The Dual Processes of a Class of Downwardly Skip-Free Processes

Abstract

We show that a class of downwardly skip-free processes can be regarded as the Siegmund dual of upwardly skip-free processes, which have been extensively studied in the literature. For such downwardly skip-free processes, using the duality method and existing results on upwardly skip-free processes, we provide explicit expressions for the extinction probability, the mean occupation time, the mean explosion time, and the mean extinction time. Additionally, we provide two examples to illustrate the main contributions of this work.

1. Introduction

Consider a continuous-time homogeneous Markov chain on ${\mathbb{Z}}_{+}:=\{0,1,2,\dots \}$, with the transition rate matrix $Q=(q_{ij}:i,j\in {\mathbb{Z}}_{+})$. Following Chen [1], we call the Markov chain a downwardly skip-free process (or single death process), if Q satisfies $q_{i,i-1}>0$ for all $i\ge 1$, and $q_{i,i-j}=0$ for all $i\ge j\ge 2$. We call the Markov chain an upwardly skip-free process (or single birth process), if Q satisfies $q_{i,i+1}>0$, $q_{i,i+j}=0$ for all $i\in {\mathbb{Z}}_{+}$ and $j\ge 2$. In this short note, we are only interested in totally stable transition rate matrices, which means that ${\sum }_{j\ne i}{q}_{ij}\le -q_{ii}<\infty$ for all $i\in {\mathbb{Z}}_{+}$.

Downwardly skip-free processes, as an important class of Markov chains, include several models such as birth–death processes and branching processes. Based on the explicit representation of the solution to the Poisson equation, various criteria for classical problems such as uniqueness, recurrence, ergodicity, exponential ergodicity, and strong ergodicity have been established for downwardly skip-free processes in [2,3,4]. However, it appears that some valuable quantities, including mean occupation time and mean explosion time, cannot be obtained using this method. Therefore, the main purpose of this note is to explore downwardly skip-free processes using a duality method, which does not rely on the representation of the solution to the Poisson equation. This duality, now known as Siegmund duality, was introduced in Siegmund [5] for Markov chains and aims to relate the entrance law of the original chain to the exit laws of its dual chain.

In this note, we observe that a downwardly skip-free process is the Siegmund dual of an upwardly skip-free process, which has been extensively studied; see, e.g., [1,2,6]. By applying the duality method, we can transform problems related to downwardly skip-free processes into those of upwardly skip-free processes. Therefore, for a class of downwardly skip-free processes, we rederive some known results and establish new quantities that have not been addressed in the literature.

Definition 1.

Let $P\left(t\right)=(p_{ij}\left(t\right):i,j\in {\mathbb{Z}}_{+})$ be a transition function. If there exists another transition function $\tilde{P}\left(t\right)=({\tilde{p}}_{ij}\left(t\right):i,j\in {\mathbb{Z}}_{+})$ satisfying

$$\begin{array}{c}\hfill \sum _{k=0}^j{\tilde{p}}_{ik}\left(t\right)=\sum _{k=i}^{\infty }{p}_{jk}\left(t\right),\forall i,j\in {\mathbb{Z}}_{+},\forall t\ge 0,\end{array}$$

(1)

then we call $\tilde{P}\left(t\right)$ the Siegmund dual of $P\left(t\right)$.

In the rest of this paper, we denote the transition rate matrix of a downwardly skip-free process by $Q=(q_{ij}:i,j\in {\mathbb{Z}}_{+})$ and the corresponding minimal process by $P\left(t\right)=(p_{ij}:i,j\in {\mathbb{Z}}_{+})$. To guarantee the existence of duality, we impose the following two conditions:

Condition 1.

$$\sum _{k=0}^j{\tilde{q}}_{ik}\ge \sum _{k=0}^j{\tilde{q}}_{i+1,k},j\ne i.$$

Condition 2.

${\sum }_{n=0}^{\infty }{\overline{m}}_n=\infty$, where $c_k=-{\tilde{q}}_{kk}-{\sum }_{j\ne k}{\tilde{q}}_{kj}$,

$${\overline{m}}_0=0,and{\overline{m}}_n={\displaystyle \frac{1}{{\tilde{q}}_{n,n-1}}}\left(1+\sum _{k=0}^{n-1}(c_k+\sum _{j=n}^{\infty }{\tilde{q}}_{kj}){\overline{m}}_k\right),n\ge 1.$$

Remark 1.

Since $c_k\ge 0$ for $k\in {\mathbb{Z}}_{+}$ and ${\sum }_{j=n}^{\infty }{\tilde{q}}_{kj}\ge 0$ for $0\le k\le n-1$, we conclude that ${\overline{m}}_n\ge {\left({\tilde{q}}_{n,n-1}\right)}^{-1}$ for $n\ge 1$. Therefore, if $\tilde{Q}$ satisfies

$$\sum _{n=1}^{\infty }{\displaystyle \frac{1}{{\tilde{q}}_{n,n-1}}}=\infty ,$$

then Condition 2 holds.

Recall that a transition function $P\left(t\right)$ is called honest if ${\sum }_{j=0}^{\infty }{p}_{ij}\left(t\right)=1$ for $i\in {\mathbb{Z}}_{+}$ and a transition rate matrix Q is called conservative if ${\sum }_{j\ne i}^{\infty }{q}_{ij}=-q_{ii}$ for $i\in {\mathbb{Z}}_{+}$. We say $P\left(t\right)$ is stochastically monotone, if ${\sum }_{j=k}^{\infty }{p}_{ij}\left(t\right)$ is a non-decreasing function of i for fixed $k\in {\mathbb{Z}}_{+}$ and $t\ge 0$. Our first main result establishes a sufficient condition for the existence of the duality between upwardly skip-free processes and downwardly skip-free processes.

Theorem 1.

Suppose that the downwardly skip-free transition rate matrix $\tilde{Q}$ satisfies Conditions 1 and 2. Then,

(i) There exists a stochastically monotone upwardly skip-free transition function $P\left(t\right)=(p_{ij}\left(t\right):i,j\in {\mathbb{Z}}_{+})$ with transition rate matrix Q defined by

$$\begin{array}{c}\hfill q_{ji}=\sum _{k=0}^j({\tilde{q}}_{ik}-{\tilde{q}}_{i+1,k}),\forall i,j\in {\mathbb{Z}}_{+},\end{array}$$

(2)

such that $\tilde{P}\left(t\right)$ is the Siegmund dual of $P\left(t\right)$.

(ii) If ${\tilde{q}}_{00}=0$, then Q is conservative, and $P\left(t\right)$ is honest.

Let ${\left({\tilde{X}}_t\right)}_{t\ge 0}$ be a downwardly skip-free process on ${\mathbb{Z}}_{+}$, with the transition rate matrix $\tilde{Q}$. Define the extinction time ${\tilde{\sigma }}_0:=inf\{t\ge 0:{\tilde{X}}_t=0\}$, the occupation time ${\tilde{T}}_j:={\int }_0^{\infty }{\mathbf{1}}_{\{{\tilde{X}}_t=j\}}dt$ for $j\ge 1$, and the explosion time $\tilde{\zeta }:={lim}_{n\to \infty }inf\{t\ge 0:{\tilde{X}}_t\ge n\}$.

