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\title[Integral-type functionals for three $Q$-processes]{Integral-type functionals for three $Q$-processes}

\author[张余辉]{\sihao 张余辉}

\date[2020年11月14日]{\wuhao (2020年11月13-15日, 北师大随机分析研讨交流会, 北师大珠海校区) }

\institute[北京师范大学]{\wuhao 北京师范大学}

%\frame{\titlepage}


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  \includegraphics[scale=0.3]{math.png}\\
\maketitle
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%----------------------------------------------------------------------------------------1
\begin{frame}
  \frametitle{CONTENTS}
  \tableofcontents%[sections]
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Background}

%----------------------------------------------------------------------------------------4
\frame{ \frametitle{Background}\small
$$Y(t)=\int_0^t f(X(s),s)\text{d}s.$$
\begin{itemize}
\item the theory of inventories and storage, moments and distributions
\item queueing, traffic jam and intersection bottlenecks
\item biology, the joint distribution of $X(t)$ and $Y(t)$
\item limiting behavior (including sub-exponentially ergodic theory and CLTs)
\end{itemize}
}
%----------------------------------------------------------------------------------------2
\frame{ \frametitle{Background}\small
\begin{itemize}
\item $Q$-process: on $\mathbb Z_+:=\{0, 1, 2, \dots\}$, suppose that sub-Markov transition probability matrix $P(t)=(p_{ij}(t), i, j\in\mathbb Z_+)$ satisfies

$Q$-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}
}

%----------------------------------------------------------------------------------------3
\frame{\frametitle{Background}\small

Define
$$\tau_i=\inf\{t\geqslant 0: X(t)=i\}, \ \sigma_i=\inf\{t\geqslant \text{the first jumping time}: X(t)=i\}.$$

Given $V\geqslant 0$ but not equal to $0$ identically. Consider the integral-type functional:
$$\xi_i=\int_0^{\tau_i}V(X(t))\text{d}t\quad\text{or}\quad\int_0^{\sigma_i}V(X(t))\text{d}t.$$

When $V\equiv 1$, $\xi_i=\tau_i$.

When $V(1)=V(2)=1$ and $V(j)=0, j\ne 1,2$, then $\xi_0$ is the total time that the process stays in the states $1$ and $2$ before hitting $0$ firstly.
}

%----------------------------------------------------------------------------------------5
\frame{\frametitle{Background}\small
\begin{itemize}
\item Z.K. Wang . On distribution of functions of birth and death processes and their applications in the theory of queues, Scientia Sinica, 1961, X(2): 160-170.
\item D.R. McNeil. Integral functionals of birth and death processes and related limiting distributions, The Annals of Mathematical Statistics, 1970, 41(2):480-485.
\item 吴立德. 齐次可数马尔科夫过程积分型泛函的分布, 数学学报, 1963, 13(1): 86-93.
\item 杨超群. 可列马氏过程的积分型泛函和双边生灭过程的边界性质, 数学进展, 1964, 7(4): 397-424.

\end{itemize}
}
%----------------------------------------------------------------------------------------6
\frame{\frametitle{Background}\small
\begin{itemize}
\item 侯振挺, 郭青峰. 齐次可列马尔可夫过程, 科学出版社, 1978.
\end{itemize}

\begin{block}{} $({\mathbb E}_i\xi_{i_0}^\ell: i\in E)$ is the minimal nonnegative solution to:
$$
y_i=\sum_{k\ne i, i_0}\frac{q_{ik}}{q_i}y_k+\frac{\ell V(i)}{q_i} {\mathbb E}_i \xi_{i_0}^{\ell-1},\qquad i\in E.
$$
\end{block}

Define $\psi_{i,i_0}(\lambda)=1-\mathbb E_i\text{e}^{-\lambda \xi_{i_0}}$.

\begin{block}{} $(\psi_{i,i_0}(\lambda): i\in E)$ is the minimal nonnegative solution to:
$$
y_i=\sum_{k\ne i, i_0}\frac{q_{ik}}{\lambda V(i)+q_i}y_k+\frac{\lambda V(i)}{\lambda V(i)+q_i},\qquad i\in E.
$$
\end{block}
}
%----------------------------------------------------------------------------------------7
\frame{\frametitle{Background}\small
\begin{itemize}
\item J.K. Zhang. On the generalized birth and death processes (I)--the numeral introduction, the functional of integral type and the distributions of runs and passage times, Acta Math Sci, 1984, 4(2): 191-209.
\item Y.Y. Liu \& Y.H. Song. Integral-type functionals of first hitting times for continuous-time Markov chains, Front. Math. China, 2018, 13(3): 619-632.
\item E. L\"ocherbach, O. Loukianav, D. Loukianova. Spectral condition, hitting times and Nash inequality, Annales de l'I.H.P. Probabilit\'es et statistiques, 2014, 50(4): 1213-1230.
\end{itemize}

