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\begin{document}
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\title[单死过程和树上生灭过程]{单死过程和树上生灭过程}

\author[张余辉]{\sihao 张余辉}

\date[2018年4月27日]{\wuhao (天津大学, 2018年4月27日-28日) }

\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
 \end{frame}

%----------------------------------------------------------------------------------------1
\begin{frame}
  \frametitle{CONTENTS}
  \tableofcontents%[sections]
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Background}


%----------------------------------------------------------------------------------------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

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

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

\item branching processes: reference model

\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{Background}\small

\begin{itemize}
\item single birth processes and dual approach.
\end{itemize}

{\blue Y.R. Li, A.G. Pakes, J. Li, A.H. Gu(2008). The limit behavior of dual Markov branching processes. J. Appl. Prob. 45, 176-189.}
\begin{itemize}

\item stationary distribution. zero-entrance.

\item quasi-stationary distribution: $\sup_{n\geqslant 1}\mathbb E_n\tau_0$.

\end{itemize}
}
\section{Hitting times, criteria on ergodicity and strong ergodicity}

%----------------------------------------------------------------------------------------6
\frame{\frametitle{Hitting times, criteria on ergodicity and strong 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, $\lim_{t\to\infty}p_{ii}(t)=\pi_i>0$.
\item $P(t)$ is strongly ergodic or uniformly ergodic if
$$\lim_{t\to\infty}\sup_i|p_{ij}(t)-\pi_j|=0.$$
\end{itemize}
}
%----------------------------------------------------------------------------------------7
\frame{\frametitle{Hitting times, criteria on ergodicity and strong ergodicity}\small
Let $Q=(q_{ij})$ be a regular irreducible $Q$-matrix and $H$ be a non-empty finite subset of $\mathbb Z_+$. Define
$\sigma_H=\inf\{t\ge\text{\,the first jump}: X(t)\in H\}$.

\begin{block}{Theorem[Issacson \& Arnold(1978)]}

(1) The $Q$-process is ergodic iff $\mathbb E_i\sigma_H<\infty$ for all $i\in H$.

(2) The $Q$-process is strongly ergodic iff $\sup_i\mathbb E_i\sigma_H<\infty$.
\end{block}

\begin{block}{Theorem[Tweedie(1981)]}
 The $Q$-process is ergodic (resp. strongly erergodic) iff the equation
 $$\begin{cases}\sum_jq_{ij}y_j\leqslant -1,\ i\notin H\\\sum_{i\in H}\sum_{j\ne i}q_{ij}y_j<\infty
 \end{cases}$$
 has a finite (resp. bounded) nonnegative solution.
\end{block}
}

%----------------------------------------------------------------------------------------8

\frame{\frametitle{Hitting times, criteria on ergodicity and strong ergodicity}\small
Assume that $Q$ is irreducible. $\pi=(\pi_i)$ is stationary distribution.
\begin{itemize}
\item Ordinary ergodicity: $\displaystyle\lim_{t\to\infty}|p_{ij}(t)-\pi_j|=0$.

\item Exponential ergodicity: $\displaystyle\lim_{t\to\infty}{\red e^{\alpha t}}|p_{ij}(t)-\pi_j|=0$.

\item Strong ergodicity: $\displaystyle\lim_{t\to\infty}\sup_i|p_{ij}(t)-\pi_j|=0$

\hskip3cm
$\Longleftrightarrow\displaystyle\lim_{t\to\infty}{\red e^{\beta t}}\sup_i|p_{ij}(t)-\pi_j|=0$.

