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\begin{document}
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\title[On classical problems]{On classical problems about jump processes \\ with finite skip}

\author[张余辉]{\sihao 张余辉}

\date[2021年6月26日]{\wuhao (2021年6月24-27日, 随机分析及相关领域学术研讨会, 武汉大学) }

\institute[北京师范大学]{\wuhao 北京师范大学}

%\frame{\titlepage}


 \begin{frame}
  \includegraphics[scale=0.3]{math.png}\\
\maketitle
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%----------------------------------------------------------------------------------------1
\begin{frame}
  \frametitle{CONTENTS}
  \tableofcontents%[sections]
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Background}
%----------------------------------------------------------------------------------------2
\frame{ \frametitle{Background}\small
\begin{itemize}
\item $Q$-process, jump process: on $\mathbb Z_+:=\{0, 1, 2, \dots\}$, suppose that sub-Markov transition probability matrix $P(t)=(p_{ij}(t), i, j\in\mathbb Z_+)$ satisfies

(1) Normal condition: $p_{ij}(t)\ge 0,\ \sum_jp_{ij}(t)\le 1,\  i, j\in\mathbb Z_+, t\ge 0$.

(2) Chapman-Kolmogorov equation: $p_{ij}(t+s)=\sum_kp_{ik}(t)p_{kj}(s)$.

(3) Jump condition: $\lim_{t\to 0}p_{ij}(t)=\delta_{ij}, \  i, j\in\mathbb Z_+.$

(4) $Q$-condition: $\lim_{t\to 0}(p_{ij}(t)-\delta_{ij})/t=q_{ij}, \  i, j\in\mathbb Z_+,$ i.e. the $Q$-matrix $Q=(q_{ij})$ is the derivative matrix at time $0$ of $P(t)$,
 $$0\leqslant q_{ij}<\infty,\ i\ne j;\ \sum_{j\ne i}q_{ij}\leqslant -q_{ii}=:q_i\leqslant \infty,\ i\in \mathbb Z_+.$$
\item totally stable: $q_i<\infty, \ i\in\mathbb Z_+$.
\item conservative: $q_i=\sum_{j\ne i}q_{ij},\ i\in\mathbb Z_+$, i.e. $\sum_{j\in\mathbb Z_+}q_{ij}=0$.
\end{itemize}
}
%----------------------------------------------------------------------------------------


\frame{\frametitle{Background}\small

\begin{itemize}
\item jump process with finite skip: $\exists m\geqslant 1$,
$$q_{i, i+m}>0, q_{ij}=0\, (j>i+m), i> 0;\ \text{\red upward}$$
or
$$q_{i, i-m}>0, q_{ij}=0\, (0\leqslant j<i-m), i> m.\ \text{\red downward}$$
\item When $m=1$, {\blue single birth, single death}.
\item Three classical problems: {\red uniqueness, recurrence, ergodicity}.
\end{itemize}
}

 \frame{\frametitle{Background}\small
\begin{itemize}
\item uniqueness: when there is only one $Q$-process $P(t)=(p_{ij}(t))$ for a given $Q$-matrix $Q=(q_{ij})$ (then the matrix $Q$ is often called regular).
\item Hou (1974, 1982), Hou \& Guo (1984), Zheng (1982), Chen \& Zheng (1983), Chen (1991).
\end{itemize}

\begin{block}{Uniqueness criterion[Feller(1957), Reuter(1957)]} For a given $Q$-matrix $Q=(q_{ij})$, the $Q$-process is unique iff the equation has only the
equation
$$(\lambda+q_i)u_i=\sum_{j\ne i}q_{ij}u_j, 0\leqslant u_i\leqslant 1, i\geqslant 0$$
has only the trivial solution $u_i\equiv0$ for some (equivalently, for all) $\lambda>0$.
\end{block}
}

\frame{\frametitle{Background}\small
For birth-death $(a_i,b_i)$: $a_i=q_{i,i-1}>0, i\ge 1; b_i=q_{i,i+1}>0, i\ge 0$.