When the downwardly skip-free process $\tilde{P}\left(t\right)$ is the Siegmund dual of the upwardly skip-free process $P\left(t\right)$ with transition rate matrix Q, we derive explicit expressions for several quantities related to the downwardly skip-free process using Q. To this end, we define $q_n^{\left(k\right)}={\sum }_{j=0}^k{q}_{nj}$ for $0\le k<n$ and

$$\begin{array}{c}\hfill m_0={\displaystyle \frac{1}{q_{01}}},m_n={\displaystyle \frac{1}{q_{n,n+1}}}\left(1+\sum _{k=0}^{n-1}{q}_n^{\left(k\right)}{m}_k\right),n\ge 1,\end{array}$$

(3)

$$\begin{array}{c}\hfill F_i^{\left(i\right)}=1,F_n^{\left(i\right)}={\displaystyle \frac{1}{q_{n,n+1}}}\sum _{k=i}^{n-1}{q}_n^{\left(k\right)}{F}_k^{\left(i\right)},0\le i<n.\end{array}$$

(4)

Theorem 2.

Suppose that the downwardly skip-free transition rate matrix $\tilde{Q}$ satisfies Condition 1, Condition 2, and ${\tilde{q}}_{00}=0$. Then:

(i) $\tilde{P}\left(t\right)$ is not strongly ergodic.

(ii) $\tilde{P}\left(t\right)$ is honest if and only if $P\left(t\right)$ is not strongly ergodic.

(iii) For any initial state $i\ge 1$, the extinction probability of the downwardly skip-free process satisfies ${\mathbf{P}}_i({\tilde{\sigma }}_0<\infty )<1$ if and only if $P\left(t\right)$ is ergodic. Indeed, the extinction probability is given by

$$\begin{array}{c}\hfill {\mathbf{P}}_i(\widetilde{{\sigma }_0}<\infty )=1-\sum _{k=0}^{i-1}{\pi }_k,i\ge 1,\end{array}$$

(5)

where ${\pi }_j:={lim}_{t\to \infty }{p}_{ij}\left(t\right)$ for $i,j\in {\mathbb{Z}}_{+}$.

(iv) The mean occupation time of the downwardly skip-free process is given by

$$\begin{array}{c}\hfill {\mathbf{E}}_i{\tilde{T}}_j=\sum _{k=0}^{i\wedge j-1}{\displaystyle \frac{F_{j-1}^{\left(k\right)}}{q_{k,k+1}}}-m_{j-1}\sum _{k=0}^{i-1}{\pi }_k,i\ge 1,j\ge 1.\end{array}$$

(6)

Moreover, if the extinction probability of the downwardly skip-free process is 1, then

$$\begin{array}{c}\hfill {\mathbf{E}}_i{\tilde{T}}_j=\sum _{k=0}^{i\wedge j-1}{\displaystyle \frac{F_{j-1}^{\left(k\right)}}{q_{k,k+1}}},i\ge 1,j\ge 1,\end{array}$$

(7)

where $i\wedge j:=min\{i,j\}$ for $i,j\in {\mathbb{Z}}_{+}$.

(v) For the downwardly skip-free process, the conditional expectation of the explosion time is given by

$$\begin{array}{c}\hfill {\mathbf{E}}_i\left(\tilde{\zeta }\right|{\tilde{\sigma }}_0=\infty )=\sum _{j=1}^{\infty }{\displaystyle \frac{{\sum }_{k=0}^{j-1}{\pi }_k}{{\sum }_{k=0}^{i-1}{\pi }_k}}\left(\sum _{k=0}^{i\wedge j-1}{\displaystyle \frac{F_{j-1}^{\left(k\right)}}{q_{k,k+1}}}-m_{j-1}\sum _{k=0}^{i-1}{\pi }_k\right),i\ge 1.\end{array}$$

(8)

(vi) For the downwardly skip-free process, the conditional expectation of the extinction time is given by

$$\begin{array}{c}\hfill {\mathbf{E}}_i\left(\widetilde{{\sigma }_0}\right|{\tilde{\sigma }}_0<\infty )=\sum _{j=1}^{\infty }{\displaystyle \frac{1-{\sum }_{k=0}^{j-1}{\pi }_k}{1-{\sum }_{k=0}^{i-1}{\pi }_k}}\left(\sum _{k=0}^{i\wedge j-1}{\displaystyle \frac{F_{j-1}^{\left(k\right)}}{q_{k,k+1}}}-m_{j-1}\sum _{k=0}^{i-1}{\pi }_k\right),i\ge 1.\end{array}$$

(9)

Moreover, if the extinction probability of the downwardly skip-free process is 1, then

$$\begin{array}{c}\hfill {\mathbf{E}}_i{\tilde{\sigma }}_0=\sum _{j=1}^{\infty }\sum _{k=0}^{i\wedge j-1}{\displaystyle \frac{F_{j-1}^{\left(k\right)}}{q_{k,k+1}}},i\ge 1.\end{array}$$

(10)

The paper is organized as follows: In Section 2, we recall some known properties of the Siegmund dual and then prove Theorem 1. In Section 3, we establish Theorem 2. Finally, in Section 4, we present two examples of downwardly skip-free processes to illustrate our main results.

2. Siegmund Dual

The following proposition is taken from Chen and Zhang [7], and it establishes a necessary and sufficient condition for the existence of Siegmund duality. This result is of independent interest, as it does not require the assumption that $P\left(t\right)$ is an upwardly skip-free process.

Proposition 1

([7]). The transition function $\tilde{P}\left(t\right)$ is the Siegmund dual of a stochastically monotone transition function $P\left(t\right)$ if and only if the following two conditions hold:

(i)

${\sum }_{k=0}^j{\tilde{p}}_{ik}\left(t\right)$ is non-increasing in i for each $j\in {\mathbb{Z}}_{+}$ and $t\ge 0$;

(ii)

${lim}_{i\to \infty }{\tilde{p}}_{ij}\left(t\right)=0$ for any $j\in {\mathbb{Z}}_{+}$ and $t\ge 0$.

Proof of Proposition 1.

Suppose that $\tilde{P}\left(t\right)$ satisfies conditions (i) and (ii). Define

$$p_{ji}\left(t\right)=\sum _{k=0}^j({\tilde{p}}_{ik}\left(t\right)-{\tilde{p}}_{i+1,k}\left(t\right)),\forall i,j\in {\mathbb{Z}}_{+}.$$

It is clear that condition (i) implies $p_{ij}\left(t\right)\ge 0$ for any $i,j\in {\mathbb{Z}}_{+}$. Furthermore, for any $i,j\in {\mathbb{Z}}_{+}$ and $t\ge 0$, we obtain that

$$\begin{array}{ccc}\hfill \sum _{k=i}^{\infty }{p}_{jk}\left(t\right)& =& \sum _{k=i}^{\infty }\sum _{l=0}^j({\tilde{p}}_{kl}\left(t\right)-{\tilde{p}}_{k+1,l}\left(t\right))\hfill \\ \hfill & =& \sum _{l=0}^j\sum _{k=i}^{\infty }({\tilde{p}}_{kl}\left(t\right)-{\tilde{p}}_{k+1,l}\left(t\right))\hfill \\ \hfill & =& \sum _{l=0}^j({\tilde{p}}_{il}\left(t\right)-\underset{k\to \infty }{lim}{\tilde{p}}_{k+1,l}\left(t\right))=\sum _{l=0}^j{\tilde{p}}_{il}\left(t\right),\hfill \end{array}$$

where the last equality follows from condition (ii). Therefore, the identity (1) holds and, in particular, ${\sum }_{k=0}^{\infty }{p}_{jk}\left(t\right)={\sum }_{l=0}^j{\tilde{p}}_{0l}\left(t\right)\le 1$. It follows from (1) that ${\sum }_{k=i}^{\infty }{p}_{jk}\left(t\right)$ is non-decreasing in j for fixed i and $t\ge 0$. Thus, $P\left(t\right)$ is stochastically monotone. Using the Chapman–Kolmogorov equation for $\tilde{P}\left(t\right)$, together with the identity (1), we have

