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\begin{document}

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\title[Single birth processes]{Unified treatment for some problems of single birth processes}

\author[Y.H. Zhang]{Yuhui Zhang}

\date[2016.05.04]{\wuhao (Jiangsu Normal University: May 3-4, 2016) }

\institute[BNU]{\sihao Beijing Normal University}


\frame{\titlepage}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%            目录            %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{目录}
  %  \frame{\frametitle{目录}\tableofcontents}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%         背景及问题        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\large

%\frame{Chen, Mu-Fa, Z. Unified representation of formulas for
%single birth processes, Frontiers of Mathematics in China, 2014,
%9(4): 761-796.}
%----------------------------------------------------------------------------------------1

\frame{
 \frametitle{Outline}%\small
\begin{enumerate}
\item Background
\item Poisson equation
\item Uniqueness
\item Recurrence, return/extinction probability
\item Laplace transform of return/extinction time
\item Summary
\item Main references
\end{enumerate}
}

%----------------------------------------------------------------------------------------2
\frame{ \frametitle{1. Background}\small
\begin{itemize}
\item $Q$-process: on $\mathbb Z_+:=\{0, 1, 2, \dots\}$, suppose that sub-Markov transition probability matrix $P(t)=(p_{ij}(t))$ satisfies

(1) Normal condition: $p_{ij}(t)\ge 0,\ \sum_jp_{ij}(t)\le 1$.

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

(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. $Q$-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}}
%----------------------------------------------------------------------------------------3
\frame{\frametitle{1. Background}\small Single birth process:
$Q=(q_{ij})$ satisfies
$$q_{i,i+1}>0,\ q_{i,i+j}=0,\ i\ge 0,j> 1.$$
%\vspace{0.3cm}
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\vskip0.3cm
$$Q=\left(%
\begin{array}{ccccc}
  - & + & 0 & 0 & \cdots \\
  * & - & + & 0 & \cdots \\
  * & * & - & + & \cdots\\
  \vdots & \vdots & \vdots & \vdots & \ddots \\
  \end{array}%
\right)$$
}
%----------------------------------------------------------------------------------------4
\frame{\frametitle{1. Background}%\small

\begin{enumerate}
\item The exit boundary of the process consists at most one single extremal point
\vskip0.1cm
\item Fundamental comparison tool in studying more complex processes, such as
infinite-dimensional reaction-diffusion processes
\vskip0.1cm
\item Usually the single birth process is non-symmetric (irreversible)
\vskip0.1cm
\item Applications in other fields
\end{enumerate}
}

%----------------------------------------------------------------------------------------5
\frame{\frametitle{1. Background}%\small
\begin{enumerate}
\item Upwardly skip-free processes, population proc. (W.J. Anderson, 1991);
\item Generalized birth and death proc. (J.K. Zhang, 1984);
\item Birth and death proc. with catastrophes; birth, death and catastrophe proc.
(Brockwell, Gani, Resnick and Pakes et al, 1982-1986; B. Cairns \& P. Pollett, 2004);
\item Single birth proc. (S.J. Yan \& M.F. Chen, 1986)
\end{enumerate}
}
%----------------------------------------------------------------------------------------6
\frame{\frametitle{1. Background}%\small

$$\aligned i&\to i+1\\
&\to i-1\\
&\to i-2\\
&\to\cdots\\
&\to 0\endaligned
\quad
\aligned \hbox{at rate}\quad & {\blue b_i}\\
& {\blue a_i}+{ c_i}{\red f_{i,i-1}}\\
& { c_i}{\red f_{i,i-2}}\\
&\cdots\\
& { c_i}{\red f_{i0}}\endaligned$$
where $\sum_{j=0}^{i-1}{\red f_{ij}}=1$.

}
%----------------------------------------------------------------------------------------7
\frame{\frametitle{1. Background}%\small

Brockwell, Gani, Resnick and Pakes et al (1982-1986) for some special
catastrophes $f_{ij}$ ($b_i=b+\lambda i$, $a_i=0$, $c_i=ci$, $i\ge 0$).