\begin{block}{Uniqueness criterion for BDP}
Given a birth-death $Q$~matrix $(a_i, b_i)$. The the birth-death process is unique iff
$$\sum_{n=0}^\infty \frac 1{\mu_nb_n}\sum_{k=0}^n\mu_k=\infty,$$
where
$$\mu_0=1, \quad \mu_i=\frac{b_0b_1\cdots b_{i-1}}{a_1a_2\cdots a_i},\quad i\ge 1.$$
\end{block}
}

\frame{\frametitle{Background}\small
\begin{itemize}
\item For single birth: $q_{i,i+1}>0,  q_{ij}=0\,(j>i+1), i\geqslant 0$.
\end{itemize}

\begin{block}{Uniqueness criterion for SBP}
Given a single birth $Q$~matrix $(a_i, b_i)$. The the single birth process is unique iff
$$\sum_{n=0}^\infty \sum_{k=0}^n\frac{F_n^{(k)}}{q_{k,k+1}}=\infty,$$
where
$$F_i^{(i)}=1, \qquad F_n^{(i)}=\frac 1{q_{n,n+1}} \sum_{k=i}^{n-1}q_n^{(k)}F_k^{(i)},\quad 0\le i<n; \quad
{\blue q_n^{(k)}=\sum_{j=0}^k q_{nj}}.$$
\end{block}
}

\frame{\frametitle{Background}\small

\begin{itemize}
\item For single death: $q_{i,i-1}>0,  q_{ij}=0\,(0\leqslant j<i-1), i\geqslant 2$. Uniqueness?

Mao, Yan \& Zhang (2021+): Three basic criteria for downwardly skip-free Markov processes.
\item Li \& Li: Down/up crossing properties of weighted Markov collision process. Front. Math. China, 2021, 16(2)： 525-542.
\end{itemize}

\begin{itemize}
\item Recurrence: if for each $h>0$, $P(h)$ is recurrent, $\int_0^\infty p_{ii}(t)dt=\infty$.
\end{itemize}

{\red embedding chain} $(\Pi_{ij})$: $\displaystyle\Pi_{ij}=I_{\{q_i\ne0\}}(1-\delta_{ij})\frac{q_{ij}}{q_i}+I_{\{q_i=0\}}\delta_{ij}.$

\begin{block}{Recurrence Theorem[Feller(1957)]} For a given totally stable and conservative $Q$-matrix $Q=(q_{ij})$,
$$\int_0^\infty p^{\text{min}}_{ij}(t)dt=\sum_{n=0}^\infty \frac{\Pi_{ij}^{(n)}}{q_j}.$$
In particular, if $Q$ is irreducible and regular, then $P(t)$ is recurrent iff so is its embedding chain.
\end{block}
}


\frame{\frametitle{Background}\small
\begin{block}{Theorem} For a regular and irreducible $Q=(q_{ij})$,  $P(t)$ is recurrent iff
$$
x_i=\sum_{j\ne 0,i}\Pi_{ij}x_j,\quad 0\leqslant x_i\leqslant 1,\,i\geqslant 0
$$
has only a trivial solution.\end{block}

%For embedding chain $(\Pi_{ij})$,
%$(f_{i,j_0}=\mathbb P_i(\sigma_{j_0}<\infty):i\in\mathbb Z_+)$ is the minimal solution to the equation
%$$x_i=\sum_{k\ne j_0}\Pi_{ik}x_k+\Pi_{i,j_0},\ i\in\mathbb Z_+.$$
%
%$(\Pi_{ij})$ is recurrent iff $f_{i,j_0}\equiv 1$.
%}
%\frame{\frametitle{Background}\small
\begin{block}{Recurrence criterion for BDP[Yan \& Chen(1986)]}
Given a regular birth-death $Q$-matrix $(a_i, b_i)$. The the birth-death is recurrent iff
$\sum_{n=0}^\infty \frac{1}{\mu_nb_n}=\infty$.
\end{block}
\begin{block}{Uniqueness criterion for SBP}
Assume the single birth $Q$-matrix $Q=(q_{ij})$ is regular and irreducible.
Then the process is recurrent iff $\sum_{n=0}^\infty F_n^{(0)}=\infty$.
\end{block}
}

\frame{\frametitle{Background}\small
\begin{itemize}
\item
$$
x_i=\sum_{j\ne 0,i}\Pi_{ij}x_j,\quad 0\leqslant x_i\leqslant 1,\,i\geqslant 0
$$