$$\xi_{i_0}=\int_0^{\tau_{i_0}}r(t)V(X(t))\text{d}t.$$
}
\section{Single birth process}
%----------------------------------------------------------------------------------------9
\frame{\frametitle{Single birth process}
$Q=(q_{ij})$ satisfies
$${\red{q_{i,i+1}>0}},\ q_{i,i+j}=0,\ i\ge 0,j> 1.$$
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\vskip0.3cm
$$Q=\left(%
\begin{array}{ccccc}
  - & + & 0 & 0 & \cdots \\
  * & - & + & 0 & \cdots \\
  * & * & - & + & \cdots\\
  \vdots & \vdots & \vdots & \vdots & \ddots \\
  \end{array}%
\right)$$
}
%----------------------------------------------------------------------------------------10
\frame{\frametitle{Single birth process}
给定在~$E$~上定义的一个函数~$c$. 定义
$$
\widetilde F_i^{(i)}=1,\qquad \widetilde F_n^{(i)}=\frac 1{q_{n,n+1}}\sum_{k=i}^{n-1}\tilde q_n^{(k)}\widetilde F_k^{(i)},\qquad 0\leqslant i<n,
$$
$$
\tilde q_n^{(k)}=q_n^{(k)}-c_n=\sum_{j=0}^kq_{nj}-c_n, \qquad 0\leqslant k<n.
$$
等价地, 也可以定义为:
$$
\widetilde F_i^{(i)}=1,\qquad  \widetilde F_n^{(i)}=\sum_{k=i+1}^{n}\frac {\widetilde F_n^{(k)}\tilde q_k^{(i)}}{q_{k,k+1}}, \qquad n\geqslant i+1.%ed
$$
}
%----------------------------------------------------------------------------------------11
\frame{\frametitle{Single birth process}
\begin{block}{Theorem (Wang \& Z., 2020)} 假定单生过程常返且对给定正整数~$\ell\geqslant 1$~和~状态~$i_0$~满足:
对所有~$i\in E$~有~$\mathbb E_i\xi_{i_0}^{\ell-1}<\infty$. 则
$$\mathbb E_n\xi_{i_0}^\ell=\begin{cases}\displaystyle\ell \sum_{n\leqslant k\leqslant i_0-1} v_k^{(\ell)}+\bigg(1-\sum_{n\leqslant k\leqslant i_0-1} u_k\bigg){\mathbb E}_{i_0}\xi_{i_0}^\ell,\  & 0\leqslant n\leqslant i_0,\\
\displaystyle-\ell\sum_{i_0\leqslant k\leqslant n-1} v_k^{(\ell)}+\bigg(1+\sum_{i_0\leqslant k\leqslant n-1}u_k\bigg){\mathbb E}_{i_0}\xi_{i_0}^\ell,\  & n\geqslant  i_0+1;\end{cases}$$
\end{block}
}
%----------------------------------------------------------------------------------------11
\frame{\frametitle{Single birth process}
\begin{block}{Theorem (continued)}
其中
$$
u_k=\begin{cases}\displaystyle
\sum_{j=i_0-1}^k \frac{F_k^{(j)} q_{ji_0}(1-\delta_{ji_0})}{q_{j, j+1}} ,\qquad &k\geqslant i_0,\nonumber\\
1, &k=i_0-1,\\
0,  &0\leqslant k\leqslant i_0-2;\end{cases}$$
$$v_k^{(\ell)}=\sum_{j=0}^k \frac{F_k^{(j)}V(j) {\mathbb E}_j\xi_{i_0}^{\ell-1}}{q_{j, j+1}},\qquad k\geqslant 0;$$
$$\aligned
{\mathbb E}_{i_0}\xi_{i_0}^\ell&=\ell\varlimsup_{n\to\infty} \frac{\sum_{i_0\leqslant k\leqslant n} v_k^{(\ell)}}{1+\sum_{i_0\leqslant k\leqslant n}u_k}\\
&=\ell\lim_{n\to\infty} \frac{v_n^{(\ell)}}{u_n}\;\; \text{\rm 若此极限存在}.\endaligned$$
\end{block}
}
%----------------------------------------------------------------------------------------12
\frame{\frametitle{Single birth process}

记其分布函数~$F_{ki}(x)=\mathbb P_k(\xi_i\leqslant x)$, Laplace~变换记为:
$$\varphi_{ki}(\lambda)=\mathbb E_k\text{e}^{-\lambda\xi_i}=\int_0^\infty\text{e}^{-\lambda x}\text{d}F_{ki}(x),\qquad \lambda>0.$$
再记~$\psi_{ki}(\lambda)=1-\varphi_{ki}(\lambda)$.
\begin{block}{Theorem (Wang \& Z., 2020)} 对单生过程, 给定状态~$i_0$, 取定~$c=-\lambda V$.  则
$$\psi_{ni_0}(\lambda)\!=\!\begin{cases}
\displaystyle \frac{\lambda \sum_{n\leqslant k\leqslant i_0-1}\tilde v_k}{1+\lambda\sum_{0\leqslant k\leqslant i_0-1}\tilde v_k}\!+\!\psi_{i_0i_0}(\lambda)\bigg(1\!-\! \sum_{n\leqslant k\leqslant i_0-1} \tilde u_k\bigg), 0\leqslant n\leqslant i_0,\\
\displaystyle -\frac{\lambda\sum_{i_0\leqslant k\leqslant n-1}\tilde v_k}{1+\lambda\sum_{0\leqslant k\leqslant i_0-1}\tilde v_k}
\!+\!\psi_{i_0i_0}(\lambda)\bigg(1\!+\!\sum_{i_0\leqslant k\leqslant n-1}\tilde u_k\bigg), n\geqslant i_0+1;\end{cases} $$
\end{block}
}
%----------------------------------------------------------------------------------------13
\frame{\frametitle{Single birth process}
\begin{block}{Theorem (continued)}
其中
\begin{gather}
\tilde u_k={\begin{cases}\displaystyle
\sum_{j=i_0-1}^k \frac{\widetilde F_k^{(j)} q_{ji_0}(1-\delta_{ji_0})}{q_{j, j+1}} ,\qquad &k\geqslant i_0,\nonumber\\
1, &k=i_0-1,\\
0,  &0\leqslant k\leqslant i_0-2;\end{cases}}\\
\tilde v_k=\sum_{j=0}^k \frac{\widetilde F_k^{(j)}V(j) }{q_{j, j+1}},\qquad k\geqslant 0;\nonumber
\end{gather}
$$\psi_{i_0i_0}(\lambda)=\frac{\lambda }{1+\lambda\sum_{0\leqslant k\leqslant i_0-1}\tilde v_k}\varlimsup_{n\to\infty} \frac{\sum_{i_0\leqslant k\leqslant n}\tilde v_k}{1+\sum_{i_0\leqslant k\leqslant n}\tilde u_k}.$$
\end{block}
}
%----------------------------------------------------------------------------------------14