$$\begin{array}{ccc}\hfill p_{ji}(t+s)& =& \sum _{k=0}^j({\tilde{p}}_{ik}(t+s)-{\tilde{p}}_{i+1,k}(t+s))\hfill \\ \hfill & =& \sum _{k=0}^j\left(\sum _{l=0}^{\infty }{\tilde{p}}_{il}\left(t\right){\tilde{p}}_{lk}\left(s\right)-\sum _{l=0}^{\infty }{\tilde{p}}_{i+1,l}\left(t\right){\tilde{p}}_{lk}\left(s\right)\right)\hfill \\ \hfill & =& \sum _{l=0}^{\infty }({\tilde{p}}_{il}\left(t\right)-{\tilde{p}}_{i+1,l}\left(t\right))\sum _{k=0}^j{\tilde{p}}_{lk}\left(s\right)\hfill \\ \hfill & =& \sum _{l=0}^{\infty }({\tilde{p}}_{il}\left(t\right)-{\tilde{p}}_{i+1,l}\left(t\right))\sum _{k=l}^{\infty }{p}_{jk}\left(s\right)\hfill \\ \hfill & =& \sum _{k=0}^{\infty }{p}_{jk}\left(s\right)\sum _{l=0}^k({\tilde{p}}_{il}\left(t\right)-{\tilde{p}}_{i+1,l}\left(t\right))\hfill \\ \hfill & =& \sum _{k=0}^{\infty }{p}_{jk}\left(s\right)\left(\sum _{l=i}^{\infty }{p}_{kl}\left(t\right)-\sum _{l=i+1}^{\infty }{p}_{kl}\left(t\right)\right)=\sum _{k=0}^{\infty }{p}_{jk}\left(s\right)p_{ki}\left(t\right).\hfill \end{array}$$

Hence, $P\left(t\right)$ satisfies the Chapman–Kolmogorov equation. Moreover, since ${lim}_{t\to 0}{\tilde{p}}_{ij}\left(t\right)={\delta }_{ij}$, where ${\delta }_{ij}=1$ if $i=j$ and ${\delta }_{ij}=0$ otherwise, it follows that for any $i,j\in {\mathbb{Z}}_{+}$,

$$\begin{array}{ccc}\hfill \underset{t\to 0}{lim}{p}_{ji}\left(t\right)& =& \underset{t\to 0}{lim}\sum _{k=0}^j({\tilde{p}}_{ik}\left(t\right)-{\tilde{p}}_{i+1,k}\left(t\right))\hfill \\ \hfill & =& \sum _{k=0}^j(\underset{t\to 0}{lim}{\tilde{p}}_{ik}\left(t\right)-\underset{t\to 0}{lim}{\tilde{p}}_{i+1,k}\left(t\right))\hfill \\ \hfill & =& \sum _{k=0}^j({\delta }_{ik}-{\delta }_{i+1,k})={\delta }_{ij}.\hfill \end{array}$$

Therefore, $\tilde{P}\left(t\right)$ is the Siegmund dual of the stochastically monotone transition function $P\left(t\right)$.

Conversely, suppose that $\tilde{P}\left(t\right)$ is the Siegmund dual of a stochastically monotone transition function $P\left(t\right)$. Then, the identity (1) holds by Definition 1. It follows immediately from (1) that condition (i) is satisfied. Furthermore, from (1), we have

$${\tilde{p}}_{ij}\left(t\right)=\sum _{k=i}^{\infty }(p_{jk}\left(t\right)-p_{j-1,k}\left(t\right)),\forall i,j\in {\mathbb{Z}}_{+},$$

which implies that ${lim}_{i\to \infty }{\tilde{p}}_{ij}\left(t\right)=0$ for any $j\in {\mathbb{Z}}_{+}$ and $t\ge 0$. Therefore, condition (ii) also holds true. □

A stochastically monotone transition function $P\left(t\right)$ and its dual $\tilde{P}\left(t\right)$ are completely determined by each other. From (1), we can derive

$$\begin{array}{c}\hfill {\tilde{p}}_{ij}\left(t\right)=\sum _{k=i}^{\infty }(p_{jk}\left(t\right)-p_{j-1,k}\left(t\right)),\forall i,j\in {\mathbb{Z}}_{+},\end{array}$$

(11)

and

$$\begin{array}{c}\hfill p_{ji}\left(t\right)=\sum _{k=0}^j({\tilde{p}}_{ik}\left(t\right)-{\tilde{p}}_{i+1,k}\left(t\right)),\forall i,j\in {\mathbb{Z}}_{+}.\end{array}$$

(12)

We now proceed to prove Theorem 1.

Proof of Theorem 1.

(i) It follows from Condition 1 and (Zhang [8], Proposition 2.4) that

$$\begin{array}{c}\hfill \sum _{k=0}^j{\tilde{p}}_{ik}\left(t\right)\ge \sum _{k=0}^j{\tilde{p}}_{i+1,k}\left(t\right),\forall i,j\in {\mathbb{Z}}_{+}.\end{array}$$

(13)

Since $\tilde{Q}$ is a downwardly skip-free transition rate matrix, we find that

$$\begin{array}{c}\hfill \underset{i\to \infty }{lim}{\tilde{q}}_{ij}=0,\forall j\in {\mathbb{Z}}_{+}.\end{array}$$

(14)

From (Wang and Zhang [9], Theorem 1.3) and (Li [10], Proposition 2.2), we obtain that Condition 2 holds true if and only if the equation

$$\lambda y_i=\sum _{j=0}^{\infty }{y}_j{\tilde{q}}_{ji},j\in {\mathbb{Z}}_{+},\sum _{i=0}^{\infty }\mid y_i\mid <\infty ,$$

has only the trivial solution for some (and hence for all) $\lambda >0$. This, together with (14) and ( Reuter and Riley [11], Theorem 8), implies that

$$\begin{array}{c}\hfill \underset{i\to \infty }{lim}{\tilde{p}}_{ij}\left(t\right)=0,\forall j\in {\mathbb{Z}}_{+}.\end{array}$$

(15)

Combining (13), (15) and Proposition 1, we conclude that there exists a stochastically monotone transition function $P\left(t\right)$ satisfying both (1) and (12). Therefore, $\tilde{P}\left(t\right)$ is the Siegmund dual of $P\left(t\right)$. Differentiating (12) at $t=0$ yields Equation (2). Since ${\tilde{q}}_{i,i-1}>0$ for all $i\ge 1$, and ${\tilde{q}}_{i,i-j}=0$ for all $i\ge j\ge 2$, it then follows from (2) that Q is an upwardly skip-free transition rate matrix.

