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\begin{document}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%            题页            %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title[Single death processes]{Notes on single death processes}

\author[Y.H. Zhang]{Yuhui Zhang(张余辉)}

\date[2017.06.30]{\wuhao (IMS-China 2017, Nanning(广西民族大学), June 29-July 1, 2017) }

\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 Motivation
\item Single birth processes
\item Dual approach
\item Zero-entrance
\item Mean hitting time
\end{enumerate}
}

%----------------------------------------------------------------------------------------2
\frame{ \frametitle{1. Motivation}\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. Motivation}\small Single death process:
$Q=(q_{ij})$ satisfies
$$q_{i,i-1}>0,\ q_{i,i-j}=0,\ i\ge 1,j> 1.$$
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\vskip0.3cm
$$Q=\left(%
\begin{array}{ccccc}
  - & * & * & * & \cdots \\
  + & - & *& * & \cdots \\
  0 & + & - & * & \cdots\\
  \vdots & \vdots & \vdots & \vdots & \ddots \\
  \end{array}%
\right)$$
}
%----------------------------------------------------------------------------------------4

\frame{ \frametitle{1. Motivation}\small
\begin{itemize}

\item branching processes: reference model

\end{itemize}

$$\aligned
 i&\to i-1\quad\text{at rate}\quad ib_0\\
  &\to i\qquad\quad \text{at rate}\quad ib_1=-i\\
  &\to i+1\quad\text{at rate}\quad ib_2\\
  &\to i+2\quad\text{at rate}\quad ib_3\\
  &\to \cdots
\endaligned$$
$$b_1=-\sum_{k\ne 1}b_k=-1$$
}
%----------------------------------------------------------------------------------------5
\frame{\frametitle{1. Motivation}
\small \begin{itemize}
\item single birth process
\end{itemize}

$$q_{i,i+1}>0,\ q_{i,i+j}=0,\ i\ge 0,j> 1.$$
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\vskip0.3cm
$$Q=\left(%
\begin{array}{ccccc}
  - & + & 0 & 0 & \cdots \\
  * & - & + & 0 & \cdots \\
  * & * & - & + & \cdots\\
  \vdots & \vdots & \vdots & \vdots & \ddots \\
  \end{array}%
\right)$$
}

%-----------------------------------------------------------6

\frame{\small
\begin{itemize}
\item Uniqueness. J.K. Zhang(1984), S.J. Yan \& M.F. 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). Chen \& Z. (2014)
\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. Brockwell(1986), Anderson(1991)
\item Poisson equation. Chen \& Z.(2014)
\item Mean occupation time. P.S. Li \& Z.(2017)
\end{itemize}
}

%----------------------------------------------------------------------------7

\frame{\frametitle{2. Single birth processes}\small

For $0\leq k<n$, define $q_n^{(k)}=\sum_{j=0}^k q_{nj}$ and
 $$F_i^{(i)}=1, \ F_n^{(i)}=\frac 1{q_{n,n+1}} \sum_{k=i}^{n-1}q_n^{(k)}F_k^{(i)},\ 0\le i<n.$$
Define
$$m_n
=\sum_{k=0}^n\frac{F_n^{(k)}}{q_{k,k+1}},\qquad n\ge
0.$$

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


}
%----------------------------------------------------------------------------8

\frame{\frametitle{2. Single birth processes}\small

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

%----------------------------------------------------------------------------------------9
\frame{\small
Define
$$d_n=\sum_{k=1}^n\frac{F_n^{(k)}}{q_{k,k+1}},\qquad n\ge 0.$$

\begin{block}{Ergodicity(Yan \& Chen(1986), Z.(2001)}
Assume the single birth $Q$-matrix $Q=(q_{ij})$ is regular and irreducible, and the process is recurrent. Then
the process is ergodic if and only if
$$d:=\varlimsup_{k\to\infty}\frac{\sum_{n=0}^k d_n}{\sum_{n=0}^k F_n^{(0)}}<\infty;$$
the process is strongly ergodic if and only if
$$\sup_{k\ge 0}\sum_{n=0}^k(F^{(0)}_nd-d_n)<\infty.$$
\end{block}


}
%----------------------------------------------------------------------------------------10
\frame{\frametitle{2. Single birth processes}\small

\begin{block}{Stationary distribution[Z.(2004)]} Assume that the process is ergodic. Define
$$c_k:=\varlimsup_{i\to\infty}\frac{\sum_{j=k}^i m_j}{\sum_{j=k}^iF_j^{(k)}}.$$
Then the stationary distribution
$$\pi_k=\frac 1{q_{k,k+1}c_k},\quad k\ge 0.$$
\end{block}