\frame{\frametitle{Single birth process}
\begin{theorem}[contin.]\rm
若还假定单生过程是常返的且极限~$\lim_{n\to\infty}\tilde v_n/\tilde u_n$~存在, 则
$$\psi_{i_0i_0}(\lambda)=\frac{\lambda }{1+\lambda\sum_{0\leqslant k\leqslant i_0-1}\tilde v_k}\lim_{n\to\infty} \frac{\tilde v_n}{\tilde u_n}.$$
\end{theorem}


定义算子~$\Omega$: $\Omega g=Qg+cg$, 其中~$(Qg)_i=\sum_{j\in E} q_{ij}(g_j-g_i)$.
当~$c\leqslant 0$~时, $\Omega$~是带杀死速率~$(-c_i)$~的单生过程对应的算子.

\begin{block}{Theorem (Chen \& Z., 2014)} 给定单生~$Q$~矩阵~$Q=(q_{ij})$~和函数~$c$~及~$f$, 则~Poisson~方程
~$\Omega g=f$~的解~$g$~有如下表示:
$$
g_n=g_0+\sum_{0\leqslant k\leqslant n-1}\sum_{0\leqslant j\leqslant k}\frac{\widetilde F_k^{(j)}(f_j-c_jg_0)}{q_{j,j+1}}, \qquad n\geqslant 0.
$$
\end{block}
}

%----------------------------------------------------------------------------------------3
\section{Single death process}
\frame{ \frametitle{Single death process}\small
Single death process:
$Q=(q_{ij})$ satisfies
$${\red q_{i,i-1}>0},\ q_{i,i-j}=0,\ i\geqslant 1,j> 1.$$
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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{Single death process}\small
\begin{itemize}

\item branching process:

\end{itemize}

$$\aligned
 i&\to i-1\quad\text{at rate}\quad \alpha ip_0\\
  &\to i\qquad\quad \text{at rate}\quad -\alpha i(1-p_1)\\
  &\to i+1\quad\text{at rate}\quad \alpha ip_2\\
  &\to i+2\quad\text{at rate}\quad \alpha ip_3\\
  &\to \cdots
\endaligned$$
}

%----------------------------------------------------------------------------------------5
\frame{ \frametitle{Single death process}\small

\begin{itemize}
%\item 2017.6.29-2017.7.1, 南宁, The 6th IMS-China International Conference on Statistics and Probability, Notes on single death processes.
%\item 2017.11.25-2017.11.26, 中南大学, 马氏过程与随机分析学术论坛,  单死过程遍历性和强遍历性.
\item Z. Criteria on ergodicity and strong ergodicity of single death processes, Frontiers of Mathematics in China, 2018, 13(5): 1215-1243.
%\item 2018.7.16-2018.7.20, The 14th Workshop on Markov Processes and Related Topics, Moments of the first hitting times for birth-death processes on trees.
%\item Z. Moments of first hitting times for birth-death processes on trees, Frontiers of Mathematics in China, 2019, 14(4): 833-854.
%\item 2018.11.10-2018.11.11, 中南大学, 随机分析与偏微分方程研讨会, 单死过程首中时的高阶矩.
\item Z. \& Zhou. High-order moments of the first hitting times for single death processes, Frontiers of Mathematics in China, 2019, 14(5): 1037-1061.
Here, {\red $V\equiv 1$}.
\end{itemize}

\vskip0.2cm
一般的~$V$?

\vskip0.2cm
{\blue 侯振挺, 郭青峰(1978). 齐次可列马尔可夫过程. 科学出版社.}

\vskip0.2cm
最小非负解方法. 极限过渡法(生灭过程, 王梓坤).


\begin{itemize}
\item J. Wang \& Z. Moments of integral-type downward functionals for single death processes, Frontiers of Mathematics in China, 2020, 15(4): 749-768.
\end{itemize}
}


%\section{Main results}
%----------------------------------------------------------------------------------------6
\frame{\frametitle{Single death process}\small
\begin{block}{} Define $q_n^{(k)}:=\sum_{j=k}^\infty q_{nj},\  k>n\geqslant 0$.
$$
G_i^{(i)}=1,\qquad  G_n^{(i)}=\frac 1{q_{n,n-1}}
\sum_{k=n+1}^{i} q_n^{(k)} G_k^{(i)},\qquad 1\leqslant n< i.
$$
\end{block}

\begin{block}{Theorem (Z., 2018) 1st moment} Assume that the single death $Q$ is irreducible and regular. Then the process is ergodic if and only if
$$D:=\sum_{k\geqslant 1}q_{0}^{(k)}\sum_{\ell\geqslant k}\frac{G_k^{(\ell)}}{q_{\ell,\ell-1}}<\infty;$$
the process is strongly ergodic if and only
$$S:=\sum_{k\geqslant 1}\sum_{\ell\geqslant k}\frac{G_k^{(\ell)}}{q_{\ell,\ell-1}}<\infty.$$
\end{block}
}