• (ii) Summing both sides of (2) over $i\ge m$ yields that for any $i,j\in {\mathbb{Z}}_{+}$,

  $$\sum _{i=m}^{\infty }{q}_{ji}=\sum _{i=m}^{\infty }\sum _{k=0}^j({\tilde{q}}_{ik}-{\tilde{q}}_{i+1,k})=\sum _{k=0}^j\sum _{i=m}^{\infty }({\tilde{q}}_{ik}-{\tilde{q}}_{i+1,k})=\sum _{k=0}^j({\tilde{q}}_{mk}-\underset{i\to \infty }{lim}{\tilde{q}}_{i+1,k}).$$

  Since $\tilde{Q}$ is a downwardly skip-free transition rate matrix, it is clear that ${lim}_{i\to \infty }{\tilde{q}}_{ij}=0$ for any $j\in {\mathbb{Z}}_{+}$. Hence, we find that

  $$\begin{array}{c}\hfill \sum _{i=m}^{\infty }{q}_{ji}=\sum _{k=0}^j{\tilde{q}}_{mk},\forall m,j\in {\mathbb{Z}}_{+}.\end{array}$$

  (16)

  Taking $m=0$ in (16) and noting that ${\tilde{q}}_{00}=0$, it follows that ${\tilde{q}}_{0k}=0$ for any $k\in {\mathbb{Z}}_{+}$. Therefore, we obtain

  $$\begin{array}{c}\hfill \sum _{k=0}^j{\tilde{q}}_{0k}=0=\sum _{i=0}^{\infty }{q}_{ji},\forall j\in {\mathbb{Z}}_{+},\forall t\ge 0,\end{array}$$

  which implies that Q is conservative. Since ${\tilde{q}}_{00}=0$, i.e., 0 is an absorbing state of $\tilde{P}\left(t\right)$, we have that

  $${\tilde{p}}_{00}\left(t\right)=1,{\tilde{p}}_{0j}\left(t\right)=0,\forall j\ge 1,\forall t\ge 0.$$

  It then follows from (1) that

  $$\sum _{k=0}^{\infty }{p}_{jk}\left(t\right)=\sum _{k=0}^j{\tilde{p}}_{0k}\left(t\right)={\tilde{p}}_{00}\left(t\right)=1,\forall i,j\in {\mathbb{Z}}_{+},\forall t\ge 0.$$

  Thus, $P\left(t\right)$ is honest. □

3. Proof of Theorem 2

Before proving Theorem 2, we first present the following useful theorem, which is taken from (Zhang et al. [12], Theorem 2.2).

Theorem 3.

Suppose $P\left(t\right)=(p_{ij}\left(t\right):i,j\in {\mathbb{Z}}_{+})$ is an ergodic and stochastically monotone transition function. Then, it is not strongly ergodic if and only if ${lim}_{i\to \infty }{p}_{ij}\left(t\right)=0$ for any $j\in {\mathbb{Z}}_{+}$ and $t\ge 0$.

Proof of Theorem 2.

(i) By Theorem 1, there exists a stochastically monotone upwardly skip-free transition function $P\left(t\right)$, such that $\tilde{P}\left(t\right)$ is the Siegmund dual of $P\left(t\right)$. Consequently, Equation (11) holds. Moreover, from (11), we deduce that

$$\underset{i\to \infty }{lim}{\tilde{p}}_{ij}\left(t\right)=0,\forall j\in {\mathbb{Z}}_{+},\forall t\ge 0.$$

It then follows from Theorem 3 that the downwardly skip-free process $\tilde{P}\left(t\right)$ is not strongly ergodic.

• (ii) Suppose that $\tilde{P}\left(t\right)$ is honest. Since ${\tilde{q}}_{00}=0$, Theorem 1 implies that $P\left(t\right)$ is honest. By (1), we see

  $$\begin{array}{c}\hfill \sum _{k=0}^j{\tilde{p}}_{ik}\left(t\right)=\sum _{k=i}^{\infty }{p}_{jk}\left(t\right)=1-\sum _{k=0}^{i-1}{p}_{jk}\left(t\right),\forall i,j\in {\mathbb{Z}}_{+},\forall t\ge 0.\end{array}$$

  (17)

  Letting $j\to \infty$ in (17), and noting that $\tilde{P}\left(t\right)$ is honest, we find that

  $$\underset{j\to \infty }{lim}\sum _{k=0}^{i-1}{p}_{jk}\left(t\right)=1-\underset{j\to \infty }{lim}\sum _{k=0}^j{\tilde{p}}_{ik}\left(t\right)=0,\forall j\in {\mathbb{Z}}_{+},\forall t\ge 0.$$

  Thus, ${lim}_{j\to \infty }{p}_{jk}\left(t\right)=0$ for any $k\in {\mathbb{Z}}_{+}$ and $t\ge 0$, and Theorem 3 then implies that $P\left(t\right)$ is not strongly ergodic.

Conversely, suppose that $P\left(t\right)$ is not strongly ergodic. Theorem 3 implies that ${lim}_{j\to \infty }{p}_{jk}\left(t\right)=0$ for $k\in {\mathbb{Z}}_{+}$ and $t\ge 0$. Letting $j\to \infty$ in (17) yields

$$\underset{j\to \infty }{lim}\sum _{k=0}^j{\tilde{p}}_{ik}\left(t\right)=\sum _{k=0}^{\infty }{\tilde{p}}_{ik}\left(t\right)=1-\underset{j\to \infty }{lim}\sum _{k=0}^{i-1}{p}_{jk}\left(t\right)=1,i\in {\mathbb{Z}}_{+}.$$

Then, it follows that $\tilde{P}\left(t\right)$ is honest.

• (iii) Letting $j=0$ in (17) gives that

  $${\tilde{p}}_{i0}\left(t\right)=1-\sum _{k=0}^{i-1}{p}_{0k}\left(t\right),i\ge 1,t\ge 0.$$
  Since ${\pi }_j={lim}_{t\to \infty }{p}_{ij}\left(t\right)$ for all $i,j\in {\mathbb{Z}}_{+}$, letting $t\to \infty$ in the above equation yields

  $$\begin{array}{c}\hfill {\mathbf{P}}_i({\tilde{\sigma }}_0<\infty )=1-\sum _{k=0}^{i-1}{\pi }_k,i\ge 1.\end{array}$$

  (18)
  Notice that $P\left(t\right)$ is ergodic if and only if ${\pi }_i>0$ for some (and hence for all) $i\in {\mathbb{Z}}_{+}$. From (18), we therefore have ${\mathbf{P}}_i({\tilde{\sigma }}_0<\infty )<1$ if and only if $P\left(t\right)$ is ergodic.
• (iv) Recall that the resolvent of $P\left(t\right)$ is given by

  $$p_{ij}\left(\lambda \right)={\int }_0^{\infty }{e}^{-\lambda t}{p}_{ij}\left(t\right)dt,i,j\in {\mathbb{Z}}_{+},\lambda >0.$$
  Since ${\tilde{q}}_{00}=0$, Theorem 1 implies that Q is conservative. Hence, the resolvent of $P\left(t\right)$ satisfies the following backward Kolmogorov equations:

  $$\begin{array}{c}\hfill \sum _{k=0}^{\infty }{q}_{ik}{p}_{kj}\left(\lambda \right)-\lambda p_{ij}\left(\lambda \right)=-{\delta }_{ij},i,j\in {\mathbb{Z}}_{+},\lambda >0.\end{array}$$
  Applying (Chen and Zhang [2], Theorem 1.1) gives

  $$\begin{array}{ccc}\hfill p_{ij}\left(\lambda \right)-p_{i-1,j}\left(\lambda \right)& =& \sum _{k=0}^{i-1}{\displaystyle \frac{{\tilde{F}}_{i-1}^{\left(k\right)}(-{\delta }_{kj}+\lambda p_{0j}\left(\lambda \right))}{q_{k,k+1}}}\hfill \\ \hfill & =& -{\mathbf{1}}_{\{j\le i-1\}}{\displaystyle \frac{{\tilde{F}}_{i-1}^{\left(j\right)}}{q_{j,j+1}}}+{\tilde{m}}_{i-1}\lambda p_{0j}\left(\lambda \right),\lambda >0,\hfill \end{array}$$