\item uniqueness:
$$u_i=\sum_{j\ne i}\frac{q_{ij}}{\lambda+q_i}u_j, \quad 0\leqslant u_i\leqslant 1, i\geqslant 0,$$
$$u=\Pi(\lambda)u.$$
\end{itemize}

We introduce a fictitious state $\Delta$ and
define on the enlarged state space $E_\Delta=E\cup\{\Delta\}$ a new transition probability
matrix
$$\Pi_{ij}^\Delta(\lambda)=\begin{cases}\Pi(\lambda), &i,j\in E;\\
\frac{\lambda}{\lambda+q_i}, &i\in E, j=\Delta;\\
p_j, &i=\Delta, j\in E,\end{cases}$$
where $(p_j, j\in E)$ is a positive probability measure.
}
\frame{\frametitle{Background}\small

\begin{block}{Theorem}
The original $Q$-process is unique iff the $\Pi^\Delta(\lambda)$-chain is recurrent.
\end{block}

\begin{itemize}
\item Spieksma, F.M.: Countable state Markov processes: non-explosiveness
and moment function. Probab. Eng. and Inform. Sci., 2015, 29: 623-637.
\item Chen, M.-F.: Practical criterion for uniqueness of $Q$-processes. Chinese J. Appl. Probab Statist. 2015, 31(2): 213-224.
\end{itemize}
$$\begin{matrix}
       &\Delta& 0    & 1    & 2    &3     &\cdots\\
\Delta &-1    & 1    & 0    &0     &0     &\cdots\\
0      &1     &-1-q_0&q_{01}&q_{02}&q_{03}&\cdots\\
1      &1     &q_{10}&-1-q_1&q_{12}&q_{13}&\cdots\\
2      &1     &q_{20}&q_{21}&-1-q_2&q_{23}&\cdots\\
3      &1     &q_{30}&q_{31}&q_{32}&-1-q_3&\cdots\\
\vdots &\vdots&\vdots&\vdots&\vdots&\vdots&\ddots
\end{matrix}$$
}

\section{Uniqueness and recurrence}

\frame{\frametitle{Uniqueness and recurrence}\small

Suppose that $Q$ is irreducible. Denote
$$\sigma_0=\inf\{t>\text{the first jumping time}: X(t)=0\},\quad\tau_{ N+}=\inf\{t>0: X(t)\geqslant N\}.$$

{\blue For recurrence, it is well known that the process is recurrent iff
$$\lim_{N\to\infty}\mathbb P_0(\tau_{N+}<\sigma_0)=0.$$}

Denote $p_i=\mathbb P_i(\tau_{N+}<\sigma_0)$ and
$$q_i^{(k)}=\sum_{j=0}^k q_{ij},\quad 0\leqslant k<i;\quad q_i^{(k)}=\sum_{j=k}^\infty q_{ij},\quad k>i\geqslant 0.$$
\begin{align}
p_0&=\sum_{j=1}^{N-1}\frac{q_{0j}}{q_0}\cdot p_j+\frac{q_{0}^{(N)}}{q_0},\nonumber\\
p_i&=\sum_{1\leqslant j\leqslant i-1}\frac{q_{ij}}{q_i}\cdot p_j+\sum_{i+1\leqslant i\leqslant N-1}\frac{q_{ij}}{q_i}\cdot p_j+\frac{q_{i}^{(N)}}{q_i},\quad 1\leqslant i\leqslant N-1,\label{e0.02}\\
p_N&=1.\label{e0.03}
\end{align}
}
\frame{\frametitle{Uniqueness and recurrence}\small
Then the equations (\ref{e0.02}) can be rewritten as
$$(I-\Pi)\mathbf p=\mathbf r^{(N)},$$
where $I$ is the unit matrix, $\mathbf p=(p_1,\cdots, p_{N-1})^{\text{T}}$ and $\mathbf r^{(N)}=(\frac{q_1^{(N)}}{q_1},\cdots, \frac{q_{N-1}^{(N)}}{q_{N-1}})^{\text{T}}$.
By the irreducibility of $Q$,  the local embedding chain is transient,
$$(I-\Pi)^{-1}=\sum_{n=0}^\infty\Pi^n<\infty\Rightarrow\mathbf p=\sum_{n=0}^\infty\Pi^n\mathbf r^{(N)},$$
there exists exactly a unique solution for (\ref{e0.02}).
The decomposition according to the first time arriving the set $\{N, N+1, \cdots\}$ before the first return to $0$ for the original embedding chain.
The recurrence criterion is presented as
$$\lim_{N\to\infty}\Big(\frac{q_{01}}{q_0},\cdots,\frac{q_{0,N-1}}{q_0}\Big)\sum_{n=0}^\infty\Pi^n\mathbf r^{(N)}=0.$$
}