\vskip0.2cm
geometric: $f_{ij}=p(1-p)^{i-1-j}\,(1\le j<i); f_{i0}=(1-p)^{i-1}$;
\vskip0.2cm
uniform: $f_{ij}=1/i\,(0\le j<i)$;
\vskip0.2cm
binomial: $f_{ij}=\binom{i-1}{j}p^j(1-p)^{i-1-j}\,(0\le j<i)$.

\vskip0.3cm
Aim: extinction times and probability of extinction.
\vskip0.3cm
Keys: generating function of $Q$-resolvent.
}
%----------------------------------------------------------------------------------------8
\frame{\frametitle{2. Poisson equation}\small For a totally stable and conservative single birth $Q$-matrix, given a function $c$,
define
$$\Omega g=Qg+cg, $$
Clearly, if $c\le 0$, then $\Omega$ is an operator corresponding to
a single birth process with killing rates $(-c_i)$. Define
$$
\widetilde F_i^{(i)}=1,\qquad  \widetilde F_n^{(i)}=\frac
1{q_{n,n+1}} \sum_{k=i}^{n-1}\tilde q_n^{(k)}\widetilde
F_k^{(i)},\qquad n>i\ge 0,$$
$$\tilde q_n^{(k)}=q_n^{(k)}-c_n:=\sum_{j=0}^kq_{nj}-c_n,\qquad 0\le k<n.$$
Once $c_i\equiv 0$, we omit the superscript $\ \widetilde{}$.
}

%----------------------------------------------------------------------------------------9
\frame{\small

\begin{block}{Theorem 1[M.F. Chen \& Z.(2014)]} Given a totally stable and conservative single birth $Q$-matrix $Q=(q_{ij})$ and functions $c$ and $f$,
the solution $g$ to the Poisson equation
$$
\Omega g=f
$$
has the following representation:
$$
g_n=g_0+\sum_{0\le k\le n-1}\sum_{0\le j\leq k}\frac{{\widetilde
F}_k^{(j)}(f_j-c_jg_0)}{q_{j,j+1}},\quad n\ge 0.
$$
In particular, the harmonic function of $\Omega$ can be represented
as
$$g_n=g_0\Bigg(1-\sum_{0\le k\le n-1}\sum_{0\le j\le k}\frac{\widetilde F_k^{(j)}c_j}{q_{j,j+1}}\Bigg),\quad n\ge 0.$$\end{block}
}
%---------------------------------------------------------10
\frame{\small \noindent{\it Proof}\quad For each $i\ge 0$, we have
\begin{align}
(\Omega g)_i&=q_{i,i+1}(g_{i+1}-g_i)-\sum_{0\le j\le i-1} q_{ij}\sum_{k=j}^{i-1}(g_{k+1}-g_k)+c_ig_i\nonumber\\
&=q_{i,i+1}(g_{i+1}-g_i)-\sum_{0\le j\le i-1}\sum_{j=0}^{k}q_{ij}(g_{k+1}-g_k)+c_ig_i\nonumber\\
&=q_{i,i+1}(g_{i+1}-g_i)-\sum_{0\le j\le i-1}\!\!\Bigg(\sum_{j=0}^{k}q_{ij}-c_i\Bigg)\!(g_{k+1}-g_k)+c_ig_0\nonumber\\
&=q_{i,i+1}(g_{i+1}-g_i)-\sum_{0\le j\le i-1}\tilde
q_i^{(k)}(g_{k+1}-g_k)+c_ig_0. \nonumber\end{align}
$$g_{i+1}-g_i=\frac 1{q_{i,i+1}}\Bigg(\sum_{0\le k \le i-1}\tilde q_{i}^{(k)}(g_{k+1}-g_k)+f_i-c_ig_0\Bigg).$$
}

%--------------------------------------------------------------------11
\frame{\small
\begin{block}{}$$g_{i+1}-g_i=\sum_{j=0}^i\frac{\widetilde
F_i^{(j)}(f_j-c_jg_0)}{q_{j,j+1}},\qquad i\ge 0.$$
\end{block}

\begin{block}{Proposition[M.F. Chen \& Z.(2014)]} For given $f$, the sequence $(h_n)$ defined successively by
$$h_n=\frac{1}{q_{n, n+1}}\bigg(f_n+\sum_{i\le k \le n-1}{\tilde q}_n^{(k)} h_k\bigg),\qquad n\ge i$$
has a unified expression as follows
$$h_n=\sum_{k=i}^n \frac{{\widetilde F}_n^{(k)}f_k}{q_{k, k+1}} ,\qquad n\ge i. $$
\end{block}
}
%-----------------------------------------------------------12