  (19)

  where

  $${\tilde{F}}_i^{\left(i\right)}=1,{\tilde{F}}_n^{\left(i\right)}={\displaystyle \frac{1}{q_{n,n+1}}}\sum _{k=i}^{n-1}{\tilde{q}}_n^{\left(k\right)}{\tilde{F}}_k^{\left(i\right)},n>i\ge 0,$$

  $${\tilde{q}}_n^{\left(k\right)}=q_n^{\left(k\right)}+\lambda =\sum _{j=0}^k{q}_{nj}+\lambda ,0\le k<n.$$

  $${\tilde{m}}_n=\sum _{k=0}^n{\displaystyle \frac{{\tilde{F}}_n^{\left(k\right)}}{q_{k,k+1}}},n\ge 0.$$
  Since for each $n\ge i\ge 0$, ${\tilde{F}}_n^{\left(i\right)}$ and ${\tilde{m}}_n$ are analytic in $\lambda \in (0,\infty )$, we have

  $$\begin{array}{c}\hfill \underset{\lambda \to 0^{+}}{lim}{\tilde{F}}_n^{\left(i\right)}=F_n^{\left(i\right)}and\underset{\lambda \to 0^{+}}{lim}{\tilde{m}}_n=m_n,n\ge i\ge 0.\end{array}$$

  (20)
  Using the method of integration by substitution, and noting that ${lim}_{t\to \infty }{p}_{ij}\left(t\right)={\pi }_j$ for all $i,j\in {\mathbb{Z}}_{+}$, it follows that for $j\in {\mathbb{Z}}_{+}$,

  $$\begin{array}{ccc}\hfill \underset{\lambda \to 0^{+}}{lim}\lambda p_{0j}\left(\lambda \right)& =& \underset{\lambda \to 0^{+}}{lim}\lambda {\int }_0^{\infty }{e}^{-\lambda t}{p}_{0j}\left(t\right)dt\hfill \\ \hfill & =& \underset{\lambda \to 0^{+}}{lim}\lambda {\int }_0^{\infty }{e}^{-x}{p}_{0j}(x/\lambda )d(x/\lambda )\hfill \\ \hfill & =& \underset{\lambda \to 0^{+}}{lim}{\int }_0^{\infty }{e}^{-t}{p}_{0j}(t/\lambda )dt\hfill \\ \hfill & =& {\pi }_j.\hfill \end{array}$$

  (21)
  Moreover, by the monotone convergence theorem and (17), we see that for $i\ge 1,j\ge 1$,

  $$\begin{array}{ccc}\hfill {\mathbf{E}}_i{\tilde{T}}_j& =& {\mathbf{E}}_i\left({\int }_0^{\infty }{\mathbf{1}}_{\{{\tilde{X}}_t=j\}}dt\right)\hfill \\ \hfill & =& {\int }_0^{\infty }{\tilde{p}}_{ij}\left(t\right)dt\hfill \\ \hfill & =& \underset{\lambda \to 0^{+}}{lim}{\int }_0^{\infty }{e}^{-\lambda t}{\tilde{p}}_{ij}\left(t\right)dt\hfill \\ \hfill & =& \underset{\lambda \to 0^{+}}{lim}{\int }_0^{\infty }{e}^{-\lambda t}\sum _{k=0}^{i-1}(p_{j-1,k}\left(t\right)-p_{jk}\left(t\right))dt\hfill \\ \hfill & =& \sum _{k=0}^{i-1}\underset{\lambda \to 0^{+}}{lim}{\int }_0^{\infty }{e}^{-\lambda t}(p_{j-1,k}\left(t\right)-p_{jk}\left(t\right))dt\hfill \\ \hfill & =& \sum _{k=0}^{i-1}\underset{\lambda \to 0^{+}}{lim}(p_{j-1,k}\left(\lambda \right)-p_{jk}\left(\lambda \right)).\hfill \end{array}$$

  (22)
  Substituting (19)–(21) into (22), we obtain for $i\ge 1,j\ge 1$,

  $$\begin{array}{ccc}\hfill {\mathbf{E}}_i{\tilde{T}}_j& =& \sum _{k=0}^{i-1}\underset{\lambda \to 0^{+}}{lim}\left({\mathbf{1}}_{\{k\le j-1\}}{\displaystyle \frac{{\tilde{F}}_{j-1}^{\left(k\right)}}{q_{k,k+1}}}-{\tilde{m}}_{j-1}\lambda p_{0k}\left(\lambda \right)\right)\hfill \\ \hfill & =& \sum _{k=0}^{i-1}\left({\mathbf{1}}_{\{k\le j-1\}}{\displaystyle \frac{F_{j-1}^{\left(k\right)}}{q_{k,k+1}}}-m_{j-1}{\pi }_k\right)\hfill \\ \hfill & =& \sum _{k=0}^{i\wedge j-1}{\displaystyle \frac{F_{j-1}^{\left(k\right)}}{q_{k,k+1}}}-m_{j-1}\sum _{k=0}^{i-1}{\pi }_k.\hfill \end{array}$$

  (23)
  If the extinction probability satisfies ${\mathbf{P}}_i(\widetilde{{\sigma }_0}<\infty )=1$, then from (5), we get ${\pi }_k=0$ for each $k\in {\mathbb{Z}}_{+}$. In that case, (23) reduces to

  $${\mathbf{E}}_i{\tilde{T}}_j=\sum _{k=0}^{i\wedge j-1}{\displaystyle \frac{F_{j-1}^{\left(k\right)}}{q_{k,k+1}}},i\ge 1,j\ge 1.$$
• (v) We begin by proving the identity

  $$\begin{array}{c}\hfill {\mathbf{E}}_i\left(\tilde{\zeta }{\mathbf{1}}_{[{\tilde{\sigma }}_0=\infty ]}\right)=\sum _{j=1}^{\infty }{\int }_0^{\infty }{\mathbf{P}}_i({\tilde{X}}_t=j,{\tilde{\sigma }}_0=\infty )dt,i\ge 1.\end{array}$$

  (24)
  Let ${\tilde{\zeta }}_n:=inf\{t\ge 0:{\tilde{X}}_t\ge n\}$ for $n\ge 0$. Recall that the explosion time $\tilde{\zeta }={lim}_{n\to \infty }inf\{t\ge 0:{\tilde{X}}_t\ge n\}$, then we know that $\tilde{\zeta }={lim}_{n\to \infty }{\tilde{\zeta }}_n$. For each $n=1,2,\dots$, define ${\tilde{X}}_t^{\left(n\right)}={\tilde{X}}_{t\wedge {\tilde{\zeta }}_n}$. Then, we get for $i\ge 1$,

  $$\begin{array}{c}\hfill {\mathbf{E}}_i\left({\tilde{\zeta }}_n{\mathbf{1}}_{[{\tilde{\sigma }}_0=\infty ]}\right)={\int }_0^{\infty }{\mathbf{P}}_i({\tilde{\zeta }}_n{1}_{[{\tilde{\sigma }}_0=\infty ]}>t)dt=\sum _{j=1}^{n-1}{\int }_0^{\infty }{\mathbf{P}}_i({\tilde{X}}_t^{\left(n\right)}=j,{\tilde{\sigma }}_0=\infty )dt.\end{array}$$