\item Strong ergodicity $\Rightarrow$ Exponential ergodicity $\Rightarrow$
Ordinary ergodicity.
\end{itemize}
}

%----------------------------------------------------------------------------------------9
\frame{\frametitle{Hitting times, criteria on ergodicity and strong ergodicity}
\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 1.} Assume that the single death $Q$ is irreducible and the process is recurrent. Then
$$\mathbb E_n\sigma_{0}=\sum_{1\leqslant k\leqslant n}\sum_{\ell\geqslant k}\frac{G_k^{(\ell)}}{q_{\ell,\ell-1}},\qquad n\geqslant 1;$$
$$\mathbb E_0\sigma_{0}=\frac 1{q_0}+\frac 1{q_0}\sum_{k\geqslant 1}q_{0}^{(k)}\sum_{\ell\geqslant k}\frac{G_k^{(\ell)}}{q_{\ell,\ell-1}}.$$
\end{block}
}

%----------------------------------------------------------------------------------------9
\frame{\frametitle{Hitting times, criteria on ergodicity and strong ergodicity}
\small

\begin{block}{Theorem 1.(continued)}
Furthermore, 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
$$S:=\sum_{k\geqslant 1}\sum_{\ell\geqslant k}\frac{G_k^{(\ell)}}{q_{\ell,\ell-1}}<\infty.$$
Actually, for the last conclusion, the recurrence assumption can be replaced by the uniqueness one.
\end{block}
}

%----------------------------------------------------------------------------------------9
\frame{\frametitle{Hitting times, criteria on ergodicity and strong ergodicity}\small
\begin{block}{}
$$G_i^{(i)}=1,\qquad  {G}_n^{(i)}=\sum_{k=n}^{i-1}
\frac{{G}_n^{(k)}{q}_k^{(i)}}{q_{k, k-1}},\qquad 1\leqslant n\leqslant i-1. $$
\end{block}

\begin{itemize}
\item Mean hitting time of single death process: $\mathbb E_n\tau_{n-1}=?$
\end{itemize}

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

}

%----------------------------------------------------------------------------------------9
\frame{\frametitle{Hitting times, criteria on ergodicity and strong ergodicity}

\begin{block}{Proposition 1. Under the conditions of Theorem 1}
$$\mathbb E_n\tau_{n-1}=\sum_{k=n}^\infty\frac{G_n^{(k)}}{q_{k,k-1}},\ n\geqslant 1.$$
\end{block}

Denote the left hand by $m_n$, the right hand by $h_n$.

$$h_n=\frac 1{q_{n,n-1}}\Bigg(1+\sum_{k\geqslant n+1}q_n^{(k)}h_k\Bigg).$$
}


%----------------------------------------------------------------------------------------10
\frame{\frametitle{Hitting times, criteria on ergodicity and strong ergodicity}

\begin{block}{Lemma 1. }
$(h_n)$ is the minimal nonnegative solution to
$$x_n=\sum_{k=n+1}^{\infty}\frac{ q_n^{(k)}}{ q_n}x_k+\frac{ q_n^{(n+1)}}{ q_n}x_n+\frac 1{q_{n}},\ n\geqslant 1.\qquad\qquad (*)$$
\end{block}

{\red $$m_n\geqslant h_n.$$}

}

%----------------------------------------------------------------------------------------11
\frame{\frametitle{Hitting times, criteria on ergodicity and strong ergodicity}\small

Sketch proof of Lemma 1:

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

%----------------------------------------------------------------------------------------12
\frame{\frametitle{Hitting times, criteria on ergodicity and strong ergodicity}\small

Sketch proof of Lemma 1(continued):

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

$$
x_N=\frac 1{(q_N\vee N)(1+G^{(N)})},\quad x_i=\frac {q_{i}^{(i+1)}}{q_i}\cdot x_i+\sum_{\ell=i+1}^N\frac {q_i^{(\ell)}}{q_i}\cdot x_\ell+\frac 1{q_i},\quad 1\leqslant i< N.
$$

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

$$h_n^{(N)}=\sum_{k=n}^{N-1}\frac{G_n^{(k)}}{q_{k,k-1}}+\frac {G_n^{(N)}}{(q_N\vee N)(1+G^{(N)})}\to \sum_{k=n}^{\infty}\frac{G_n^{(k)}}{q_{k,k-1}}=h_n\ \text{as}\ N\to\infty.$$
}