\frame{
\begin{itemize}
\item Uniqueness. J.K. Zhang(1984), Yan \& Chen(1986).
\item Recurrence. Yan \& Chen(1986).
\item Extinction/return probability. P.J. Brockwell(1986), W.J. Anderson(1991), Z.T. Hou
\& Q.F. Guo(1978, 1988).
\item Ergodicity. Yan \& Chen(1986).
\item Strong ergodicity. Z.(2001).
\item Polynomial moments and stationary dist. Z.(2003, 2004, 2013)
\item Exponential moments. Z.(2003)
\item Laplace transform. P. J. Brockwell(1986), W.J. Anderson(1991)
\end{itemize}
}

%----------------------------------------------------------------------------13

\frame{\frametitle{3. Uniqueness}\small

$Q$-process: $P'(t)|_{t=0}=Q$.

Backward Kolmogorov equation: $P'(t)=QP(t)$.

Forward Kolmogorov equation: $P'(t)=P(t)Q$.

$$p_{ij}(t)=\sum_{k\ne i}\int_0^tq_{ik}e^{-q_i(t-s)}p_{kj}(s)ds+\delta_{ij}e^{-q_it},$$
$$p_{ij}(t)=\sum_{k\ne j}\int_0^tp_{ik}(s)q_{kj}e^{-q_j(t-s)}ds+\delta_{ij}e^{-q_jt},$$

\begin{itemize}
\item $Q$-process exist? unique?
\end{itemize}

\begin{block}{Theorem} Let $Q=(q_{ij})$ be totally stable and conservative. Every $Q$-process satisfies the backward Kolmogorov equation.
\end{block}
}
%----------------------------------------------------------------------------------------14
\frame{\frametitle{3. Uniqueness}\small


\begin{block}{Existence Theorem} Given a totally stable and conservative $Q$-matrix, there always exist a $Q$-process. In details, the minimal solution is a $Q$-process. Moreover, BE and FE have the same minimal solution.
\end{block}

Laplace transform: $p_{ij}(\lambda)=\int_0^\infty e^{-\lambda
t}p_{ij}(t)dt$.
$$p_{ij}(\lambda)=\sum_{k\ne i}\frac{q_{ik}}{\lambda+q_i}p_{kj}(\lambda)+\frac{\delta_{ij}}{\lambda+q_i},$$
$$p_{ij}(\lambda)=\sum_{k\ne j}p_{ik}(\lambda)\frac{q_{kj}}{\lambda+q_j}+\frac{\delta_{ij}}{\lambda+q_i}.$$

Denote the minimal process by $P^{\text{min}}(\lambda)$. Consider
$P(\lambda)-P^{\text{min}}(\lambda)$: }
%----------------------------------------------------------------------------------------15

\frame{\frametitle{3. Uniqueness} \begin{block}{}$$u_i=\sum_{k\ne
i}\frac{q_{ik}}{\lambda+q_i}u_{k}, \quad 0\le u_i\le 1, i\in\mathbb
Z_+.$$\end{block}

\begin{block}{Uniqueness Theorem[Feller(1957), Reuter(1957)]} For a given totally stable and conservative $Q$-matrix $Q=(q_{ij})$, the $Q$-process is unique iff the equation has only the trivial solution $u_i\equiv0$ for some (equivalently, for all)
$\lambda>0$.
\end{block}
}
%----------------------------------------------------------------------------------------16
\frame{\frametitle{3. Uniqueness}
\small
Define
$${\widetilde m}_0=\frac{1}{q_{01}},\quad
{\widetilde m}_n=\frac 1{q_{n,n+1}} \bigg(1+\sum_{k=0}^{n-1}{\tilde
q}_n^{(k)}{\widetilde m}_k\bigg),\quad n\ge 1.$$

$${\widetilde m}_n
=\sum_{k=0}^n\frac{{\widetilde F}_n^{(k)}}{q_{k,k+1}},\qquad n\ge
0.$$

\begin{block}{Uniqueness[J.K. Zhang(1984), Yan \& Chen(1986)]}
Given a totally stable and conservative single birth $Q$-matrix
$Q=(q_{ij})$, the process is unique $($non-explosive$)$ iff
$\sum_{n=0}^\infty m_n=\infty$.
\end{block}