%----------------------------------------------------------------------------------------6
\frame{\frametitle{Single death process}\small
\begin{block}{Theorem (Zhou \& Z., 2019) high order} Assume that the single death $Q$ is irreducible and the corresponding process is recurrent. Given $i_0\in \mathbb Z_+$ and a positive integer $n\geqslant 1$ arbitrarily. Then
$$\mathbb E_i\tau_{i_0}^n=n\sum_{i_0+1\leqslant k\leqslant i}\sum_{\ell\geqslant k}\frac{G_k^{(\ell)}\mathbb E_\ell\tau_{i_0}^{n-1}}{q_{\ell,\ell-1}},\qquad i\geqslant i_0+1.$$
\end{block}

\begin{block}{Theorem 1 (Wang \& Z., 2020)} Assume that the single death $Q$ is irreducible and the corresponding process is recurrent. Given $i_0\in \mathbb Z_+$ and a positive integer $n\geqslant 1$ arbitrarily. Then
$$\mathbb E_i\xi_{i_0}^n=n\sum_{i_0+1\leqslant k\leqslant i}\sum_{\ell\geqslant k}\frac{G_k^{(\ell)}V(\ell)\mathbb E_\ell\xi_{i_0}^{n-1}}{q_{\ell,\ell-1}},\qquad i\geqslant i_0+1.$$
\end{block}
}
%----------------------------------------------------------------------------------------7
\frame{\frametitle{Single death process}\small

%Let $w_i^{(n)}:=\mathbb E_i\xi_{i-1}^n$ and $w_i:=w_i^{(1)}$.  $w_i^{(0)}=1$.
%
%\begin{block}{Theorem 2} Under the conditions of Theorem 1, then
%$$\aligned \mathbb E_i\xi_{i_0}^n &=(w_{i_0+1}+\cdots +w_i)^{(n)}\\
%&=:\sum_{n_{i_0+1}+\cdots+n_{i}=n}\frac{n!}{n_{i_0+1}!\cdots n_i!}w_{i_0+1}^{(n_{i_0+1})}\cdots w_i^{(n_i)},\qquad i> i_0, n\geqslant 1.\endaligned$$
%\end{block}

\begin{block}{}
$$\mathbb E_i\xi_{i_0}^n=\sum_{i_0+1\leqslant \ell\leqslant i}\mathbb E_\ell\xi_{\ell-1}^n+\sum_{i_0+2\leqslant\ell\leqslant i}\sum_{s=1}^{n-1}\binom{n}{s}\mathbb E_\ell\xi_{\ell-1}^{n-s}\mathbb E_{\ell-1}\xi_{i_0}^s.$$
\end{block}
%}
%%----------------------------------------------------------------------------------------7
%\frame{\frametitle{Single death processes. Sketch of proof.}\small

%Let

$$M_{ik}^{(n-1)}:=\sum_{i+1\leqslant\ell\leqslant k}\sum_{s=1}^{n-1}\binom{n}{s}\mathbb E_\ell\xi_{\ell-1}^{n-s}\mathbb E_{\ell-1}\xi_{i-1}^s,$$

$$M_i^{(n-1)}:=V(i)\mathbb E_i\xi_{i-1}^{n-1}+\frac1n\sum_{k\geqslant i+1}q_{ik}M_{ik}^{(n-1)},\qquad i\geqslant 1.$$

\begin{block}{}
$$\mathbb E_i\xi_{i_0}^n=\sum_{i_0+1\leqslant \ell\leqslant i}\mathbb E_\ell\xi_{\ell-1}^n+M_{i_0+1,i}^{(n-1)}, \qquad i\geqslant i_0+1,$$
$$M_{ik}^{(n-1)}=M_{ij}^{(n-1)}+M_{j+1,k}^{(n-1)}+\sum_{s=1}^{n-1}\binom{n}{s}\mathbb E_k\xi_{j}^{n-s}\mathbb E_{j}\xi_{i-1}^s,\qquad 1\leqslant i\leqslant j\leqslant k.$$
\end{block}
}


%----------------------------------------------------------------------------------------9
\frame{\frametitle{Single death process. Sketch of proof.}\small

\begin{itemize}
\item Single death property: $\mathbb E_i\xi_{i-1}^n=?$
\end{itemize}

\begin{block}{}
$$x_i=\frac{ q_i^{(i+1)}}{ q_i}x_i+\sum_{k\geqslant i+1}\frac{ q_i^{(k)}}{ q_i}x_k+\frac n{q_{i}}M_i^{(n-1)},\ i\geqslant 1.\qquad\qquad (*)$$
\end{block}

\begin{block}{Theorem 3. Under the conditions of Theorem 1}
$${\red m_i:=} \mathbb E_i\xi_{i-1}^n=n\sum_{k\geqslant i}\frac{G_i^{(k)}M_k^{(n-1)}}{q_{k,k-1}}{\red=: h_i},\qquad i\geqslant 1, n\geqslant 1.$$
\end{block}

%\vskip0.3cm
%Denote the left hand side by $m_i$, the right hand side by $h_i$.