  (25)
  On the one hand, fix $m>0$. For sufficiently large n, (25) yields that

  $${\mathbf{E}}_i\left({\tilde{\zeta }}_n{\mathbf{1}}_{[{\tilde{\sigma }}_0=\infty ]}\right)\ge \sum _{j=1}^m{\int }_0^{\infty }{\mathbf{P}}_i({\tilde{X}}_t^{\left(n\right)}=j,{\tilde{\sigma }}_0=\infty )dt,i\ge 1.$$
  Letting $n\to \infty$, and applying the monotone convergence theorem on the left hand side and Fatou’s lemma on the right hand side, then we see

  $$\begin{array}{c}\hfill {\mathbf{E}}_i\left(\tilde{\zeta }{\mathbf{1}}_{[{\tilde{\sigma }}_0=\infty ]}\right)\ge \sum _{j=1}^m{\int }_0^{\infty }{\mathbf{P}}_i({\tilde{X}}_t=j,{\tilde{\sigma }}_0=\infty )dt,i\ge 1.\end{array}$$
  Letting $m\to \infty$ in the above equation shows that

  $$\begin{array}{c}\hfill {\mathbf{E}}_i\left(\tilde{\zeta }{\mathbf{1}}_{[{\tilde{\sigma }}_0=\infty ]}\right)\ge \sum _{j=1}^{\infty }{\int }_0^{\infty }{\mathbf{P}}_i({\tilde{X}}_t=j,{\tilde{\sigma }}_0=\infty )dt,i\ge 1.\end{array}$$

  (26)
  On the other hand, it follows from (25) that

  $${\mathbf{E}}_i\left({\tilde{\zeta }}_n{\mathbf{1}}_{[{\tilde{\sigma }}_0=\infty ]}\right)\le \sum _{j=1}^{n-1}{\int }_0^{\infty }{\mathbf{P}}_i({\tilde{X}}_t=j,{\tilde{\sigma }}_0=\infty )dt,i\ge 1.$$
  Letting $n\to \infty$ in the above equation gives

  $$\begin{array}{c}\hfill {\mathbf{E}}_i\left(\tilde{\zeta }{\mathbf{1}}_{[{\tilde{\sigma }}_0=\infty ]}\right)\le \sum _{j=1}^{\infty }{\int }_0^{\infty }{P}_i({\tilde{X}}_t=j,{\tilde{\sigma }}_0=\infty )dt,i\ge 1.\end{array}$$

  (27)
  Combining (26) and (27) immediately yields (24). When ${\mathbf{P}}_i({\tilde{\sigma }}_0=\infty )>0$, dividing both sides by ${\mathbf{P}}_i({\tilde{\sigma }}_0=\infty )$ gives

  $$\begin{array}{ccc}\hfill {\mathbf{E}}_i\left(\tilde{\zeta }\right|{\tilde{\sigma }}_0=\infty )& =& \sum _{j=1}^{\infty }{\int }_0^{\infty }{\displaystyle \frac{{\mathbf{P}}_i({\tilde{X}}_t=j,{\tilde{\sigma }}_0=\infty )}{{\mathbf{P}}_i({\tilde{\sigma }}_0=\infty )}}dt,i\ge 1.\hfill \end{array}$$

  (28)
  Using the Markov property, we conclude that for $i\ge 1,j\ge 1$,

  $$\begin{array}{c}\hfill {\mathbf{P}}_i({\tilde{X}}_t=j,{\tilde{\sigma }}_0=\infty )={\mathbf{P}}_i({\tilde{X}}_t=j){\mathbf{P}}_i({\tilde{\sigma }}_0=\infty |{\tilde{X}}_t=j)={\mathbf{P}}_i({\tilde{X}}_t=j){\mathbf{P}}_j({\tilde{\sigma }}_0=\infty ).\end{array}$$

  (29)
  Substituting (5), (6), and (29) into (28), we obtain for $i\ge 1$,

  $$\begin{array}{ccc}\hfill {\mathbf{E}}_i\left(\tilde{\zeta }\right|{\tilde{\sigma }}_0=\infty )& =& \sum _{j=1}^{\infty }\left({\displaystyle \frac{{\mathbf{P}}_j({\tilde{\sigma }}_0=\infty )}{{\mathbf{P}}_i({\tilde{\sigma }}_0=\infty )}}{\int }_0^{\infty }{\mathbf{P}}_i({\tilde{X}}_t=j)dt\right)\hfill \\ \hfill & =& \sum _{j=1}^{\infty }{\displaystyle \frac{{\mathbf{P}}_j({\tilde{\sigma }}_0=\infty )}{{\mathbf{P}}_i({\tilde{\sigma }}_0=\infty )}}{\mathbf{E}}_i{\tilde{T}}_j\hfill \\ \hfill & =& \sum _{j=1}^{\infty }{\displaystyle \frac{{\sum }_{k=0}^{j-1}{\pi }_k}{{\sum }_{k=0}^{i-1}{\pi }_k}}\left(\sum _{k=0}^{i\wedge j-1}{\displaystyle \frac{F_{j-1}^{\left(k\right)}}{q_{k,k+1}}}-m_{j-1}\sum _{k=0}^{i-1}{\pi }_k\right).\hfill \end{array}$$
• (vi) For $i\ge 1$, we have

  $${\mathbf{E}}_i\left({\tilde{\sigma }}_0{\mathbf{1}}_{[{\tilde{\sigma }}_0<\infty ]}\right)={\int }_0^{\infty }{\mathbf{P}}_i({\tilde{\sigma }}_0{\mathbf{1}}_{[{\tilde{\sigma }}_0<\infty ]}>t)dt=\sum _{j=1}^{\infty }{\int }_0^{\infty }{\mathbf{P}}_i({\tilde{X}}_t=j,{\tilde{\sigma }}_0<\infty )dt.$$
  When ${\mathbf{P}}_i({\tilde{\sigma }}_0<\infty )>0$, dividing both sides by ${\mathbf{P}}_i({\tilde{\sigma }}_0<\infty )$ gives

  $$\begin{array}{c}\hfill {\mathbf{E}}_i\left({\tilde{\sigma }}_0\right|{\tilde{\sigma }}_0<\infty )=\sum _{j=1}^{\infty }{\int }_0^{\infty }{\displaystyle \frac{{\mathbf{P}}_i({\tilde{X}}_t=j,{\tilde{\sigma }}_0<\infty )}{{\mathbf{P}}_i({\tilde{\sigma }}_0<\infty )}}dt,i\ge 1.\end{array}$$

  (30)
  By the Markov property, we can get for $i\ge 1,j\ge 1$,

  $$\begin{array}{c}\hfill {\mathbf{P}}_i({\tilde{X}}_t=j,{\tilde{\sigma }}_0<\infty )={\mathbf{P}}_i({\tilde{X}}_t=j){\mathbf{P}}_i({\tilde{\sigma }}_0<\infty |{\tilde{X}}_t=j)={\mathbf{P}}_i({\tilde{X}}_t=j){\mathbf{P}}_j({\tilde{\sigma }}_0<\infty ).\end{array}$$