}

%----------------------------------------------------------17
\frame{\small Sketch of proof.  unique iff the solution $(u_i)$ to the
equation
$$(\lambda +q_i)u_i=\sum_{j\ne i}q_{ij}u_j, \qquad i\ge 0;\qquad u_0=1
$$
is unbounded for some (equivalently for all) $\lambda>0$. Rewrite it
as
$$\Omega u=Qu- \lambda u=0;\qquad u_0=1.$$
Applying Theorem 1 to $c_i\equiv -\lambda$ and $f_i\equiv 0$, we
obtain:
$$u_n=1+\lambda\sum_{0\le k \le n-1}\sum_{j=0}^k\frac{\widetilde F_k^{(j)}}{q_{j,j+1}}=1+\lambda\sum_{0\le k \le n-1}\widetilde m_k,\qquad n\ge 0.$$
Clearly, $u_n$ is increasing in $n$ and then is unbounded iff
$\sum_n {\widetilde  m}_n=\infty$. Thus, it remains to show that
$\sum_n {\widetilde  m}_n=\infty$ iff $\sum_n {m}_n=\infty$.
}
%----------------------------------------------------------------------------------------18
\frame{\frametitle{4. Recurrence, return/extinction probability} \small
\begin{itemize}
\item $P(t)$ is recurrent if for each $h>0$, $P(h)$ is recurrent. Equivalently, $\int_0^\infty p_{ii}(t)dt=\infty$.
\end{itemize}

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}

regular: totally stable, conservative and $Q$-process is unique. }
%----------------------------------------------------------------------------------------19
\frame{\frametitle{4. Recurrence, return/extinction probability} \small
\begin{block}{Theorem} For a regular and irreducible $Q=(q_{ij})$,  $P(t)$ is recurrent iff for some
(equivalently, all) $j_0$,
$$
x_i=\sum_{j\ne j_0,i}\frac{q_{ij}}{q_i}x_j,\qquad 0\le x_i\le
1,\,i\ge 0
$$
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 $f_{i,j_0}\equiv 1$. }
%----------------------------------------------------------------------------------------20
\frame{\frametitle{4. Recurrence, return/extinction probability}\small

\begin{block}{Recurrence Theorem[Yan \& Chen(1986)]}
Assume the single birth $Q$-matrix $Q=(q_{ij})$ is non-explosive and
irreducible. Then the process is recurrent iff $\sum_{n=0}^\infty
F_n^{(0)}=\infty$.\end{block}

Sketch of proof. recurrent iff the equation
$$
x_i=\sum_{k\ne 0}\Pi_{ik}x_k,\qquad 0\le x_i\le 1,\qquad i\ge 0
$$
has only zero solution, where $\Pi_{ik}=(1-\delta_{ik})q_{ik}/q_i$.
It is equivalent to that the solution to the equation
$$x_i=\sum_{k\ne 0}\Pi_{ik}x_k,\qquad i\ge 0; \qquad x_0=1$$
is unbounded. }

%----------------------------------------------------------21
\frame{\frametitle{4. Recurrence, return/extinction probability}\small

(contin.)

Rewrite the previous equation as
$$(Qx)_0=0,\qquad (Qx)_i=q_{i0},\qquad i\ge 1;\qquad x_0=1.$$
Applying Theorem 1 to $c_i\equiv 0$ and $f_i=q_{i0}(1-\delta_{i0})$,
we obtain the unique solution as follows
$$x_0=1,\quad x_n=1+\sum_{k=1}^{n-1}F_k^{(0)}=\sum_{k=0}^{n-1}F_k^{(0)},\quad n\ge 1.$$
Clearly, $(x_n)$ is unbounded iff $\sum_{k=0}^\infty
F_k^{(0)}=\infty$. In other words, the equation has only a trivial
solution iff $\sum_{k=0}^\infty F_k^{(0)}=\infty$. }