}
%----------------------------------------------------------------11
\frame{\frametitle{2. Single birth processes}\small
Define $\sigma_0=\inf\{t\ge\text{the first jumping time:}\ X_t=0\}.$

\begin{block}{Return/extinction probability[Brockwell(1986), Chen \& Z.(2014)]}
 Let the single birth $Q$-matrix $Q=(q_{ij})$ be non-explosive and irreducible. Then the return/extinction probability is as follows:
$$\mathbb P_0(\sigma_0<\infty)=\frac{\sum_{k=1}^{\infty}F_k^{(0)}}{\sum_{k=0}^\infty F_k^{(0)}}, \mathbb P_n(\sigma_0<\infty)=\frac{\sum_{k=n}^{\infty}F_k^{(0)}}{\sum_{k=0}^\infty F_k^{(0)}}, n\ge 1.$$
\end{block}

\begin{block}{[Z.(2004)]}
$$\mathbb E_i\sigma_0=\sum_{k=0}^{i-1}(F_k^{(0)}d-d_k),\quad i\ge 1.$$
\end{block}
}
%----------------------------------------------------------------------------12

\frame{\frametitle{2. Single birth processes}\small
$$\text{Poisson}: Qg+cg=f$$

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 Chen \& Z.(2014) to get the solution to Poisson equation.
\item[(c)] Work out the answer for the problem using the solution obtained in (b).
\end{itemize}

{\blue M.F. Chen \& Z.(2014): Unified representation of formulas for single birth processes, Frontiers of Mathematics in China,
9(4): 761-796.}
}
%---------------------------------------------13
\frame{\frametitle{2. Single birth processes}\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 $f_i^{(\ell)}=q_{ii_0}(1-\delta_{ii_0})g_{i_0}-\ell\mathbb
E_i\sigma_{i_0}^{\ell-1}$.
}

%-----------------------------------------------------------------------------------------14
\frame{\frametitle{3. Dual approach }
\small

Given a $Q$-matrix $Q$, denote by $P(t)=(P_{ij}(t))$ the corresponding minimal $Q$-process. The $Q$-process $P(t)$
is said to be {\blue stochastically monotone} if
$$
\sum_{j\geq k} P_{ij}(t)\leq \sum_{j\geq k} P_{mj}(t), \quad i\le m, k=0,1,2,\cdots.
$$
{\blue$\Rightarrow$}
$$
\sum_{j\geq k} q_{ij}\leq \sum_{j\geq k} q_{mj},\quad i\le m, k\in\{0,\cdots,i\}\cup\{m+1, m+2,\cdots\}.
$$
We also say a $Q$-matrix $Q=(q_{ij})$ is {\blue monotone} if the inequality above holds. If a $Q$-matrix is regular and monotone, then the minimal $Q$-process is also {\blue stochastically monotone}.
}
%-----------------------------------------------------------------------------------------15
\frame{\frametitle{3. Dual approach }
\small

\begin{block}{Propositioin[Siegmund(1976)]}
A transition function $P(t)$ is stochastically monotone if and only if there exists a dual process $\widetilde{P}(t)$ of $P(t)$ such that
$$
\widetilde{P}(t)\Lambda=\Lambda P(t)^{\text{T}},
$$
where
$$
\Lambda(i, j)=1_{\{i\leq j\}}.
$$
\end{block}

\vskip0.2cm{\blue
D. Siegmund(1976): The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes. Ann. Probab. 41, 914-924.}
}

%-----------------------------------------------------------------------------------------16
\frame{\frametitle{3. Dual approach }
\small

\begin{block}{Theorem[M.F. Chen, P.S. Li \& Z.(2017+)]}
Let $Q=(q_{ij})$ be a regular stochastically monotone single birth $Q$-matrix and  let $\tilde{Q}$ be the dual of $Q$: $\tilde Q=\Lambda Q^{\text{T}}\Lambda^{-1}$. Denote the single birth process by $(X_t)_{t\geq 0}$ and denote the (dual) single death process by $(\widetilde{X}_t)_{t\geq 0}$. Then we have:

(1) The (dual) single death matrix $\widetilde{Q}$ is regular if and only if the single birth process $(X_t)_{t\geq 0}$ is not strongly ergodic, i.e.,
$$\sup_{i\geq 1}\sum^{i-1}_{j=0}(F_j^{(0)}d-d_j)=\infty.$$