  (31)
  Substituting (5), (6), and (31) into (30), we obtain for $i\ge 1$,

  $$\begin{array}{ccc}\hfill {\mathbf{E}}_i\left({\tilde{\sigma }}_0\right|{\tilde{\sigma }}_0<\infty )& =& \sum _{j=1}^{\infty }\left({\displaystyle \frac{{\mathbf{P}}_j({\tilde{\sigma }}_0<\infty )}{{\mathbf{P}}_i({\tilde{\sigma }}_0<\infty )}}{\int }_0^{\infty }{\mathbf{P}}_i({\tilde{X}}_t=j)dt\right)\hfill \\ \hfill & =& \sum _{j=1}^{\infty }{\displaystyle \frac{{\mathbf{P}}_j({\tilde{\sigma }}_0<\infty )}{{\mathbf{P}}_i({\tilde{\sigma }}_0<\infty )}}{\mathbf{E}}_i{\tilde{T}}_j\hfill \\ \hfill & =& \sum _{j=1}^{\infty }{\displaystyle \frac{1-{\sum }_{k=0}^{j-1}{\pi }_k}{1-{\sum }_{k=0}^{i-1}{\pi }_k}}\left(\sum _{k=0}^{i\wedge j-1}{\displaystyle \frac{F_{j-1}^{\left(k\right)}}{q_{k,k+1}}}-m_{j-1}\sum _{k=0}^{i-1}{\pi }_k\right).\hfill \end{array}$$
  If the extinction probability satisfies ${\mathbf{P}}_i(\widetilde{{\sigma }_0}<\infty )=1$, it then follows from (7) that for $i\ge 1$,

  $${\mathbf{E}}_i\widetilde{{\sigma }_0}={\int }_0^{\infty }{\mathbf{P}}_i({\tilde{\sigma }}_0>t)dt=\sum _{j=1}^{\infty }{\int }_0^{\infty }{\mathbf{P}}_i({\tilde{X}}_t=j)dt=\sum _{j=1}^{\infty }{E}_i{\tilde{T}}_j=\sum _{j=1}^{\infty }\sum _{k=0}^{i\wedge j-1}{\displaystyle \frac{F_{j-1}^{\left(k\right)}}{q_{k,k+1}}}.$$

□

4. Examples

In this section, we consider two examples of the downwardly skip-free processes and provide the extinction probability, the mean occupation time, the mean explosion time, and the mean extinction time for the two downwardly skip-free processes.

Example 1.

Let $\tilde{Q}$ be a downwardly skip-free transition rate matrix on ${\mathbb{Z}}_{+}$, and let $\tilde{P}\left(t\right)$ be the corresponding minimal process. Suppose that $\tilde{Q}=\left({\tilde{q}}_{ij}\right)$ is given as follows:

$${\tilde{q}}_{i,i-1}={\displaystyle \frac{1}{2^{i-1}}},i\ge 1;{\tilde{q}}_{ii}=-{\displaystyle \frac{3}{2^i}},i\ge 1;$$

$${\tilde{q}}_{ij}={\displaystyle \frac{1}{2^j}},j\ge i+1,i\ge 1;{\tilde{q}}_{ij}=0,\mathrm{otherwise}.$$

It is clear that ${\sum }_{k=0}^j{\tilde{q}}_{ik}\ge {\sum }_{k=0}^j{\tilde{q}}_{i+1,k}$ for $j\ne i$, thus, Condition 1 holds. Since

$$\sum _{i=1}^{\infty }{\displaystyle \frac{1}{{\tilde{q}}_{i,i-1}}}=\sum _{i=1}^{\infty }{2}^{i-1}=\infty ,$$

Remark 1 implies that Condition 2 also holds.

Then, from Theorem 1, there exists a stochastically monotone upwardly skip-free transition function $P\left(t\right)$, such that $\tilde{P}\left(t\right)$ is the Siegmund dual of $P\left(t\right)$. Let $Q=\left(q_{ij}\right)$ be the transition rate matrix of $P\left(t\right)$. Equation (2) then yields

$$q_{01}=-q_{00}=1;q_{i,i+1}=q_{i0}={\displaystyle \frac{1}{2^i}},i\ge 1;$$

$$q_{ii}=-{\displaystyle \frac{1}{2^{i-1}}},i\ge 1;q_{ij}=0,\mathrm{otherwise}.$$

Using ${\tilde{q}}_{00}=0$ and Theorem 1, we have that Q is conservative. Noting that $\{-q_{ii},i\in {\mathbb{Z}}_{+}\}$ is bounded, we find that Q is regular. Define

$$\begin{array}{c}\hfill d_0=0,d_n={\displaystyle \frac{1}{q_{n,n+1}}}\left(1+\sum _{k=0}^{n-1}{q}_n^{\left(k\right)}{d}_k\right),n\ge 1.\end{array}$$

(32)

From (3), (4), and (32), we can compute that

$$\begin{array}{c}\hfill F_i^{\left(j\right)}=\left\{\begin{array}{ccc}1,\hfill & & i=j,\hfill \\ 2^{i-j-1},\hfill & & i>j.\hfill \end{array}\right.\end{array}$$

$$m_n=n{2}^{n-1}+2^n,n\ge 0.$$

$$d_0=0,d_n=(n-1)2^{n-1}+2^n,n\ge 1.$$

Noting that Q is regular and ${\sum }_{n=0}^{\infty }{F}_n^{\left(0\right)}=1+{\sum }_{n=1}^{\infty }{2}^{n-1}=\infty$, it follows from (Chen [1], Theorem 4.52) that $P\left(t\right)$ is recurrence. Since

$$d=\underset{k\in {\mathbb{Z}}_{+}}{sup}\left({\displaystyle \frac{{\sum }_{n=0}^k{d}_n}{{\sum }_{n=0}^k{F}_n^{\left(0\right)}}}\right)=\underset{k\in {\mathbb{Z}}_{+}}{sup}{\displaystyle \frac{{\sum }_{n=1}^k[(n-1)2^{n-1}+2^n]}{1+{\sum }_{n=1}^k{2}^{n-1}}}=\underset{k\in {\mathbb{Z}}_{+}}{sup}k=\infty ,$$

([1], Theorem 4.52) implies that $P\left(t\right)$ is not ergodic. Then, we find that ${lim}_{t\to \infty }{p}_{ij}\left(t\right)={\pi }_j=0$ for $i,j\in {\mathbb{Z}}_{+}$, and $P\left(t\right)$ is not strongly ergodic. Applying Theorem 2, we conclude that $\tilde{P}\left(t\right)$ is honest, $\tilde{P}\left(t\right)$ is not strongly ergodic, and the extinct probability of $\tilde{P}\left(t\right)$ is 1.

Using (7) yields

$$\begin{array}{c}\hfill {\mathbf{E}}_i{\tilde{T}}_j=\sum _{k=0}^{i\wedge j-1}{\displaystyle \frac{F_{j-1}^{\left(k\right)}}{q_{k,k+1}}}=\sum _{k=0}^{i\wedge j-1}{\displaystyle \frac{2^{j-k-2}}{\frac{1}{2^k}}}=(i\wedge j)2^{j-2},i\ge 1,j\ge 1.\end{array}$$

(33)

Substituting (33) into (10) shows

$${\mathbf{E}}_i{\tilde{\sigma }}_0=\sum _{j=1}^{\infty }\sum _{k=0}^{i\wedge j-1}{\displaystyle \frac{F_{j-1}^{\left(k\right)}}{q_{k,k+1}}}=\sum _{j=1}^{\infty }(i\wedge j)2^{j-2}=\infty ,i\ge 1.$$

This example demonstrates how our duality-based approach efficiently computes mean occupation and extinction times for a downwardly skip-free process. The explicit expressions obtained through Theorems 2(iv) and 2(vi) would be challenging to derive using conventional methods for skip-free processes like those in Chen [1], while Zhang [3,4] established recurrence/transience criteria for similar processes, our method provides exact quantitative results for occupation and extinction times that complement these qualitative classifications. The infinite mean extinction time aligns with known results for processes with unbounded transition rates.

Example 2.

Let $\tilde{Q}$ be a downwardly skip-free transition rate matrix on ${\mathbb{Z}}_{+}$, and let $\tilde{P}\left(t\right)$ be the corresponding minimal process. Suppose that $\tilde{Q}=\left({\tilde{q}}_{ij}\right)$ is given as follows:

$${\tilde{q}}_{i,i-1}={\displaystyle \frac{1}{2}},i\ge 1;{\tilde{q}}_{ii}=-1,i\ge 1;{\tilde{q}}_{ij}=0,\mathrm{otherwise}.$$

It is clear that ${\sum }_{k=0}^j{\tilde{q}}_{ik}\ge {\sum }_{k=0}^j{\tilde{q}}_{i+1,k}$ for $j\ne i$, thus, Condition 1 holds. Since

$$\sum _{i=1}^{\infty }{\displaystyle \frac{1}{{\tilde{q}}_{i,i-1}}}=\sum _{i=1}^{\infty }2=\infty ,$$

Remark 1 implies that Condition 2 also holds.