%----------------------------------------------------------------------------------------13
\frame{\frametitle{Hitting times, criteria on ergodicity and strong ergodicity}\small

\begin{block}{Lemma 2. Under the conditions of Theorem 1}
$(\mathbb E_i\tau_{i_0}: i\ge i_0)$ is the minimal nonnegative solution to
$$
x_{i_0}=0,\qquad x_i=\sum_{j\ne i,j> i_0}\frac {q_{ij}}{q_i} x_j+\frac 1{q_i},\qquad  i> i_0.
$$
\end{block}

$\sum_{i_0+1\leqslant n\leqslant i}h_n$ satisfies the equation above.

$$\mathbb E_i\tau_{i_0}\leqslant \sum_{i_0+1\leqslant n\leqslant i}h_n \Rightarrow  {\red m_n\leqslant h_n}.$$
}

%----------------------------------------------------------------------------------------13
\frame{\frametitle{Hitting times, criteria on ergodicity and strong ergodicity}\small

\begin{block}{Proposition 1. Under the conditions of Theorem 1}
$$m_n=h_n:=\sum_{k=n}^\infty\frac{G_n^{(k)}}{q_{k,k-1}},\ n\geqslant 1.$$
\end{block}

$$\aligned
\mathbb E_0\sigma_0&=\frac 1{q_0}+\sum_{j\geqslant 1}\frac {q_{0j}}{q_0} \mathbb E_j\tau_0
=\frac 1{q_0}+\sum_{j\geqslant 1}\frac {q_{0j}}{q_0}\sum_{1\leqslant k\leqslant j}m_k\\
&=\frac 1{q_0}+\frac 1{q_0}\sum_{k\geqslant 1}q_{0}^{(k)}h_k=: \frac 1{q_0}+\frac 1{q_0}D\ {\red \Leftrightarrow\text{`ergodicity iff} \ D<\infty\text{'}}\endaligned$$

$$\sup_{i\geqslant 1}\mathbb E_i\tau_0=\sup_{i\geqslant 1}\sum_{1\leqslant k\leqslant i} m_k=\sum_{k\geqslant 1} h_k=:S, \qquad  D\leqslant q_0S$$

$${\red\Leftrightarrow \text{`strong ergodicity iff}\ S<\infty\text{'}}$$

}
%----------------------------------------------------------------------------------------13
\frame{\frametitle{Hitting times, criteria on ergodicity and strong ergodicity}\small
\begin{block}{$Q$-matrix is zero-entrance} for some (equivalently, for all) $\lambda>0$, the following equation has only trivial solution:
$$
\lambda y_i=\sum_{j=0}^\infty y_jq_{ji},\qquad y_i\ge 0,\ i\geqslant 0\quad\text{ and }\quad \sum_{i=0}^\infty y_i<\infty.
$$
\end{block}

\begin{block}{Proposition} Given single death $Q$-matrix. Then it is zero-entrance if and only if $\sum_{\ell=0}^\infty \overline m_\ell=\infty$, where
$$\overline m_0=0,\qquad \overline m_\ell=\frac 1{q_{\ell,\ell-1}}\bigg(1+\sum_{n=0}^{\ell-1}q_n^{(\ell)}\overline m_n\bigg),\qquad \ell\geqslant 1.$$
\end{block}
}

%----------------------------------------------------------------------------------------13
\frame{\frametitle{Hitting times, criteria on ergodicity and strong ergodicity}\small
\begin{block}{}
$$
\overline m_\ell=\frac 1{q_{\ell,\ell-1}}\sum_{k=1}^\ell G_k^{(\ell)},\qquad \ell\geqslant 1.
$$
\end{block}
\begin{block}{Corollary 1.} Assume that the single death $Q$ is irreducible and regular. Then
the process is strongly ergodic if and only if the $Q$-matrix is not zero-entrance.
\end{block}
}

