,l_0)$ will be derived. Denote by $\nu^{(i)}_{t}$ the stochastic intensity of $\{N_i(t),\ t\geq0\}$ , $i=1,2, \cdots,m$ . Then $\nu^{(1)}_{t}$ can be specified as

\begin{align*}\nu^{(1)}_{t} &\equiv \lim_{\Delta t\rightarrow 0} \frac{\mathbb{P}\left(N_1(t,t+\Delta t)=1|{\mathcal{H}}^{(1)}_{t-}\right)}{\Delta t}\\[2pt]&= \mathbb{E}_{\big({\mathcal{H}}^{(2)}_{t-}, {\mathcal{H}}^{(3)}_{t-},\cdots, {\mathcal{H}}^{(m)}_{t-}|{\mathcal{H}}^{(1)}_{t-}\big)}\Bigg[\lim_{\Delta t\rightarrow 0} \frac{\mathbb{P}\left(N_1(t,t+\Delta t)=1|{\mathcal{H}}^{(1)}_{t-}, {\mathcal{H}}^{(2)}_{t-}, \cdots, {\mathcal{H}}^{(m)}_{t-}\right)}{\Delta t}\Bigg],\end{align*}

where, from Definition 3,

\begin{eqnarray*}\lim_{\Delta t\rightarrow 0} \frac{\mathbb{P}\left(N_1(t,t+\Delta t)=1|{\mathcal{H}}^{(1)}_{t-}, {\mathcal{H}}^{(2)}_{t-}, \cdots, {\mathcal{H}}^{(m)}_{t-}\right)}{\Delta t}=\Bigg(\!-\alpha\Bigg(\sum_{i=1}^{m}N_i(t-\!)\Bigg)+\alpha l_0\Bigg)\lambda_1(t).\end{eqnarray*}

Thus, $\nu^{(1)}_{t}$ is obtained by

\begin{eqnarray*}\nu^{(1)}_{t}=\Bigg(\!-\alpha \Bigg(N_1(t-\!)+\sum_{i=2}^{m}\mathbb{E}\Big[N_i(t-\!)| {\mathcal{H}}^{(1)}_{t-}\Big] \Bigg)+\alpha l_0\Bigg)\lambda_1(t).\end{eqnarray*}

We will now derive the conditional distribution of $\big(N_2(t-\!)| {\mathcal{H}}^{(1)}_{t-}\big)$ ; the other conditional distributions of $\big(N_i(t-\!)| {\mathcal{H}}^{(1)}_{t-}\big)$ , $i=3,4,\cdots,m$ , can be obtained similarly. From (3), the joint distribution of $(T_{11},T_{12},\cdots,T_{1N_1(t-\!)}, N_1(t-\!), N_2(t-\!))$ is obtained by

\begin{eqnarray*}&&\frac{l_0!}{n_2!(l_0-n_1-n_2)!}\cdot \left(\prod_{j=1}^{n_{1}} \alpha \lambda_1(t_{1j})\exp\{-\alpha\Lambda(t_{1j})\}\right) \cdot \Bigg(\int_0^t \alpha \lambda_2(x)\exp\{-\alpha\Lambda(x)\}dx\Bigg)^{n_2} \\&&\ \times \Bigg(\sum_{i=3}^{m}\int_0^t \alpha \lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx+\exp\{-\alpha\Lambda(t)\}\Bigg)^{l_0-n_1-n_2}.\end{eqnarray*}

Then, by similar arguments as those described in the proof of Theorem 2, it can be shown that the conditional distribution of $( N_2(t-\!)|T_{11},T_{12},\cdots,T_{1N_1(t-\!)}, N_1(t-\!))$ does not depend on $(T_{11},T_{12},\cdots,T_{1N_1(t-\!)})$ , and accordingly,

\begin{eqnarray*}( N_2(t-\!)|T_{11},T_{12},\cdots,T_{1N_1(t-\!)}, N_1(t-\!)) {\buildrel D \over =} ( N_2(t-\!)| N_1(t-\!)),\end{eqnarray*}

where ‘ ${\buildrel D \over =}$ ’ stands for equality in distribution. From Theorem 4,

\begin{eqnarray*}&&\mathbb{P}(N_1(t)=n_1)=\left(\begin{array}{c}{l_0 }\\ \\[-7pt] {n_1} \end{array}\right) \Bigg(\alpha\int_{0}^{t}\lambda_1(x)\exp\{-\alpha\Lambda(x)\}dx\Bigg)^{n_1}\\ &&\ \times \Bigg(\sum_{i=2}^{m}\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx+\exp\{-\alpha\Lambda(t)\}\Bigg)^{l_0-n_1},\end{eqnarray*}

\begin{eqnarray*}&&\mathbb{P}(N_1(t)=n_1, N_2(t)=n_2)=\left(\begin{array}{c}{l_0 }\\ \\[-7pt] {n_1, n_2, l_0-n_1-n_2} \end{array}\right) \Bigg(\alpha\int_{0}^{t}\lambda_1(x)\exp\{-\alpha\Lambda(x)\}dx\Bigg)^{n_1}\\ &&\ \times \Bigg(\alpha\int_{0}^{t}\lambda_2(x)\exp\{-\alpha\Lambda(x)\}dx\Bigg)^{n_2}\Bigg(\sum_{i=3}^{m}\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx+\exp\{-\alpha\Lambda(t)\}\Bigg)^{l_0-n_1-n_2},\end{eqnarray*}