%$$h_i=\frac 1{q_{i,i-1}}\Bigg(nM_i^{(n-1)}+\sum_{k\geqslant i+1}q_i^{(k)}h_k\Bigg).$$
%}
%
%
%%----------------------------------------------------------------------------------------11
%\frame{\frametitle{Single death process. Sketch of proof.}

\begin{block}{Lemma 1. ({\red $m_i\geqslant h_i$})}
$(h_i)$ is the minimal nonnegative solution to $(*)$.
%$$x_i=\frac{ q_i^{(i+1)}}{ q_i}x_i+\sum_{k\geqslant i+1}\frac{ q_i^{(k)}}{ q_i}x_k+\frac n{q_{i}}M_i^{(n-1)},\ i\geqslant 1.\qquad\qquad (*)$$
\end{block}
}

%----------------------------------------------------------------------------------------12
\frame{\frametitle{Single death process. Sketch of proof.}\small

Sketch proof of Lemma 1:

Fix $N\geqslant 2$, define $Q$-matrix $Q^{(N)}=(\tilde q_{ij})$ on $\{0,1,2,\cdots, N\}$:
$$\tilde q_{ij}=\begin{cases} q_{ij} &\text{if}\ i<N, j<N;\\ q_i^{(N)} &\text{if}\ i<N,j=N;\\ (q_N\vee N)(1+nG^{(N)}a_N) &\text{if}\ i=N, j=N-1;\\
-(q_N \vee N)(1+nG^{(N)}a_N) &\text{if}\ i=N, j=N;\\
0, &\text{if}\ i=N, j<N-1,\end{cases}$$
where $G^{(N)}=\max_{1\leqslant i\leqslant N}G_i^{(N)}$ and
$$a_N=\begin{cases} M_{N}^{(n-1)}&\text{if}\ M_{N}^{(n-1)}<\infty; \\ 1 & \text{if}\ M_{N}^{(n-1)}=\infty.\end{cases}$$
Define $\tilde q_n^{(k)}=\sum_{j=k}^N \tilde q_{nj},\ 0\leqslant n< k\leqslant N,$
$$
\widetilde G_i^{(i)}=1,\qquad   \widetilde G_n^{(i)}=\frac 1{\tilde q_{n,n-1}}
\sum_{k=n+1}^{i} \tilde q_n^{(k)} \widetilde G_k^{(i)},\qquad 1\leqslant n< i\leqslant N.
$$
}

%----------------------------------------------------------------------------------------13
\frame{\frametitle{Single death process. Sketch of proof.}\small

Sketch proof of Lemma 1(continued):

$$
h_n^{(N)}:=n\sum_{k=i}^N\frac{\widetilde G_i^{(k)}M_k^{(n-1)}}{\tilde q_{k,k-1}}\quad (1\leqslant i\leqslant N)
$$
is a unique solution (the minimal non-negative solution) to the following equations
$$
x_i=\frac {\tilde q_{i}^{(i+1)}}{\tilde q_i}\cdot x_i+\sum_{k=i+1}^N\frac {\tilde q_{i}^{(k)}}{\tilde q_i}\cdot x_k+\frac n{\tilde q_i}M_i^{(n-1)},\qquad 1\leqslant i\leqslant N.
$$

$$
x_N=\frac {nM_N^{(n-1)}}{(q_N\vee N)(1+nG^{(N)}a_N)},\ x_i=\frac {q_{i}^{(i+1)}}{q_i}\cdot x_i+\sum_{k=i+1}^N\frac {q_i^{(k)}}{q_i}\cdot x_k+\frac n{q_i}M_i^{(n-1)}.
$$

$ (h_n^{(N)})$ is increasing to the minimal non-negative solution of ($*$) as $N\to \infty$.

$$h_n^{(N)}=n\sum_{k=i}^{N-1}\frac{G_i^{(k)}M_k^{(n-1)}}{q_{k,k-1}}+\frac {nG_i^{(N)}M_N^{(n-1)}}{(q_N\vee N)(1+nG^{(N)}a_N)}\to n\sum_{k=i}^{\infty}\frac{G_i^{(k)}M_k^{(n-1)}}{q_{k,k-1}}=h_i.$$
}

%----------------------------------------------------------------------------------------14
\frame{\frametitle{Single death process. Sketch of proof.}\small

\begin{block}{Lemma 2 ($\tau_i$)}
Under the conditions of Theorem 1, $(\mathbb E_i\xi_{i_0}^n: i\geqslant i_0)$ is the minimal nonnegative solution to
$$
x_{i_0}=0,\qquad x_i=\sum_{j\ne i,j> i_0}\frac {q_{ij}}{q_i} x_j+\frac {nV(i)}{q_i}\mathbb E_i\xi_{i_0}^{n-1},\qquad  i> i_0.
$$
\end{block}
\vskip0.4cm
$\displaystyle {\red n}\sum_{i_0+1\leqslant k\leqslant i}{\red \sum_{\ell\geqslant k}\frac {G_k^{(\ell)}M_{\ell}^{(n-1)}}{q_{\ell,\ell-1}}}+M_{i_0+1,i}^{(n-1)}$ satisfies the equation above.

$$\mathbb E_i\xi_{i_0}^n\leqslant n\sum_{i_0+1\leqslant k\leqslant i}\sum_{\ell\geqslant k}\frac {G_k^{(\ell)}M_{\ell}^{(n-1)}}{q_{\ell,\ell-1}}+M_{i_0+1,i}^{(n-1)}\Rightarrow  {\red m_i\leqslant h_i}.$$
}

%----------------------------------------------------------------------------------------16
\frame{\frametitle{Single death process. Sketch of proof.}\small