%----------------------------------------------------------------22
\frame{\frametitle{4. Recurrence, return/extinction probability}\small
Define $\sigma_0=\inf\{t\ge\text{the first jumping time:}\ X_t=0\}.$

\begin{block}{Proposition}
 Let the single birth $Q$-matrix $Q=(q_{ij})$ be non-explosive and irreducible. Then the return/extinction probability is as follows:
$$\mathbb P_0(\sigma_0<\infty)=\frac{\sum_{k=1}^{\infty}F_k^{(0)}}{\sum_{k=0}^\infty F_k^{(0)}}, \mathbb P_n(\sigma_0<\infty)=\frac{\sum_{k=n}^{\infty}F_k^{(0)}}{\sum_{k=0}^\infty F_k^{(0)}}, n\ge 1.$$
Furthermore, $\mathbb P_n(\sigma_0<\infty)=1$ for all $n\ge 0$ iff $\mathbb P_0(\sigma_0<\infty)=1$, equivalently iff $\sum_{n=0}^\infty F_n^{(0)}=\infty$.
\end{block}

P. J. Brockwell(1986), W. J. Anderson(1991)
}
%--------------------------------------------------------23

\frame{\small Sketch of proof. $(\mathbb P_i(\sigma_0<\infty): i\in E)$ is the minimal nonnegative solution to the equation
$$x_i=\sum_{k\ne 0, i}\frac{q_{ik}}{q_i}x_k + \frac{q_{i0}}{q_i}(1-\delta_{i0}),\qquad i\in E.$$
Then, we have $(Qx)_i=q_{i0}(1-\delta_{i0})(x_0-1),i\ge 0 $.
Applying Theorem 1 to $c_i\equiv 0$ and $f_i= q_{i0}(1-\delta_{i0})(x_0-1)$, we get
$$x_n=x_0 \bigg(1+\sum_{1\le k\le n-1} F_k^{(0)}\bigg)-\sum_{1\le k\le n-1}F_k^{(0)},\quad n\ge 0.$$
Because $x_n>0$, it follows that
$$x_0\ge \sup_{n\ge 1}\frac{\sum_{k=1}^{n-1}F_k^{(0)}}{\sum_{k=0}^{n-1}F_k^{(0)}}=1-\frac 1{\sum_{k=0}^\infty F_k^{(0)}}.$$
From here, we obtain the minimal nonnegative solution:
$$x_0^*=1-\frac 1{\sum_{k=0}^\infty F_k^{(0)}},\qquad x_n^*=1-\frac{\sum_{k=0}^{n-1}F_k^{(0)}}{\sum_{k=0}^\infty F_k^{(0)}},\quad n\ge 1.$$
}

%----------------------------------------------------------------24
\frame{\frametitle{4. Recurrence, return/extinction probability}\small
Consider the occupation time of the state $j$:
 $$T_j:=\int_0^\infty \mathbf 1_{\{X(t)=j\}}\textrm{d}\,t.$$
  By the return probability and the embedding chain, we get

 \begin{block}{Proposition[P.S. Li \& Z.(2016)]} Let the single birth $Q$-matrix $Q=(q_{ij})$ be regular and irreducible. Then the mean occupation time
$$\mathbf E_iT_j= \frac{1}{q_{j,j+1}}\sum^\infty_{n=i\vee j} F_n^{(j)}.$$
\end{block}
}

%----------------------------------------------------------------25
\frame{\frametitle{5. Laplace transform of return/extinction time}
Take $c_i\equiv -\lambda<0$. Define $\big(\widetilde F_k^{(i)}\big)$ and

$$\tilde d_0=0,\ \tilde d_n=\frac 1{q_{n,n+1}}\Bigg(1+\sum_{k=0}^{n-1}\tilde q_n^{(k)}\tilde d_k\Bigg),\ n\ge 1.$$
$$\text{Equivalently,}\ \tilde d_n=\sum_{1\le j\le n}\frac{\widetilde F_n^{(j)}}{q_{j,j+1}},\  n\ge 0.$$