%----------------------------------------------------------------------------------------13
\frame{\frametitle{Hitting times, criteria on ergodicity and strong ergodicity}\small
Define
$$
\kappa_0=1,\qquad \kappa_k=\frac 1{q_{k,k-1}}\sum_{n=0}^{k-1}q_n^{(k)}\kappa_n,\qquad k\geqslant 1.
$$

\begin{block}{Proposition}  Assume that the single death $Q$ is irreducible and regular. Then
the process is ergodic if and only if
$$\kappa:=\sum_{i\geqslant 0}\kappa_i<\infty,$$
the distribution $(\pi_i)$:
$$ \pi_i=\frac{\kappa_i}{\kappa},\qquad i\geqslant 0.$$
\end{block}
\begin{block}{}
$$\kappa_k=\frac 1{q_{k,k-1}}\sum_{\ell=1}^k q_0^{(\ell)}G_\ell^{(k)},\qquad k\geqslant 1.$$
$$\kappa=1+D.$$
\end{block}
}

%----------------------------------------------------------------------------------------14
\frame{\frametitle{Hitting times, criteria on ergodicity and strong ergodicity}\small

{\blue S. Martinez, J. S. Martin , D. Villemonais(2014). Existence and uniqueness of a quasi-
stationary distribution for Markov Processes with fast return from infinity. Journal of
Applied Probability, 51(3): 756-768.}

\begin{block}{Theorem 2} Assume that the single death $Q$ is `irreducible' and regular such that $0$ is an absorbing state and  absorption occurs almost surely. If
$$\sum_{n=1}^\infty\sum_{k=n}^\infty\frac{G_n^{(k)}}{q_{k,k-1}}<\infty,$$
then there exists a quasi-stationary distribution uniquely.
\end{block}

}
\section{Three examples}

%----------------------------------------------------------------------------------------15
\frame{\frametitle{Three examples }\small

\begin{block}{Example 1.} For regular birth-death chain $(a_i, b_i)$, define
$$\mu_0=1,\quad \mu_i=\frac{b_0b_1\cdots b_{i-1}}{a_1a_2\cdots a_i},\qquad i\geqslant 1;\qquad \mu[i,+\infty)=\sum_{k\geqslant i}\mu_k,\qquad i\geqslant 0.$$
\end{block}
Then
$$G_n^{(i)}=\frac{\mu_ia_i}{\mu_na_n},\qquad 1\leqslant n \leqslant i; \qquad m_n=\frac{\mu[n,+\infty)}{\mu_na_n},\qquad n\geqslant 1.$$
Therefore, the birth-death process is ergodic if and only if $D=\mu[1,+\infty)<\infty$, equivalently, $\mu:=\mu[0,+\infty)<\infty$; the birth-death process is strongly ergodic if and only if
$$S=\sum_{n\geqslant 1}\frac{\mu[n,+\infty)}{\mu_na_n}=\sum_{n\geqslant 0}\frac{\mu[n+1,+\infty)}{\mu_nb_n}<\infty.$$
}

%----------------------------------------------------------------------------------------16
\frame{\frametitle{Three examples }\small

\begin{block}{Example 2.} Given a constant $b>2$ (for regularity only needed that $b>1$). Let
$$q_{ij}=\frac{b-1}{b^{j-i+2}},\quad j\geqslant i+1;\quad q_{i,i-1}=\frac{b-1}b,\quad q_i=-q_{ii}=\frac {b^2-b+1}{b^2},\quad i\geqslant 1;$$
$$q_{0j}=\frac{b-1}{b^{j+1}},\qquad j\geqslant 1;\qquad q_0=-q_{00}=\frac 1b.$$
\end{block}

Then $q_n^{(k)}=1/b^{k-n+1}(1\leqslant n<k)$,  $q_0^{(k)}=1/b^{k}(k\geqslant 1)$,
$$G_n^{(i)}=\frac1 {b(b-1)^{i-n}},\qquad 1\leqslant n <i; \qquad m_n=\frac{b-1}{b-2},\qquad n\geqslant 1.$$
Therefore, by $D=1/(b-2)$ and $S=+\infty$, we know that the process is ergodic but not strongly ergodic.
}

%----------------------------------------------------------------------------------------17
\frame{\frametitle{Three examples }\small