\frame{\frametitle{Uniqueness and recurrence: $m=2$ downwardly}\small

Let $w_k=p_k-p_{k-1}$ for all $2\leqslant k\leqslant N$. From (\ref{e0.02}) and (\ref{e0.03}) it follows that
\begin{equation}\label{e0.04}
q_{i0}p_1+\sum_{2\leqslant k\leqslant i}q_i^{(k-1)}w_k=\sum_{i+1\leqslant k\leqslant N}q_i^{(k)}w_k,\quad 1\leqslant i\leqslant N-1.
\end{equation}

Consider the {\red double death process with possible catastrophes:
$$q_{i,i-2}>0, \quad i\geqslant 2;\quad q_{ij}=0,\quad 0\leqslant j<i-2.$$}
$$q_{i0}+\sum_{(i-1)\vee 2\leqslant k\leqslant i}\Big(q_i^{(k-1)}-q_{i0}\Big)w_k=\sum_{i+1\leqslant k\leqslant N}\Big(q_i^{(k)}+q_{i0}\Big)w_k,\quad 1\leqslant i\leqslant N-1.$$
{\blue $(w_{N-1}, w_N)$}. For $3\leqslant i\leqslant N-1$,
\begin{equation}\label{e01.02}
w_{i-1}=\frac1{q_{i, i-2}}\Bigg(\sum_{i+1\leqslant k\leqslant N}\Big(q_i^{(k)}+q_{i0}\Big)w_k-\Big(q_i^{(i-1)}-q_{i0}\Big)w_i-q_{i0}\Bigg).
\end{equation}

{\blue $$q_{i0}+\sum_{(i-1)\vee 2\leqslant k\leqslant i}\Big(q_i^{(k-1)}-q_{i0}\Big)w_k=\sum_{i+1\leqslant k\leqslant N}\Big(q_i^{(k)}+q_{i0}\Big)w_k,\quad 1\leqslant i\leqslant 2.$$}

}

\frame{\frametitle{Uniqueness and recurrence: $m=2$ downwardly}\small
For $2\leqslant i\leqslant N-1$, define
$$g_N^{(N)}=1,\ g_{N-1}^{(N)}=0,\ g_{i-1}^{(N)}=\frac1{q_{i, i-2}}\Bigg(\sum_{i+1\leqslant k\leqslant N}q_i^{(k)}g_k^{(N)}-q_i^{(i-1)}g_i^{(N)}\Bigg),$$
$$g_N^{(N-1)}=0,\ g_{N-1}^{(N-1)}=1,\ g_{i-1}^{(N-1)}=\frac1{q_{i, i-2}}\Bigg(\sum_{i+1\leqslant k\leqslant N}q_i^{(k)}g_k^{(N-1)}-q_i^{(i-1)}g_i^{(N-1)}\Bigg).$$
Define
$$g_{0}^{(N)}=\sum_{2\leqslant k\leqslant N}q_1^{(k)}g_k^{(N)}-q_1^{(0)}g_1^{(N)},\quad  g_{0}^{(N-1)}=\sum_{2\leqslant k\leqslant N}q_1^{(k)}g_k^{(N-1)}-q_1^{(0)}g_1^{(N-1)}.$$
}