and thus

\begin{align*}\mathbb{P}(N_2(t)&=n_2|N_1(t)=n_1)=\left(\begin{array}{c}{l_0-n_1 }\\ \\[-7pt] {n_2}\end{array}\right) \\[3pt]&\quad \times\Bigg(\frac{\alpha\int_{0}^{t}\lambda_2(x)\exp\{-\alpha\Lambda(x)\}dx}{\sum_{i=2}^{m}\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx+\exp\{-\alpha\Lambda(t)\}} \Bigg)^{n_2}\\[3pt]&\qquad\qquad\qquad \times\Bigg(\frac{\sum_{i=3}^{m}\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx+\exp\{-\alpha\Lambda(t)\}}{\sum_{i=2}^{m}\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx+\exp\{-\alpha\Lambda(t)\}} \Bigg)^{l_0-n_1-n_2},\end{align*}

$n_2=0,1,\cdots,l_0-n_1$ . Therefore,

\begin{align*}\mathbb{E}[N_2(t-\!)| {\mathcal{H}}^{(1)}_{t-}] &=\mathbb{E}[N_2(t-\!)| N_1(t-\!)]\\[3pt]&=(l_0-N_1(t-\!))\frac{\alpha\int_{0}^{t}\lambda_2(x)\exp\{-\alpha\Lambda(x)\}dx}{\sum_{i=2}^{m}\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx+\exp\{-\alpha\Lambda(t)\}},\end{align*}

and, symmetrically,

\begin{align*}\mathbb{E}\big[N_i(t-\!)| {\mathcal{H}}^{(1)}_{t-}\big] &=\mathbb{E}[N_i(t-\!)| N_1(t-\!)]\\&=(l_0-N_1(t-\!))\frac{\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx}{\sum_{i=2}^{m}\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx+\exp\{-\alpha\Lambda(t)\}},\end{align*}

$i=3,4,\cdots,m$ . Therefore,

\begin{align*}\nu^{(1)}_{t}&=\Bigg(\!-\alpha \Bigg(N_1(t-\!)+\sum_{i=2}^{m}\mathbb{E}\Big[N_i(t-\!)| {\mathcal{H}}^{(1)}_{t-}\Big] \Bigg)+\alpha l_0\Bigg)\lambda_1(t)\\&= \Bigg(\!-\alpha \Bigg(N_1(t-\!)+\sum_{i=2}^{m}\mathbb{E}\Big[N_i(t-\!)| N_1(t-\!)\Big] \Bigg)+\alpha l_0\Bigg)\lambda_1(t)\\&=(\!-\alpha N_1(t-\!)+\alpha l_0) \frac{\lambda_1(t)\exp\{-\alpha\Lambda(t)\}}{\sum_{i=2}^{m}\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx+\exp\{-\alpha\Lambda(t)\}}.\end{align*}

By noting that

(5) \begin{eqnarray} &&\alpha\int_{0}^{t}\lambda_1(x)\exp\{-\alpha\Lambda(x)\}dx+\sum_{i=2}^{m}\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx \nonumber \\&&=\int_{0}^{t}\alpha\lambda(x)\exp\{-\alpha\Lambda(x)\}dx=1-\exp\{-\alpha\Lambda(t)\},\end{eqnarray}

we have the desired result. The stochastic intensity $\nu^{(i)}_{t}$ , $i=2,3,\cdots,m$ , can be obtained symmetrically.

From the relationship in (5),

\begin{eqnarray*}{\tilde \lambda}_1(t)= \frac{\lambda_1(t)\exp\{-\alpha \Lambda(t)\}}{\sum_{i=2}^{m}\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx+\exp\{-\alpha\Lambda(t)\}},\end{eqnarray*}

and

\begin{eqnarray*}\int_{0}^{t}{\tilde \lambda}_1(x)dx=-\frac{1}{\alpha} \ln\Bigg(\sum_{i=2}^{m}\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx+\exp\{-\alpha\Lambda(t)\}\Bigg).\end{eqnarray*}

As

\begin{eqnarray*}\alpha\int_{0}^{t}\lambda_1(x)\exp\{-\alpha\Lambda(x)\}dx+\sum_{i=2}^{m}\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx+\exp\{-\alpha\Lambda(t)\}=1,\qquad \forall\ t >0,\end{eqnarray*}

we have

\begin{eqnarray*}0<\sum_{i=2}^{m}\alpha\int_{0}^{\infty}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx+\exp\{-\alpha\Lambda(\infty)\} < 1,\end{eqnarray*}

whether $\Lambda(\infty)=\infty$ or $\Lambda(\infty)<\infty$ . This implies that the marginal process $\{N_i(t),\ t\geq 0\}$ of MEGPP $(\lambda_1(t), \lambda_2(t),\cdots,\lambda_m(t), \alpha,l_0)$ is an incomplete EGPP. Thus, applying Corollary 1, it can be characterized as the mixture of the complete EGPPs.