The original branching process can be described as follows. Let $\alpha>0$ and $(p_j:j\in \mathbb Z_+)$ be a probability distribution. Then the process has death rate
$\alpha ip_0:i\to i-1\,(i\ge 1)$ and growth rate $\alpha ip_{k+1}:i\to i+k\,(k\ge 1,i\in \mathbb Z_+)$. An extended class of branching processes:

\begin{block}{Example 3.}
$$
q_{ij}=\begin{cases} q_{0j},\quad &j>i=0;\\-q_0,&j=i=0;\\r_ip_0,&j=i-1,\,i\ge 1;\\
r_ip_{k+1},&j=i+k,\,i,k\ge 1;\\-r_i(1-p_1),&j=i\ge 1;\\
0,&\text{else},\quad i,j\in \mathbb Z_+.\end{cases}$$
where $r_i> 0$ for all $i\ge 1$ and $0<q_0:=\sum_{j\ge 1}q_{0j}<\infty$.
\end{block}


{\blue R.R. Chen(1997). An extended class of time-continuous branching processes. J. Appl. Prob., 34(1), 14-23.}
}

%----------------------------------------------------------------------------------------18
\frame{\frametitle{Three examples }\small

\begin{block}{Theorem 3.} Assume that the extended branching $Q$ is irreducible and regular. Then
the process is ergodic if and only if
$$\sum_{\ell\geqslant 1}\frac 1{r_\ell }\bigg(q_0^{(\ell)}+\sum_{1\leqslant k\leqslant \ell-1}\frac {(\mathbf f^{*k}*\mathbf q)_{\ell-k+1}}{p_0^k}\bigg)<\infty; $$
it is strongly ergodic if and only if
$$\sum_{\ell\geqslant 1}\frac 1{r_\ell }\bigg(1+\sum_{1\leqslant k\leqslant \ell-1}\frac {(\mathbf f^{*k}*\mathbf 1)_{\ell-k+1}}{p_0^k}\bigg)<\infty.$$
where $\mathbf f=(\sum_{k\geqslant n}p_k,n\geqslant 2)$, $\mathbf 1=(1,1,\cdots)$ and $\mathbf q=(q_0^{(n-1)}, n\geqslant 2)$ and
$$(\mathbf a*\mathbf b)_n=\sum_{2\leqslant m\leqslant n}a_{n+2-m}b_m,\ n\geqslant 2.$$
\end{block}
}

%----------------------------------------------------------------------------------------19
\frame{\frametitle{Three examples }\small
Come back to Example 2. Fix a positive constant $a$ such that $a<1-1/(b^2-b+1)$. Then Example 2 is the special case of Example 3:
$$r_i=(b-1)/(ab), \ p_0=a, \ p_j=a/b^j(j\geqslant 2),\ p_1=1-a-a/(b^2-b).$$
$$
f_n=\frac a{(b-1)b^{n-1}},\qquad n\geqslant 2;\qquad q_0^{(\ell)}=\frac 1{b^\ell},\qquad \ell\geqslant 1.
$$
$$(\mathbf f^{*k}*\mathbf q)_{\ell-k+1}
=\frac {a^k}{(b-1)^kb^\ell}C_{\ell-1}^k, \qquad 1\leqslant k\leqslant \ell-1.$$

$$\sum_{1\leqslant k\leqslant \ell-1}\frac{(\mathbf f^{*k}*\mathbf q)_{\ell-k+1}}{p_0^k}=\frac 1{b(b-1)^{\ell-1}}-\frac 1{b^\ell}.$$

$$\sum_{\ell\geqslant 1}\frac 1{r_\ell }\bigg(q_0^{(\ell)}+\sum_{1\leqslant k\leqslant \ell-1}\frac {(\mathbf f^{*k}*\mathbf q)_{\ell-k+1}}{p_0^k}\bigg)=\frac a{b-2}<\infty. $$
}

%----------------------------------------------------------------------------------------20
\frame{\frametitle{Three examples }\small