\frame{\frametitle{Uniqueness and recurrence: $m=2$ downwardly}\small

\begin{block}{Recurrence criterion} Assume that the double death $Q$-matrix $Q=(q_{ij})$ without catastrophes is non-explosive and irreducible. Then the process is recurrent iff
\begin{equation}\label{e01.08}
\lim_{N\to\infty}\frac{\sum_{k=1}^{N}q_0^{(k)}\big(g_0^{(N-1)}g_k^{(N)}-g_0^{(N)}g_k^{(N-1)}\big)}
{g_0^{(N-1)}\sum_{k=1}^{N}g_k^{(N)}-g_0^{(N)}\sum_{k=1}^{N}g_k^{(N-1)}}=0.
\end{equation}
 holds. If the irreducibility is kept when only resetting $\bar q_{01}>0$ and $\bar q_{0j}=0$ for all $j\geqslant 2$, i.e., the $Q$-matrix constrained on $\mathbb N:=\{1, 2, \cdots\}$ is irreducible, then the process is recurrent iff
$$\lim_{N\to\infty}\frac{g_0^{(N-1)}g_1^{(N)}-g_0^{(N)}g_1^{(N-1)}}
{g_0^{(N-1)}\sum_{k=1}^{N}g_k^{(N)}-g_0^{(N)}\sum_{k=1}^{N}g_k^{(N-1)}}=0.
$$
\end{block}
}
\frame{\frametitle{Uniqueness and recurrence: $m=2$ upwardly}\small
Consider the {\red double birth process:
$$q_{i,i+2}>0, \quad i\geqslant 1;\quad q_{ij}=0,\quad j>i+2.$$}
Define
$$
f_0^{(0)}=1,\ f_1^{(0)}=0, \ f_{i+1}^{(0)}=\frac 1{q_{i,i+2}}\Bigg(\sum_{k=0}^{i-1} q_i^{(k)} f_k^{(0)}-q_i^{(i+1)} f_i^{(0)}\Bigg),\ 1\leqslant i\leqslant N-2,
$$
$$
f_0^{(1)}=0,\ f_1^{(1)}=1,\ f_{i+1}^{(1)}=\frac 1{q_{i,i+2}}\Bigg(\sum_{k=0}^{i-1} q_i^{(k)} f_k^{(1)}-q_i^{(i+1)} f_k^{(1)}\Bigg),\ 1\leqslant i\leqslant N-2,
$$
$$f_N^{(0)}=\sum_{k=0}^{N-2} q_{N-1}^{(k)} f_k^{(0)}-q_{N-1}^{(N)} f_{N-1}^{(0)}, \ f_N^{(1)}=\sum_{k=0}^{N-2} q_{N-1}^{(k)} f_k^{(1)}-q_{N-1}^{(N)} f_{N-1}^{(1)}.$$
}

\frame{\frametitle{Uniqueness and recurrence: $m=2$ upwardly}\small

\begin{block}{Recurrence criterion for double birth} Assume that the $2$-birth $Q$-matrix $Q=(q_{ij})$ is non-explosive and irreducible.
Then the process is recurrent iff
$$\lim_{N\to\infty}\frac{\sum_{k=0}^{N-1}q_0^{(k+1)}(f_N^{(0)}f_k^{(1)}-f_N^{(1)}f_k^{(0)}) }{f_N^{(0)}\sum_{k=0}^{N-1}f_k^{(1)}-f_N^{(1)}\sum_{k=0}^{N-1}f_k^{(0)}}
=0.$$
If the irreducibility is kept when only resetting $\bar q_{01}>0$ and $\bar q_{0j}=0$ for all $j\geqslant 2$,
then the original process is recurrent iff
$$\lim_{N\to\infty}\frac{f_N^{(1)}}{f_N^{(0)}\sum_{k=0}^{N-1}f_k^{(1)}-f_N^{(1)}\sum_{k=0}^{N-1}f_k^{(0)}}=0.$$
\end{block}
}