By induction and Dominated Convergence Theorem, for all $n\geqslant 1$,

\begin{block}{}
$$\mathbb E_i(\xi_{i_0}^{(N)})^n\quad\big\uparrow\quad n\sum_{i_0+1\leqslant k\leqslant i}\sum_{\ell\geqslant k}\frac {G_k^{(\ell)}V(\ell)\mathbb E_\ell\xi_{i_0}^{n-1}}{q_{\ell,\ell-1}},$$
\end{block}

\begin{block}{}
$$
n\sum_{i_0+1\leqslant k\leqslant i}\sum_{\ell\geqslant k}\frac {G_k^{(\ell)}V(\ell)\mathbb E_\ell\xi_{i_0}^{n-1}}{q_{\ell,\ell-1}}=\mathbb E_i\xi_{i_0}^n,\qquad i\geqslant i_0.$$
\end{block}
}
%----------------------------------------------------------------------------------------15
\frame{\frametitle{Single death processes. Sketch of proof.}\small

\begin{block}{Lemma 3. Under the conditions of Theorem 1}
$$\mathbb E_i\xi_{i_0}^n\leqslant n\sum_{i_0+1\leqslant k\leqslant i}\sum_{\ell\geqslant k}\frac {G_k^{(\ell)}V(\ell)\mathbb E_\ell\xi_{i_0}^{n-1}}{q_{\ell,\ell-1}},\qquad i\geqslant i_0, n\geqslant 1.$$
\end{block}

\begin{block}{Lemma 4. Under the conditions of Theorem 1}
$$\mathbb E_i\xi_{i_0}= \sum_{i_0+1\leqslant k\leqslant i}\sum_{\ell\geqslant k}\frac {G_k^{(\ell)}V(\ell)}{q_{\ell,\ell-1}},\qquad i\geqslant i_0.$$
\end{block}

\begin{block}{Lemma 5 }
$$\mathbb E_i(\xi_{i_0}^{(N)})^n= n\sum_{i_0+1\leqslant k\leqslant i}\sum_{k\leqslant\ell\leqslant N}\frac {G_k^{(N,\ell)}V(\ell)\mathbb E_\ell(\xi_{i_0}^{(N)})^{n-1}}{q_{\ell,\ell-1}^{(N)}},\qquad i_0\leqslant i\leqslant N.$$
\end{block}
}

%----------------------------------------------------------------------------------------15
\frame{\frametitle{Single death process. Sketch of proof.}\small
$n=1$:
\begin{block}{}
$$\mathbb E_i\xi_{i_0}^{(N)}=\sum_{i_0+1\leqslant k\leqslant i}\sum_{k\leqslant\ell\leqslant N}\frac {G_k^{(\ell)}V(\ell)}{q_{\ell,\ell-1}}
\quad\big\uparrow\quad \sum_{i_0+1\leqslant k\leqslant i}\sum_{\ell\geqslant k}\frac {G_k^{(\ell)}V(\ell)}{q_{\ell,\ell-1}},$$
\end{block}

\begin{block}{}
$$
\sum_{i_0+1\leqslant k\leqslant i}\sum_{\ell\geqslant k}\frac {G_k^{(\ell)}V(\ell)}{q_{\ell,\ell-1}}=\mathbb E_i\xi_{i_0},\qquad i\geqslant i_0.$$
\end{block}
}

%----------------------------------------------------------------------------------------15
\frame{\frametitle{Single death process. Sketch of proof.}\small
Assume that the assertions hold until $n-1$.

By Monotone Convergence Theorem,

\begin{block}{}
$$\mathbb E_i(\xi_{i_0}^{(N)})^n\quad\big\uparrow\quad n\sum_{i_0+1\leqslant k\leqslant i}\sum_{\ell\geqslant k}\frac {G_k^{(\ell)}V(\ell)\mathbb E_\ell\xi_{i_0}^{n-1}}{q_{\ell,\ell-1}},$$
\end{block}

\begin{block}{}
$$
n\sum_{i_0+1\leqslant k\leqslant i}\sum_{\ell\geqslant k}\frac {G_k^{(\ell)}V(\ell)\mathbb E_\ell\xi_{i_0}^{n-1}}{q_{\ell,\ell-1}}\geqslant\mathbb E_i\xi_{i_0}^n,\qquad i\geqslant i_0.$$
\end{block}

\begin{block}{}
$$\mathbb E_i(\xi_{i_0}^{(N)})^n\leqslant n \sum_{i_0+1\leqslant k\leqslant i}\sum_{k\leqslant\ell\leqslant N}\frac {G_k^{(\ell)}M_\ell^{(N,n-1)}}{q_{\ell,\ell-1}^{(N)}}+M_{i_0+1,i}^{(N,n-1)},$$
\end{block}

By Dominated Convergence Theorem.
}
%%-----------------------------
%\frame{\frametitle{Single death process}\small
%Distribution?
%
%\begin{block}{}
%$$\mathbb E_i\text{e}^{-\lambda\xi_{i_0}^{(N)}}\to \mathbb E_i\text{e}^{-\lambda\xi_{i_0}},\quad i>i_0?$$
%\end{block}
%\begin{block}{} Define $\tilde q_n^{(k)}:=\sum_{j=k}^\infty q_{nj}+\lambda V(n),\  k>n\geqslant 0$.
%$$
%\widetilde G_i^{(i)}=1,\qquad  \widetilde G_n^{(i)}=\frac 1{q_{n,n-1}}
%\sum_{k=n+1}^{i} \tilde q_n^{(k)} \widetilde G_k^{(i)},\qquad 1\leqslant n< i.
%$$
%\end{block}
%
%\begin{block}{}
%$$1-\mathbb E_i\text{e}^{-\lambda\xi_{i_0}^{(N)}}=\frac{\lambda\sum_{i_0+1\leqslant k\leqslant i}\sum_{k\leqslant \ell\leqslant N}\frac{\widetilde G_k^{(\ell)}V(\ell)}{q_{\ell,\ell-1}}}
%{1+\lambda\sum_{i_0+1 \leqslant k\leqslant N }\sum_{k\leqslant\ell\leqslant N}\frac{\widetilde G_k^{(\ell)}V(\ell)}{q_{\ell,\ell-1}}},\qquad i\geqslant i_0+1.$$
%\end{block}
%}
%-----------------------------
\frame{\frametitle{Single death process}\small
Define
$$\kappa_0=1,\quad \kappa_i=\frac 1{q_{i,i-1}}\sum_{k=1}^iq_0^{(k)}G_k^{(i)},\qquad i\geqslant 1,\quad \kappa=\sum_{i\in E}\kappa_i.$$
In the ergodic case, the stationary distribution $\pi=(\pi_i)$: $\pi_i=\kappa_i/\kappa$.