}
%----------------------------------------------------------------26
\frame{\frametitle{5. Laplace transform of return/extinction
time}\small

\begin{block}{Proposition}
Let the single birth process be
recurrent. Then the Laplace transform of $\sigma_0$ is given by
$$\mathbb E_0\text{e}^{-\lambda\sigma_0}=\frac {q_{01} (1-\lambda\tilde d)}{q_{01}+\lambda},
 \mathbb E_n\text{e}^{-\lambda\sigma_0}=1- \lambda \sum_{k=0}^{n-1}\Big(\widetilde F_k^{(0)}\tilde d-\tilde d_k\Big), n\ge 1,$$
where
$$\tilde d=\lim_{n\to\infty}\frac{\sum_{k=0}^{n-1}\tilde d_k}{\sum_{k=0}^{n-1}\widetilde F_k^{(0)}}
=\lim_{n\to\infty}\frac{{\tilde d}_n}{ {\widetilde F}_n^{(0)}} \quad
\text{\sf if the limit exists}.$$
\end{block}

P. J. Brockwell(1986), W. J. Anderson(1991)
}
%--------------------------------------------------------27

\frame{\small Sketch of proof. $(\mathbb E_i\text{e}^{-\lambda\sigma_0}: i\in
E)$ is the minimal solution $(x_i^*)$ of the following equation
$$x_i=\frac{1}{q_i+\lambda}\sum_{k\notin \{0,i\}}
  q_{ik}x_k+\frac{q_{i0}(1-\delta_{i0})}{q_i+\lambda}, \qquad x_i\ge1,\qquad i\in E. $$
Then, we have
$$
(Qx)_i-\lambda x_i=q_{i0}(x_0-1),\quad i\ge 1;\quad (Qx)_0-\lambda
x_0=0.
$$
Applying Theorem 1 to $c_i\equiv -\lambda$ and
$f_i=q_{i0}(1-\delta_{i0})(x_0-1)$ for all $i\ge 0$,
$$\aligned x_n&=x_0\bigg(1+\frac \lambda{q_{01}}\bigg)\sum_{k=0}^{n-1}\widetilde F_k^{(0)}-
\sum_{k=0}^{n-1}\big(\widetilde F_k^{(0)}-\lambda\tilde
d_k\big)+1\\
&=:x_0 \alpha_{n-1}-\beta_{n-1}, \quad n\ge 1.\endaligned$$ }
%----------------------------------------------------28
\frame{\small

By the minimal nonnegative property, $x_0^*=\sup_{n\ge
1}\beta_{n}/\alpha_{n}$, then
$$x_0^*=\varlimsup_{n\to\infty}\frac{\beta_{n}}{\alpha_{n}}.$$
We can replace $\varlimsup_{n\to\infty}$ by $\lim_{n\to\infty}$. On
the one hand, since $x_n\in (0, 1]$,
$$\frac{\beta_{n}}{\alpha_{n}}< x_0\le \frac{\beta_{n}+1}{\alpha_{n}},\qquad n\ge 1.$$
On the other hand,  we can prove that $\sum_{k}{\widetilde
F}_k^{(0)}=\infty$ since $\sum_{k}{F}_k^{(0)}=\infty$ by the
recurrent assumption. Therefore, we have
$$x_0^*=
\frac{q_{01}}{q_{01}+\lambda}\lim_{n\to\infty}\bigg[1-\lambda
\frac{\sum_{k=0}^{n-1} {\tilde d}_k}{\sum_{k=0}^{n-1}{\widetilde
F}_k^{(0)}}\bigg]= \frac{q_{01}}{q_{01}+\lambda}
\big[1-\lambda{\tilde d}\,\big].$$ \vskip-0.3cm
$$\hskip-2.2cm
x_n^*=(1-\lambda {\tilde d})\sum_{k=0}^{n-1}\widetilde
F_k^{(0)}-\sum_{k=0}^{n-1}\big(\widetilde F_k^{(0)}-\lambda\tilde
d_k\big)+1 $$
 \vskip-0.3cm
$$\hskip-3.2cm=1- \lambda
\sum_{k=0}^{n-1}\Big(\widetilde F_k^{(0)}\tilde d-\tilde
d_k\Big),\quad n\ge 1. $$}
%---------------------------------------------29
\frame{
\begin{block}{Example [uniform catastrophes]} Let
$q_{i,i+1}=b\, i$, $i\ge 0$; $q_{ij}=a$, $j=0,1,\dots, i-1$; and
$q_{ij}=0$ for other $j>i+1$, where $a$ and $b$ are positive
constants. Then the extinction of the process has an exponential
distribution
$${\mathbb E}_n\text{e}^{-\lambda\tau_0}=\frac{a}{a+\lambda},
\qquad \lambda>0,\; n\ge 1.$$ It is surprising that the distribution
is independent of $b$ and the starting point $n$. Redefine
$q_{01}=1$. Then the irreducible process is indeed strongly ergodic.
\end{block}