Note that
$$(\mathbf f*\mathbf 1)_{\ell}=\frac a{(b-1)^2}\bigg(1-\frac 1{b^{\ell-1}}\bigg).$$
From the equality above, it follows that
$$ \sum_{\ell\geqslant 1}\frac 1{r_\ell }\bigg(1+\sum_{1\leqslant k\leqslant \ell-1}\frac {(\mathbf f^{*k}*\mathbf 1)_{\ell-k+1}}{p_0^k}\bigg)
\geqslant  \sum_{\ell\geqslant 2}\frac 1{r_\ell }\cdot\frac {(\mathbf f*\mathbf 1)_{\ell}}{p_0}=\infty.
$$
So the process in Example 2 is ergodic but not strong ergodic. This is the case of $b>2$. Note that
$$M_1:=\sum_{k=1}^\infty kp_k=\frac a{(b-1)^2}+1-a.$$
So $M_1\leqslant 1$ if and only if $b\geqslant 2$. Hence, it is easy to be known that the process is unique and null recurrent when $b=2$; the process is unique and transient when $1<b<2$.
}

\section{Exponential ergodicity}

%----------------------------------------------------------------------------------------21
\frame{\frametitle{Exponential ergodicity }\small
\begin{block}{}
Define $G_n=\varliminf_{m\to\infty}{G_n^{(m)}}/{G_1^{(m)}},\ n\geqslant 1$.
Then
$$
G_n\geqslant\frac 1{q_{n,n-1}}\sum_{k=n+1}^\infty q_n^{(k)}G_k,\quad n\geqslant 1.$$
\end{block}

\begin{block}{Theorem 4.} Let the single death $Q$-matrix be regular and irreducible.
Assume that
$$
\sum_{k\geqslant 1} q_0^{(k)}G_k<\infty.$$
If
$$q:=\inf_{n\geqslant 0}q_n>0\quad\text{and}\quad M:=\sup_{n\geqslant 1}\Bigg[\sum_{k=1}^{n}G_k\Bigg]\Bigg[\sum_{j=n}^\infty\frac 1{q_{j,j-1}G_j}\Bigg]<\infty,$$
then the process is exponentially ergodic.
\end{block}
}

%----------------------------------------------------------------------------------------22
\frame{\frametitle{Exponential ergodicity }\small

\begin{block}{Theorem [Tweedie(1981)]} Given a regular irreducible $Q$-matrix $Q=(q_{ij})$. Then the process
$P(t)$ is exponentially ergodicity if and only if for some $\lambda>0$ with $\lambda <q_i$ for all $i$,
$$\begin{cases}y_i\geqslant 1,\quad i\in E&\\ \sum_j q_{ij}y_j\leqslant -\lambda y_i,\quad i\notin H&\\
\sum_{i\in H}\sum_{j\ne i}q_{ij}y_j<\infty&
\end{cases}$$
has a finite solution $(y_i)$.
\end{block}
}
%----------------------------------------------------------------------------------------22
\frame{\frametitle{Exponential ergodicity }\small
Sketch of proof: Define an operator
$$I\!I_i(f)=\frac 1{f_i}\sum_{j=1}^{i}G_j\sum_{k=j}^\infty \frac {f_k}{q_{k,k-1}G_k},
\quad i\geqslant 1$$
and
$$\varphi_i=\sum_{j=1}^{i}G_j, \qquad i\geqslant 1.$$
Then $\varphi$ is increasing in $i$ and $\varphi_1=G_1$.
Let $f=cq_{10}\sqrt{\varphi}/\sqrt{G_1}$ for some $c> 1$. Then $f$ is increasing and $f_1=cq_{10}$. Finally, define
$g=fI\!I(f)$ and $g_0=1$. Then $g$ is increasing, $1\leqslant g_i<\infty$ for all $i\geqslant 0$ and
$\sum_{j\geqslant 1}q_{0j}g_j<\infty$.
Then we can take any $\lambda$:
$$
0<\lambda<\Bigg(\frac {c-1}c\,I\!I_1(f)^{-1}\Bigg)\bigwedge\Bigg(\inf_{i\geqslant 2}I\!I_i(f)^{-1}\Bigg)\bigwedge q.
$$
provided the right-hand side is positive or, equivalently,
$\sup_{i\geqslant 1}I\!I_i(f)<\infty$.
}