\begin{block}{Theorem 4 (Wang \& Z. 2020+) ({\red $\sigma_i$}) Under the conditions of Th. 1,}

$$\mathbb E_{i_0}\xi_{i_0}^n=\frac 1{\kappa_{i_0}}\sum_{i=0}^{i_0}\frac{\kappa_i\widetilde H_i^{(n,i_0)}}{q_i},\qquad i_0\geqslant 0,$$
$$
\mathbb E_i\xi_{i_0}^n=\frac{D_{i_0}^{(i_0)}}{C_{i_0}^{(i_0)}}C_{i_0-1-i}^{(i_0)}-D_{i_0-1-i}^{(i_0)},\qquad 0\leqslant i\leqslant i_0-1,$$
where
$$
H_i^{(n,i_0)}= nV(i)\mathbb E_i\xi_{i_0}^{n-1},\qquad i \geqslant 0,$$
$$ \widetilde H_i^{(n,i_0)}=\sum_{j\geqslant i_0+1}q_{ij}\mathbb E_j\xi_{i_0}^n+ H_i^{(n,i_0)},\qquad 0\leqslant i\leqslant i_0.$$
\end{block}
}

%-----------------------------
\frame{\frametitle{Single death process}\small
\begin{block}{Theorem 4 (continued)}
$$C^{(i_0)}_{i_0}=\frac{q_{i_0,i_0-1}}{q_{i_0}}\kappa_{i_0},\qquad D^{(i_0)}_{i_0}=\sum_{i=0}^{i_0-1}\frac{\kappa_i\widetilde H_i^{(n,i_0)}}{q_i},$$
and
$$C_{i}^{(i_0)}=\sum_{i_0-i\leqslant k\leqslant i_0}G_{k}^{(i_0)},\
D_{i}^{(i_0)}=\sum_{i_0-i\leqslant \ell\leqslant i_0-1}\sum_{k=i_0-i}^{\ell}\frac{G_{k}^{(\ell)}\widetilde H_{\ell}^{(n,i_0)}}{q_{\ell,\ell-1}},
\  0\leqslant i\leqslant i_0-1.$$
\end{block}
}

\section{Birth-death process on trees}
%----------------------------------------------------------------------------------------28
\frame{\frametitle{Birth-death process on trees}\small

The tree $T$ is a connected graph without cycles. We fix a point on $T$ as the root, denoted by $o$. For any vertex $i\in T\setminus\{o\}$, there is a unique simple path from $i$ to the root $o$.
\begin{itemize}
\item $\mathscr P(i)$: the set of all the vertices on this path (the root $o$ is excluded).

\item $|i|$: the number of segments of this path is the length of $i$.

\item $i\sim j$: two vertices $i$ and $j$ are called adjacent if they are joined by a segment.

\item When $|j|=|i|+1$ and $i\sim j$, $j$ is called one offspring of $i$ and the set of all the offsprings of $i$ is denoted by $J(i)$.

\item When $|j|=|i|-1$ and $i\sim j$, $j$ is called the father of $i$ and denoted by $i^*$.

\item Denote $T_i$ the subtree with $i$ as its root, including all the descendants of $i$.

\item Assume that any vertex has finite offsprings.
\end{itemize}
}

%----------------------------------------------------------------------------------------28
\frame{\frametitle{Birth-death process on trees}\small

We consider a birth-death process on this tree which $Q$-matrix satisfies: $q_{ij}>0$ if and only if $i\sim j$, i.e., $j=i^*$, or $j\in J(i)$.

\vskip0.5cm
$q_i:=-q_{ii}=q_{ii^*}+\sum_{j\in J(i)}q_{ij}<\infty$ for all $i\in T$.

\vskip0.5cm
Define a measure $\mu$ on $T$ as follows.
$$\mu_o=1,\qquad \mu_i=\prod_{j\in\mathscr P(i)}\frac{q_{j^*j}}{q_{jj^*}},\qquad i\in T\setminus\{0\},$$
which is invariant with respect to $Q$.
In fact, $\mu$ satisfies the so-called detailed balance equation:
$$\label{e0.01} \mu_iq_{ij}=\mu_jq_{ji},\qquad i\sim j.$$
}

%----------------------------------------------------------------------------------------28
\frame{\frametitle{Birth-death process on trees}\small
\begin{itemize}
\item Z. Moments of first hitting times for birth-death processes on trees, Frontiers of Mathematics in China, 2019, 14(4): 833-854.
\end{itemize}

\begin{block}{Theorem (Z., 2019)}
Assume that the $Q$-matrix on $T$ is regular and the process is recurrent. Then
$$\mathbb E_i\tau_k^n=n\sum_{j\in\mathscr P(i)\setminus\mathscr P(k)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell\mathbb E_\ell\xi_k^{n-1},
\qquad k\in \mathscr P(i),$$
$$\mathbb E_i\tau_o^n=n\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell \mathbb E_\ell\xi_o^{n-1}.
$$
\end{block}