Proposition 5. Let $\{N_i(t),\ t\geq 0\}$ , $i=1,2,\cdots,m$ , be the marginal processes of MEGPP $(\lambda_1(t), \lambda_2(t),\cdots,\lambda_m(t), \alpha,l_0)$ . Let $L_i$ be a binomial random variable: $L_i\sim {\mbox{Bin}} \Big(l_0, \alpha\int_{0}^{\infty}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx\Big),\ i=1,2,\cdots,m$ . For $t>0$ , let

\begin{eqnarray*}\phi_i(t) \equiv \frac{\lambda_i(t)\exp\{-\alpha\Lambda(t)\}} {\alpha\int_{t}^{\infty}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx}.\end{eqnarray*}

If $\{N_i^*(t),\ t\geq0\}$ satisfies $\{N_i^*(t),\ t\geq0\}|L_i \sim {\mbox{EGPP}} (\phi_i(t), \alpha, L_i)$ , then it is stochastically equivalent to $\{N_i(t),\ t\geq0\}$ , $i=1,2,\cdots,m$ .

Proof. Observe that ${\tilde \Lambda_i}(t)\equiv \int_{0}^{t}{\tilde \lambda}_i(s)ds$ is given by

\begin{eqnarray*}{\tilde \Lambda}_i(t)=-\frac{1}{\alpha}\ln\Bigg(1-\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx \Bigg),\end{eqnarray*}

and

\begin{eqnarray*}\exp\{-\alpha {\tilde \Lambda}_i(t)\}=1-\alpha\int_{0}^{t}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx.\end{eqnarray*}

Furthermore, we have

\begin{eqnarray*}\exp\{-\alpha {\tilde \Lambda}_i(\infty)\}=1-\alpha\int_{0}^{\infty}\lambda_i(x)\exp\{-\alpha\Lambda(x)\}dx.\end{eqnarray*}

Then, by applying Corollary 1, we have the desired result.

5. Some remarks for reliability applications

In reliability, various repair types have been developed based on different counting process models, and they have been applied to maintenance modeling and optimization. See [Reference Sherif and Smith24], [Reference Valdez-Flores and Feldman26], and [Reference Wang27] for surveys on various practical maintenance models. More theoretical and sophisticated models have also been suggested in the literature (see, e.g., [Reference Mi20, Reference Aven2, Reference Aven and Jensen4, Reference Badía, Berrade and Campos5, Reference Badía, Berrade and Campos6, Reference Finkelstein12, Reference Arriaza, Navarro and Suárez-Llorens1, Reference Navarro, Arriaza and Suárez-Llorens21]).

The two important basic repair types which have been most frequently applied in reliability are ‘perfect’ and ‘minimal’ repairs. By a perfect repair, the system is returned to a state that is as good as new, whereas by a minimal repair, the system is returned to a state that is as bad as old (see [Reference Finkelstein13] and Section 5 of [Reference Cha8]). The failure process of a repairable system following the perfect repair process is modeled by the renewal process, and that following the minimal repair process is modeled by the NHPP.

As mentioned in Section 5 of [Reference Cha8], a counting process model specifies the corresponding repair type. The GPP can define a new repair type which is worse than minimal repair (see Section 5 of [Reference Cha8] and [Reference Lee and Cha17]), in the sense that the state of the system after the repair is worse than that just before the failure. On the other hand, if we define the counting process $\{N(t),\ t\geq 0\}$ as the failure process of a repairable system, where N(t) represents the total number of failures of the system by time t, the stochastic intensity model of the EGPP in Definition 2 implies that the state of the system after the repair is better than that just before the failure, and, in this sense, the EGPP can be used to model a repair process which is better than the minimal repair process. More specifically, if we assume that the failure process $\{N(t),\ t\geq 0\}$ follows EGPP $(\lambda(t),\alpha,l_0)$ , then this implies that, on each system failure, a repair which is better than the minimal repair is applied until $(l_0-1)$ th failure. Note that, as mentioned earlier, in the EGPP, no event occurs after the occurrence of the $l_0$ th event, whereas, in practice, the failure of the system recurs without limit. Thus, from the $l_0$ th failure, the minimal repair process can be employed. From the counting-process point of view, the whole failure process in this case can be viewed as the combination of an EGPP and an NHPP, where the NHPP starts at a random time $T_{l_0}$ which is the occurrence time of the $l_0$ th event in EGPP $(\lambda(t),\alpha,l_0)$ . Thus, under this type of repair process, before the $l_0$ th failure of the system, the system state after the repair is better than after a minimal repair, and from $l_0$ th failure, the system is minimally repaired. Interpreting this type of repair process from a practical point of view, in the initial stage of system operation, the reliability state of the system after a repair can be improved compared with the state just before the failure, but, after this initial stage, repairs cannot improve the reliability of the system, only restore the system to the as-bad-as-old condition. Employing this type of repair process, various maintenance models could be developed.