\end{block}
}
%-----------------------------------------------------------------------------------------17
\frame{\frametitle{3. Dual approach }
\small

\begin{block}{Theorem(continued)}
(2) The single birth process is ergodic if and only if for any initial state $i> 0$, the extinction probability of the (dual) single death process $\mathbf P_i(\tilde{\sigma}_0<\infty)<1$. Indeed the extinction probability is given by
$$\mathbf P_i(\tilde{\sigma}_0<\infty)=1-\sum^{i-1}_{j=0} \pi_j,\quad i=1,2,\cdots.$$
\end{block}
}
%-----------------------------------------------------------------------------------------18
\frame{\frametitle{3. Dual approach }
\small

\begin{block}{Theorem(continued)}
(3) Denote the occupation time of the dual single birth process by $\widetilde{T}_j=\int^\infty_0 1_{\{\widetilde{X}_t=j\}}dt$ $j=1,2,\ldots$. Then
$$
\mathbf E_i(\tilde{T}_j)=\int^\infty_0 \mathbf P_{i}(\tilde{X}_t= j)dt=\sum^{i-1}_{k=0}\Bigg(1_{\{k\leq j-1\}}\frac{F^{(k)}_{j-1}}{q_{k, k+1}}-m_{j-1}\pi_k\Bigg).
$$
Moreover, when the extinction probability of the (dual) single death process is $1$, then
$$
\mathbf E_i\tilde{T}_j=\sum^{i-1}_{k=0}1_{\{k\leq j-1\}}\frac{F^{(k)}_{j-1}}{q_{k, k+1}}=\sum^{i\wedge j-1}_{k=0}\frac{F^{(k)}_{j-1}}{q_{k, k+1}}.
$$
\end{block}
}
%-----------------------------------------------------------------------------------------19
\frame{\frametitle{3. Dual approach }
\small

\begin{block}{Theorem(continued)}
(4) Define the explosion time of the (dual) single death process $\tilde{\zeta}=\lim_{n\rightarrow\infty} \tilde{\zeta}_n$, where $\tilde{\zeta}_n= \inf\{t\geq0: \widetilde{X}_t\geq n\}$. The conditional expectation of the explosion time and the extinction time are given by
$$
\mathbf E_i(\tilde{\zeta}|\tilde{\sigma}_0=\infty)
=
\sum^\infty_{j=1}\frac{\sum^{j-1}_{k=0} \pi_k}{\sum^{i-1}_{k=0} \pi_k}\sum^{i-1}_{k=0}\Bigg(1_{\{k\leq j-1\}}\frac{F^{(k)}_{j-1}}{q_{k, k+1}}-m_{j-1}\pi_k\Bigg),
$$
$$\mathbf E_i(\tilde{\sigma_0}|\tilde{\sigma}_0< \infty)\!\!= \!\!\sum^\infty_{j=1}\frac{1\!-\!\sum^{j-1}_{k=0} \pi_k}{1\!-\!\sum^{i-1}_{k=0} \pi_k}\sum^{i-1}_{k=0}\Bigg(1_{\{k\leq j-1\}}\frac{F^{(k)}_{j-1}}{q_{k, k+1}}-m_{j-1}\pi_k\Bigg).
$$

\end{block}
}
%-----------------------------------------------------------------------------------------20
\frame{\frametitle{3. Dual approach }
\small

\begin{block}{Theorem(continued)}
If the extinction probability of the (dual) single death process is $1$, then the mean extinction time of the (dual) single death process is given by
$$
\mathbf E_i\tilde{\sigma}_0= \sum^\infty_{j=1}\sum^{i\wedge j-1}_{k=0}\frac{F_{j-1}^{(k)}}{q_{k,k+1}},\qquad i=1,2,\cdots.
$$
\end{block}

\vskip0.3cm
{\blue H.J. Zhang, A.Y. Chen, X. Lin, Z.T. Hou(2001): Strong ergodicity of monotone transition functions, Statist. Probab. Lett. 55: 63-69.
}

}

%-----------------------------------------------------------------------------------------21
\frame{\frametitle{4. Zero-entrance }\small

Define $c_i:=q_i-\sum_{j\ne i}q_{ij}$ (non-conservative quantity).

\begin{block}{[L.D. Wang \&Z.(2014)]} Given a totally stable single birth $Q$-matrix $Q=(q_{ij})$. Then the single birth $Q$-matrix is zero-exit(equivalently, $BQ$-process is unique) if and only if
$$R:=\sum_{n=0}^\infty m_n=\infty,$$
$$
m_0=\frac {1+c_0}{q_{01}},\ m_n=\frac 1{q_{n,n+1}}\bigg(1+c_n+\sum_{k=0}^{n-1}(c_n+q_n^{(k)})m_k\bigg),\ n\ge 1.
$$
\end{block}

\begin{block}{}$$\lambda x=Qx,\ 0\le x\le 1.$$\end{block}
}
%-----------------------------------------------------------------------------------------21
\frame{\frametitle{4. Zero-entrance }\small
\begin{block}{[L.D. Wang \& Z.(2014)]}