\frame{\frametitle{Uniqueness and recurrence}\small
\begin{block}{Recurrence criterion for finite skip downward}Assume that non-explosive and irreducible. Then the process is recurrent iff
$$\lim_{N\to\infty}\sum_{k=2}^{N}\Big(q_0^{(1)}-q_0^{(k)}\Big)\Bigg(g_k^{(2)}+\frac{\sum_{N-m+1\leqslant j\leqslant N}\text{det}(\mathbf D_j^{(N)})g_k^{(j)}}{\text{det}(\mathbf D^{(N)})}\Bigg)
=q_0.$$
\end{block}

\begin{block}{Recurrence criterion for finite skip upward}
Assume that non-explosive and irreducible.
Then the process is recurrent iff
$$\lim_{N\to\infty}\frac{\sum_{k=0}^{N-1}q_0^{(k+1)}\sum_{j=0}^{m-1}\text{det}(\mathbf C_j^{(N)})f_k^{(j)}}{\text{det}(\mathbf C^{(N)})}
=0.$$
If the irreducibility is kept when only resetting $\bar q_{01}>0$ and $\bar q_{0j}=0$ for all $j\geqslant 2$,
$$\text{recurrence iff}\ \lim_{N\to\infty}\frac{\text{det}(\mathbf C_0^{(N)})}{\text{det}(\mathbf C^{(N)})}=0.$$
\end{block}
}

\frame{\frametitle{Uniqueness and recurrence}\small
\begin{block}{Uniqueness criterion for finite skip downward}
Assume that the $m$-death $Q$-matrix $\widetilde Q=(\tilde q_{ij})$ without catastrophes is irreducible.
Then the process is unique iff
$$\lim_{N\to\infty}\sum_{k=2}^{N}\Bigg(g_k^{(2)}+\frac{\sum_{N-m+1\leqslant j\leqslant N}\text{det}(\mathbf D_j^{(N)})g_k^{(j)}}{\text{det}(\mathbf D^{(N)})}\Bigg)
=1.$$
\end{block}

\begin{block}{Uniqueness criterion for finite skip upward}
Assume that the $m$-birth $Q$-matrix $\widetilde Q=(\tilde q_{ij})$ is irreducible.
Then the process is unique iff
$$\lim_{N\to\infty}\frac{\text{det}(\mathbf C_{0}^{(N)})}{\text{det}(\mathbf C^{(N)})}=0.$$
\end{block}
}

\section{Examples}
\frame{\frametitle{Examples}\small
\begin{block}{Example 1} Given a non-explosive, irreducible single death $Q$-matrix $Q=(q_{ij})$ satisfying:
$$q_{i,i-1}=a>0,\quad q_{i,i+1}=b\geqslant 0,\quad q_{i,i+2}=d\geqslant 0,  \quad i\geqslant 1,$$
where $b+d>0$, and $q_{ij}=0$ for other $i, j\geqslant 1$, $i\ne j$. Then the process is recurrent iff $a\geqslant b+2d$.
\end{block}

\begin{block}{Example 2} Given a non-explosive, irreducible double death $Q$-matrix $Q=(q_{ij})$ satisfying:
$$q_{i,i-2}=c>0,\quad q_{i,i-1}=a\geqslant 0,\quad q_{i,i+1}=b> 0, \quad i\geqslant 1,$$
and $q_{ij}=0$ for other $i, j\geqslant 1$, $i\ne j$. Then the process is recurrent iff $a+2c\geqslant b$. This example with $a=0$ and $c=b=1$ is taken from [Mao \& Z. 2004] in which it is proven that the process is exponentially ergodic but not strongly ergodic.
\end{block}
}

\frame{\frametitle{Examples}\small
\begin{block}{Example 3} Given a non-explosive, irreducible double death $Q$-matrix $Q=(q_{ij})$ satisfying:
$$q_{i,i-2}=c>0,\quad i\geqslant 2;\quad q_{i,i+2}=d> 0, \quad i\geqslant 0;\quad q_{01}=b>0,\quad q_{10}=a>0,$$
and $q_{ij}=0$ for other $i\ne j$. Then
the process is recurrent iff $c\geqslant d$.\end{block}

\begin{block}{Example 4} Given a non-explosive, irreducible double death $Q$-matrix $Q=(q_{ij})$ satisfying:
$$q_{i,i-2}=q_{i,i-1}=1,\quad i\geqslant 2;\quad q_{i,i+2}=q_{i,i+1}=1, \quad i\geqslant 0;\quad q_{10}=1,$$
and $q_{ij}=0$ for other $i\ne j$. Then
the process is recurrent.\end{block}