P. J. Brockwell et al(1982)
}
%----------------------------------------------------------------------------30

\frame{\frametitle{6. Summary}

Roughly speaking, the unified treatment consists of the following
three steps.
\begin{itemize}
\item[(a)] Find out the Poisson equation corresponding to the problem .
\item[(b)] Apply Theorem 1 to get the solution to Poisson equation.
\item[(c)] Work out the answer for the problem using the solution obtained in (b).
\end{itemize}
}
%---------------------------------------------31
\frame{\small\vskip0.5cm
\begin{center}{\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|c|c|c|}
  \hline
  Problem                  & $c_i\in {\mathbb R}$                   & $f_i\in {\mathbb R}$
  \\ \hline
  Harmonic function               & $c_i\in {\mathbb R}$   &
  $f_i\equiv 0$ \\ \hline
  Uniqueness                & $c_i\equiv -\lambda<0$   & $f_i\equiv 0$ \\ \hline
  Recurrence                & $c_i\equiv 0$            & $f_i=q_{i0}(1-\delta_{i0})$\\ \hline
  Ext./return probability    &$c_i\equiv 0$             & $f_i=q_{i0}(1-\delta_{i0})(g_0-1)$ \\ \hline
  Ergodicity                & $c_i\equiv 0$            & $f_i=q_{i0}(1-\delta_{i0})g_0-1$\\ \hline
  Strong ergodicity         & $c_i\equiv 0$            & $f_i=q_{i0}(1-\delta_{i0})g_0-1$\\ \hline
  Polynomial moment         & $c_i\equiv 0$            & $f_i^{(\ell)}$\\ \hline
  Exp. moment/ergod.        & $c_i\equiv\lambda>0$     & $f_i=q_{i0}(1-\delta_{i0})(g_0-1)$ \\ \hline
  Laplace transform  & $c_i\equiv -\lambda<0$ & $f_i=q_{i0}(1-\delta_{i0})(g_0-1)$ \\ \hline
\end{tabular}}\end{center}
where $f_i^{(\ell)}=q_{ii_0}(1-\delta_{ii_0})g_{i_0}-\ell\mathbb
E_i\sigma_{i_0}^{\ell-1}$.
}

%-----------------------------------------------------------------------------------------32
\frame{%\frametitle{6. Stationary distribution of single death proc.}
\small

Let the single death $Q$-matrix $Q=(q_{ij})$ be regular and irreducible. Assume that the corresponding process is ergodic.
To get the stationary distribution of the process, solve
$$\pi Q=\mathbf 0.$$

\begin{block}{Proposition[L.D. Wang \& Z.(2014)]} The stationary distribution of the single death process is represented as follows:
$\pi_n={\mu_n}/{\mu},\ n\ge 0$, where
$$\mu_0=1,\ \mu_n=\frac 1{q_{n,n-1}}\sum_{k=0}^{n-1}\bar q_k^{(n)}\mu_k,\ n\ge 1;$$
$$\mu=\sum_{n=0}^\infty \mu_n;\ \bar q_k^{(n)}=\sum_{j=n}^\infty q_{kj},\ n>k.$$
\end{block}
}
%----------------------------------------------------------------------------------------33

\frame{%\frametitle{6. Stationary distribution of single death proc.}
\small


Transpose the matrix equation $\pi Q=\mathbf 0$ to
$$Q^{\text{T}}\pi^{\text{T}} =\mathbf 0^{\text{T}}.$$
Then
$$Q^{\text{T}}g=:\Omega g=Q^*g+cg=\mathbf 0^{\text{T}},$$
where $Q^*$ is a totally stable and conservative single birth $Q$-matrix and
\begin{block}{}
$$c_i=\sum_{j=0}^{i-1}q_{ji}+q_{i+1,i}-\sum_{j= i+1}^\infty q_{ij}-q_{i,i-1},\ i\ge0; $$
$$f_i\equiv  0.$$\end{block}
}