Then, from Theorem 1, there exists a stochastically monotone upwardly skip-free transition function $P\left(t\right)$, such that $\tilde{P}\left(t\right)$ is the Siegmund dual of $P\left(t\right)$. Let $Q=\left(q_{ij}\right)$ be the transition rate matrix of $P\left(t\right)$. Equation (2) then yields

$$q_{01}=-q_{00}={\displaystyle \frac{1}{2}};q_{i,i+1}=q_{i0}={\displaystyle \frac{1}{2}},i\ge 1;$$

$$q_{ii}=-1,i\ge 1;q_{ij}=0,\mathrm{otherwise}.$$

Using ${\tilde{q}}_{00}=0$ and Theorem 1, we have that Q is conservative. Noting that $\{-q_{ii},i\in {\mathbb{Z}}_{+}\}$ is bounded, we find that Q is regular.

From (3), (4) and (32), we can compute that

$$F_n^{\left(k\right)}=2^{n-k-1},n>k\ge 0;F_k^{\left(k\right)}=1.$$

$$m_n=2^{n+1},n\ge 0.$$

$$d_0=0,d_n=2^n,n\ge 1.$$

Since Q is regular and

$$d=\underset{k\in {\mathbb{Z}}_{+}}{sup}\left({\displaystyle \frac{{\sum }_{n=0}^k{d}_n}{{\sum }_{n=0}^k{F}_n^{\left(0\right)}}}\right)=\underset{k\in {\mathbb{Z}}_{+}}{sup}{\displaystyle \frac{{\sum }_{n=1}^k{2}^n}{1+{\sum }_{n=1}^k{2}^{n-1}}}=\underset{k\in {\mathbb{Z}}_{+}}{sup}\left(2-{\displaystyle \frac{2}{2^k}}\right)=2<\infty ,$$

([1], Theorem 4.52) implies that $P\left(t\right)$ is ergodic. Let $\pi =\left({\pi }_i\right)$ be the stationary distribution of $P\left(t\right)$. From $\pi Q=0$, we get ${\pi }_n=2^{-n-1}$ for $n\ge 0$. Noting that

$$\underset{k\in {\mathbb{Z}}_{+}}{sup}\sum _{n=0}^k(F_n^{\left(0\right)}d-d_n)=\underset{k\in {\mathbb{Z}}_{+}}{sup}\left(\sum _{n=1}^k(2^{n-1}·2-2^n)+2\right)=2<\infty ,$$

it follows from ([1], Theorem 4.52) that $P\left(t\right)$ is strongly ergodic. Applying Theorem 2, we conclude that $\tilde{P}\left(t\right)$ is not honest, $\tilde{P}\left(t\right)$ is not strongly ergodic, and the extinct probability of $\tilde{P}\left(t\right)$ is given by

$${\mathbf{P}}_i({\tilde{\sigma }}_0<\infty )=1-\sum _{k=0}^{i-1}{\pi }_k={\displaystyle \frac{1}{2^i}}<1,i\ge 1.$$

When $1\le i<j$, using (6) yields

$$\begin{array}{c}\hfill {\mathbf{E}}_i{\tilde{T}}_j=\sum _{k=0}^{i-1}{\displaystyle \frac{F_{j-1}^{\left(k\right)}}{q_{k,k+1}}}-m_{j-1}\sum _{k=0}^{i-1}{\pi }_k=2\sum _{k=0}^{i-1}{2}^{j-k-2}-2^j\sum _{k=0}^{i-1}{\displaystyle \frac{1}{2^{k+1}}}=0.\end{array}$$

(34)

When $i\ge j\ge 1$, using (6) yields

$$\begin{array}{c}\hfill {\mathbf{E}}_i{\tilde{T}}_j=\sum _{k=0}^{j-1}{\displaystyle \frac{F_{j-1}^{\left(k\right)}}{q_{k,k+1}}}-m_{j-1}\sum _{k=0}^{i-1}{\pi }_k=2\left(\sum _{k=0}^{j-2}{2}^{j-k-2}+1\right)-2^j\sum _{k=0}^{i-1}{\displaystyle \frac{1}{2^{k+1}}}=2^{j-i}.\end{array}$$

(35)

Substituting (34) and (35) into (8) shows

$${\mathbf{E}}_i\left(\tilde{\zeta }\right|{\tilde{\sigma }}_0=\infty )=\sum _{j=1}^{\infty }{\displaystyle \frac{{\sum }_{k=0}^{j-1}{\pi }_k}{{\sum }_{k=0}^{i-1}{\pi }_k}}{\mathbf{E}}_i{\tilde{T}}_j=\sum _{j=1}^i{\displaystyle \frac{1-\frac{1}{2^j}}{1-\frac{1}{2^i}}}{2}^{j-i}=2-{\displaystyle \frac{i}{2^i-1}},i\ge 1.$$

Substituting (34) and (35) into (9) shows

$${\mathbf{E}}_i\left(\widetilde{{\sigma }_0}\right|{\tilde{\sigma }}_0<\infty )=\sum _{j=1}^{\infty }{\displaystyle \frac{1-{\sum }_{k=0}^{j-1}{\pi }_k}{1-{\sum }_{k=0}^{i-1}{\pi }_k}}{\mathbf{E}}_i{\tilde{T}}_j=\sum _{j=1}^i{\displaystyle \frac{\frac{1}{2^j}}{\frac{1}{2^i}}}{2}^{j-i}=i,i\ge 1.$$

This example illustrates how our duality framework yields closed-form expressions for conditional explosion and extinction times in a constant-rate downwardly skip-free process.

5. Conclusions

In this note, we have established that a class of downwardly skip-free processes can be viewed as the Siegmund dual of upwardly skip-free processes. By leveraging this duality relationship and existing results on upwardly skip-free processes, we derived explicit expressions for several important quantities associated with downwardly skip-free processes. Specifically, under Conditions 1 and 2, we showed the following:

1. The extinction probability of a downwardly skip-free process is given by ${\mathbf{P}}_i({\tilde{\sigma }}_0<\infty )=1-{\sum }_{k=0}^{i-1}{\pi }_k$ for $i\ge 1$, where ${\pi }_k$ is the stationary distribution of the dual upwardly skip-free process.

2. The mean occupation time has the following explicit form:

$${\mathbf{E}}_i{\tilde{T}}_j=\sum _{k=0}^{i\wedge j-1}{\displaystyle \frac{F_{j-1}^{\left(k\right)}}{q_{k,k+1}}}-m_{j-1}\sum _{k=0}^{i-1}{\pi }_k$$

for $i,j\ge 1$, with simplified expressions when extinction is certain.

3. The conditional expectations of explosion and extinction times are provided in Theorems 2(v) and 2(vi), generalizing previous results.

Through two concrete examples, we demonstrated how these formulas can be applied to compute key quantities for specific downwardly skip-free processes. The duality approach developed in this work provides a powerful framework for analyzing downwardly skip-free processes by transferring problems to their upwardly skip-free duals, for which comprehensive theory already exists.

Future work should explore extensions to more general state spaces, applications to specific stochastic models in queueing theory and population dynamics, and connections to other duality relationships in stochastic processes.