}

\frame{\frametitle{Examples}\small
\begin{block}{Example 5} Given a irreducible single death $Q$-matrix $Q=(q_{ij})$ without catastrophes satisfying:
$$q_{i,i-1}>0,\quad q_{ij}=0,\quad i\geqslant 1, j\ne i, i-1; \quad q_{0j}>0,\quad j\geqslant 1, \quad \sum_{j\geqslant 1}q_{0j}<\infty.$$
Then the process is unique.\end{block}
}

\section{Ergodicity}

\frame{\frametitle{Ergodicity}\small

Let $Q=(q_{ij})$ be a regular irreducible $Q$-matrix. Then the limit
$$\lim_{t\to\infty}p_{ij}(t)=:\pi_j$$
exists and it is independent of $i$. Moreover, we have either
$\sum_j\pi_j=1$ or $\sum_j\pi_j=0$.

\begin{itemize}
\item $P(t)$ is positive recurrent or ergodic if so is $P(h)$ for every $h>0$. Equivalently, $\lim_{t\to\infty}p_{ii}(t)=\pi_i>0$.
\end{itemize}

\begin{block}{Theorem[Issacson \& Arnold(1978)]}
The $Q$-process is ergodic iff $\mathbb E_0\sigma_0<\infty$.
\end{block}
}

\frame{\frametitle{Ergodicity}\small

Assume that the process is recurrent.
$$\lim_{N\to\infty}\mathbb E_i\sigma_0\mathbbold 1_{\{\sigma_0<\tau_{N+}\}}=\mathbb E_i\sigma_0\mathbbold 1_{\{\sigma_0<\infty\}}=\mathbb E_i\sigma_0, \qquad i\geqslant 0.$$
Define
$h_i=\mathbb E_i\sigma_0\mathbbold 1_{\{\sigma_0<\tau_{ N+}\}}$.
Then for all $i\geqslant N$, $h_i=0$,  and for $0\leqslant i<N$,
So for $0\leqslant i<N$,
\begin{align}
h_0&=\frac 1{q_0}\sum_{0<j<N}\frac{q_{0j}}{q_0} \mathbb P_j(\sigma_0<\tau_{ N+})
+\sum_{0<j<N}\frac{q_{0j}}{q_0}h_j\nonumber\\
&=:s_0^{(N)}+\sum_{0<j<N}\frac{q_{0j}}{q_0}h_j,\label{e19}\\
h_i&=\frac 1{q_i}\sum_{i\ne j<N}\frac{q_{ij}}{q_i} \mathbb P_j(\sigma_0<\tau_{N+})
+\sum_{j\ne i, 0<j<N}\frac{q_{ij}}{q_i}h_j\nonumber\\
&=:s_i^{(N)}+\sum_{j\ne i, 0<j<N}\frac{q_{ij}}{q_i}h_j.\label{e20}
\end{align}
}
\frame{\frametitle{Ergodicity}\small
Set $\mathbf h=(h_1,\cdots, h_{N-1})^{\text{T}}$ and $\mathbf s^{(N)}=(s_1^{(N)},\cdots, s_{N-1}^{(N)})^{\text{T}}$.
Then
$$(I-\Pi)\mathbf h=\mathbf s^{(N)}.$$
Hence
$$\mathbf h=(I-\Pi)^{-1}\mathbf s^{(N)}=\sum_{n= 0}^{\infty}\Pi^n\mathbf s^{(N)},$$
and
$$h_0=s_0^{(N)}+\Big(\frac{q_{01}}{q_0},\cdots, \frac{q_{0,N-1}}{q_0}\Big)\sum_{n= 0}^{\infty}\Pi^n\mathbf s^{(N)}.$$

Assume that $\mathbf h$ is finite and let $w_k=h_k-h_{k-1}$ for all $1\leqslant k\leqslant N$. From (\ref{e20}) it follows that
\begin{equation}\label{e21}
q_{i0}h_1+\sum_{2\leqslant k\leqslant i}q_i^{(k-1)}w_k=q_is_i^{(N)}+\sum_{i+1\leqslant k\leqslant N}q_i^{(k)}w_k,\quad 1\leqslant i<N.
\end{equation}
}

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\end{CJK*}

\end{document}