%----------------------------------------------------------------------------------------34
\frame{%\frametitle{6. Stationary distribution of single death proc.}
\small
\begin{block}{Proposition[Z.(2016)]} The stationary distribution of the single death process is represented as follows:
$\pi_n={\mu_n}/{\mu},\ n\ge 0$, where
$$\mu_0=1,\  \mu_n=1-\sum_{k=0}^{n-1}\sum_{j=0}^k\frac{\widetilde F_k^{(j)}c_j}{q_{j+1,j}}, n\ge 1;\ \mu:=\sum_{n=0}^\infty \mu_n;$$
\begin{align}
&\widetilde F_i^{(i)}=1,\quad  \widetilde F_n^{(i)}=\frac 1{q_{n+1,n}}
\sum_{k=i}^{n-1}\tilde q_n^{(k)}\widetilde F_k^{(i)},\quad n>i\ge 0,\nonumber\\
&\tilde q_n^{(k)}=\sum_{j=0}^kq_{jn}-c_n,\quad n>k\ge 0.\nonumber\end{align}
\end{block}
}

%---------------------------------------------------------------------- 35
\frame{\frametitle{ 7. Main references}\small
\begin{itemize}\small

\item  Anderson W J. Continuous-Time Markov Chains: An Applica\-tions-Oriented Approach. New York: Springer-Verlag, 1991\\[-6.5mm]

\item Brockwell P J. The extinction time of a general birth and death processes with catastrophes. J Appl Prob, 1986, 23: 851--858\\[-6.5mm]

\item Brockwell P J, Gani J, Resnick S I. Birth, immigration and catastrophe processes. Adv Appl Prob, 1982, 14: 709--731\\[-6.5mm]

\item Chen M F. From Markov Chains to Non-Equilibrium Particle Systems (2nd Edition). Singapore: World Scientific, 2004\\[-6.5mm]

\item Chen M F. Single birth processes. Chinese Ann Math, 1999, 20B: 77--82\\[-6.5mm]

\item Chen M F. Explicit criteria for several types of ergodicity. Chinese J Appl Prob Stat, 2001, 17(2): 1--8\\[-6.5mm]

%\item Chen M F, Zhang X. Isospectral operators. 2014, preprint\\[-6.5mm]

\end{itemize}
}
%---------------------------------------------------------------------- 36
\frame{%\frametitle{6. Main references(contin.)}
\small

\begin{itemize}
\item {\blue Chen M F, Zhang Y H. Unified representation of formulas for single birth processes, Frontiers of Mathematics in China, 2014,
9(4): 761-796}\\[-6.5mm]

\item Mao Y H, Zhang Y H. Exponential ergodicity for single-birth processes. J Applied Probab, 2004, 41: 1022--1032\\[-6.5mm]

\item Yan S J, Chen M F. Multidimensional $Q$-processes. Chinese Ann Math, 1986, 7B: 90--110\\[-6.5mm]

\item Zhang J K. On the generalized birth and death processes \text{\rm (I)}. Acta Math Sci, 1984, 4: 241--259\\[-6.5mm]

\item Zhang Y H. Strong ergodicity for single-birth processes. J Appl Prob, 2001, 38(1): 270--277\\[-6.5mm]

\item Zhang Y H. Moments of the first hitting time for single birth processes. J BNU, 2003, 39(4): 430--434 \\[-6.5mm]

\item Zhang Y H. The hitting time and stationary distribution for single birth processes. J BNU, 2004, 40(2): 157--161\\[-6.5mm]
\end{itemize}
}


%----------------------------------------------------37
\frame{

\centerline{ \emph{\textbf{\color{blue} Thank you for your attention!}}}
\vskip0.5cm
\centerline{Homepage: http://math0.bnu.edu.cn/${}^\sim$zhangyh/}

}

\end{CJK*}

\end{document}