Given a totally stable single death $Q$-matrix $Q=(q_{ij})$. Define $\bar q_n^{(k)}=\sum_{j=k}^\infty q_{nj}$ $(k>n)$. Then the single death $Q$-matrix is zero-entrance if and only if
~$\overline R:=\sum_{n=0}^\infty \overline m_n=\infty$, where
$$\overline m_0=0,\ \overline m_n=\frac 1{q_{n,n-1}}\bigg(1+\sum_{k=0}^{n-1}(c_k+\bar q_n^{(k)})\overline m_k\bigg),\ n\ge 1.$$
If the single death $Q$-matrix is non-conservative, then $FQ$-process is unique if and only if $\overline R=\infty$.
\end{block}

\begin{block}{}$$\lambda y=yQ,\  y\ge 0, ||y||_1<\infty.$$\end{block}
}
%-----------------------------------------------------------------------------------------22
\frame{\frametitle{4. Zero-entrance }\small
\begin{block}{[L.D. Wang \& Z.(2014)]}

Given a totally stable single death $Q$-matrix $Q=(q_{ij})$. Then there exists a unique invariant measure as follows:
$$\mu_0=1,\ \mu_n=\frac 1{q_{n,n-1}}\sum_{k=0}^{n-1}(c_k+\bar q_n^{(k)})\mu_k,\ n\ge 1.$$
\end{block}

The stationary distribution of the single death process is represented as follows:
$\pi_n={\mu_n}/{\mu},\ n\ge 0$, where
$$\mu_0=1,\ \mu_n=\frac 1{q_{n,n-1}}\sum_{k=0}^{n-1}\bar q_n^{(k)}\mu_k,\ n\ge 1;$$
$$\mu=\sum_{n=0}^\infty \mu_n.$$
}
%-----------------------------------------------------------------------------------------23
\frame{\frametitle{4. Zero-entrance }\small
\small

$$\pi Q=\mathbf 0.$$

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}
}

%----------------------------------------------------------------------------------------24
\frame{\frametitle{4. Zero-entrance }
\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}
}

%----------------------------------------------------------------------------------------24
\frame{\frametitle{5. Mean hitting time }

\small \begin{itemize}
\item single birth process
\end{itemize}
$$m_n
=\sum_{k=0}^n\frac{F_n^{(k)}}{q_{k,k+1}},\ n\ge
0.$$

$$\mathbb E_n\tau_{n+1}=m_n=\sum_{k=0}^n\frac{F_n^{(k)}}{q_{k,k+1}},\ n\ge
0.$$

\begin{block}{}
$$x_n=\frac 1{q_{n}}\bigg(1+\sum_{k=0}^{n-1}q_n^{(k)}x_k+q_n^{(n-1)}x_n\bigg),\ n\ge 0.$$
\end{block}

\begin{block}{}
$$F_i^{(i)}=1, \ F_n^{(i)}=\frac 1{q_{n,n+1}} \sum_{k=i}^{n-1}q_n^{(k)}F_k^{(i)},\ 0\le i<n.$$
\end{block}
}
%----------------------------------------------------------------------------------------25
\frame{\frametitle{5. Mean hitting time }
\small \begin{itemize}
\item single death process: $\mathbb E_n\tau_{n-1}=?$
\end{itemize}
\begin{block}{}
$$x_n=\frac 1{q_{n}}\bigg(1+\sum_{k=n+1}^{\infty}\bar q_n^{(k)}x_k+\bar q_n^{(n+1)}x_n\bigg),\ n\ge 1.$$
\end{block}

$$\bar q_n^{(k)}:=\sum_{j=k}^\infty q_{nj},\  k>n\ge 0$$

\begin{block}{}
$$
G_i^{(i)}=1,\  G_n^{(i)}=\frac 1{q_{n,n-1}}
\sum_{k=n+1}^{i} q_n^{(k)} G_k^{(i)},\ 0\le n< i.
$$
\end{block}
}
%----------------------------------------------------------------------------------------25
\frame{\frametitle{5. Mean hitting time }\small

\begin{block}{}
$$x_n=\frac 1{q_{n}}\bigg(1+\sum_{k=n+1}^{\infty}\bar q_n^{(k)}x_k+\bar q_n^{(n+1)}x_n\bigg),\ n\ge 1.$$
\end{block}

\begin{block}{}
$$\mathbb E_n\tau_{n-1}=\sum_{k=n}^\infty\frac{G_n^{(k)}}{q_{k,k-1}} (n\ge 1)?$$
\end{block}

\begin{block}{The minimal nonnegative solution}
$$h_n=\sum_{k=n}^\infty\frac{G_n^{(k)}}{q_{k,k-1}} (n\ge 1).$$
\end{block}


\begin{block}{}
$$\sum_{n=1}^\infty\sum_{k=n}^\infty\frac{G_n^{(k)}}{q_{k,k-1}}<\infty \Rightarrow \exists\ \text{a QSD uniquely}.$$
\end{block}


}

%----------------------------------------------------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}