Corollary.  ergodic $\Leftrightarrow\mu<\infty$; strongly ergodic $\Leftrightarrow$
$$
\sup_{i\in T\setminus\{o\}}\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell<\infty.
$$
}

\frame{\frametitle{Birth-death process on trees}\small

\begin{block}{Theorem (Wang \& Z., 2020+)} Assume that the $Q$-matrix on the tree $T$ is regular and the process is recurrent. Then
$$\mathbb E_i\xi_k^n=n\sum_{j\in\mathscr P(i)\setminus\mathscr P(k)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell V(\ell)\mathbb E_\ell\xi_k^{n-1},
\qquad k\in \mathscr P(i)$$
$$\mathbb E_i\xi_o^n=n\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell V(\ell)\mathbb E_\ell\xi_o^{n-1}. $$
\end{block}
}

\section{Application}
%-----------------------------
\frame{\frametitle{Application}\small
\begin{itemize}
\item Y.Y. Liu \& Y.H. Song. Integral-type functionals of first hitting times for continuous-time Markov chains, Front. Math. China, 2018, 13(3): 619-632.
\end{itemize}

$$\xi_{i_0}=\int_0^{\sigma _{i_0}}r(t)V(X(t))\text{d}t.$$

\begin{block}{}
Assume that $r, r'\geqslant 0$. Then $({\mathbb E}_i\xi_{i_0}: i\in E)$ is the minimal nonnegative solution to:
$$
y_i=\sum_{k\ne i, i_0}\frac{q_{ik}}{q_i}y_k+\frac{ \overline V(i)}{q_i} ,\qquad i\in E,
$$
where
$$\overline V=\widetilde V+r(0)V,\qquad\widetilde V(i)=\mathbb E_i\int_0^{\sigma_{i_0}}r'(t)V(X(t))\text{d}t.$$
\end{block}
}
%-----------------------------
\frame{\frametitle{Application. Single death process for example.}\small
\begin{block}{}
$$\mathbb E_i\xi_{i_0}=\mathbb E_i\int_0^{\sigma_{i_0}}\overline V(X(t))\text{d}t.$$
\end{block}
\begin{block}{}
$(\mathbb E_i\xi_{i_0}: i\geqslant i_0)$ is the minimal nonnegative solution to
$$
x_{i_0}=0,\qquad x_i=\sum_{k\ne i,k> i_0}\frac {q_{ik}}{q_i} x_j+\frac {\overline V(i)}{q_i},\qquad  i> i_0.
$$
\end{block}

\vskip0.5cm

For single death process,
\begin{block}{}
$$\mathbb E_i\xi_{i_0}=\sum_{i_0+1\leqslant k\leqslant i}\sum_{\ell\geqslant k}\frac{G_k^{(\ell)}\overline V(\ell)}{q_{\ell,\ell-1}},\qquad i\geqslant i_0+1.$$


\end{block}}

%-----------------------------
\frame{\frametitle{Application. Single death process for example.}\small

In particular, when $r(t)=t^n$, denote
$$\xi_{i_0}=\xi_{i_0}^{(n)}=\int_0^{\sigma _{i_0}}t^nV(X(t))\text{d}t,\
\widetilde V(i)=n\mathbb E_i\xi_{i_0}^{(n-1)}.$$

$$\mathbb E_i\xi_{i_0}^{(n)}=n\sum_{i_0+1\leqslant k\leqslant i}\sum_{\ell\geqslant k}\frac{G_k^{(\ell)}\mathbb E_\ell\xi_{i_0}^{(n-1)}}{q_{\ell,\ell-1}},\qquad i\geqslant i_0+1.$$
\begin{itemize}
\item subexponential ergodicity and polynomial ergodicity
\item CLT:
$$t^{1/2}\bigg(\frac 1t\int_0^tf(X(s))\text{d}s-\pi(f)\bigg)\ {\overset{D}\to}\ N(0,\sigma^2(f)),\quad t\to\infty.$$
If $\pi(|f|)<\infty$, then a CLT holds iff
$$\mathbb E_0\bigg(\int_0^{\sigma_0}(f-\pi(f))(X(t))\text{d}t\bigg)^2<\infty \  \Leftarrow \ \mathbb E_0\bigg(\int_0^{\sigma_0}|f-\pi(f)|(X(t))\text{d}t\bigg)^2<\infty.$$
\end{itemize}}

\section{References}
\frame{\frametitle{References}\small

\begin{itemize}
\item Z. Criteria on ergodicity and strong ergodicity of single death processes, Frontiers of Mathematics in China, 2018, 13(5): 1215-1243.
\item Z. Moments of first hitting times for birth-death processes on trees, Frontiers of Mathematics in China, 2019, 14(4): 833-854.
\item Z. \& X.F. Zhou. High order moments of first hitting times for single death processes, Frontiers of Mathematics in China, 2019, 14(5): 1037-1061.
\item Z. 单死过程的稳定性, 中国科学: 数学, 2020, 50(1): 211-230.
\item J. Wang \& Z. Moments of integral-type downward functionals for single death processes, Frontiers of Mathematics in China, 2020, 15(4): 749-768.
\item 王婧, Z. 单生过程积分型泛函的矩和分布, 北京师范大学学报, 2020, 56(4): 493-499.
\item 王婧, Z. 单死过程的向下积分型泛函, 应用概率统计, 2020, 36(4): 393-414.
%\item
\end{itemize}
}

%----------------------------------------------------83
\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}