%----------------------------------------------------------------------------------------22
\frame{\frametitle{Exponential ergodicity }\small

To prove the last property, define another operator
$$I_i(f)=\frac {G_{i+1}}{f_{i+1}-f_i}\sum_{k=i+1}^\infty \frac {f_k}{q_{k,k-1}G_k}.
\quad i\geqslant 1,$$
By the proportion property, we get
$$\sup_{i\geqslant 1}I\!I_i(f)\leqslant \sup_{i\geqslant 1}I_i(f).$$
By the condition $M<\infty$, it follows that
$$I_i(f)=\frac {G_{i+1}}{\sqrt{\varphi_{i+1}}-\sqrt{\varphi_i}}\sum_{k=i+1}^\infty
\frac {\sqrt{\varphi_k}}{q_{k,k-1}G_k}\leqslant\frac{2MG_{i+1}}
{(\sqrt{\varphi_{i+1}}-\sqrt{\varphi_i})\sqrt{\varphi_{i+1}}}\leqslant 4M$$
for all $i\geqslant 1$. Therefore, $\sup_{i\geqslant 1}I\!I_i(f)\leqslant 4M<\infty$ as required. We have thus constructed a solution
$(g_i)$ to the equation with $1\leqslant g_i<\infty$ for all $i\geqslant 0$ and $\sum_{j\geqslant 1}q_{0j}g_j<\infty$. This implies
the exponential ergodicity of the single death process.
}

%----------------------------------------------------------------------------------------23
\frame{\frametitle{Exponential ergodicity }\small
For the birth-death process $(a_i, b_i)$, $G_n^{(m)}/G_1^{(m)}=\mu_1a_1/(\mu_na_n)$. Thus
$$G_n=\lim_{m\to\infty}\frac {G_n^{(m)}}{G_1^{(m)}}=\frac {\mu_1a_1}{\mu_na_n},\qquad n\geqslant 1$$
and  $\sum_{k\geqslant 1} q_0^{(k)}G_k=b_0<\infty$. The equality holds. Now
$$M=\sup_{n\geqslant 1}\Bigg[\sum_{k=1}^{n}\frac 1{\mu_ka_k}\Bigg]\Bigg[\sum_{j=n}^\infty\mu_j\Bigg]=\sup_{n\geqslant 1}\Bigg[\sum_{k=0}^{n-1}\frac 1{\mu_kb_k}\Bigg]\Bigg[\sum_{j=n+1}^\infty\mu_j\Bigg].
$$
The birth-death process is exponentially ergodic if and only if $M<\infty$.
}

%----------------------------------------------------------------------------------------24
\frame{\frametitle{Exponential ergodicity }\small
Come back to Example 2 for $b>2$. We see that $G_n^{(m)}/G_1^{(m)}=(b-1)^{n-1}$. Then
$$G_n=\lim_{m\to\infty}\frac {G_n^{(m)}}{G_1^{(m)}}=(b-1)^{n-1},\qquad n\geqslant 1,$$
and $\sum_{k\geqslant 1} q_0^{(k)}G_k=1<\infty$. The equality holds. Note that  $q=1/b>0$ and
$$M=\frac{b(b-1)}{(b-2)^2}<\infty.$$
Hence, the process is exponentially ergodic.
}

%----------------------------------------------------------------------------------------25
\frame{\frametitle{Exponential ergodicity }\small
Define
$$a_i=q_{i,i-1},\qquad b_i=\frac{q_{i,i-1}G_i}{G_{i+1}},\qquad i\geqslant 1;\qquad b_0=\frac{q_{10}}{G_1}.$$
Then $\mathbb E_i\tau_0^n\leqslant \mathbb E_i\bar\tau_0^n, \ n\geqslant 0, i\geqslant 1.$

\begin{block}{Theorem 5.} Let the single death $Q$-matrix be regular and irreducible.
Assume that $\sum_{k\geqslant 1} G_k=\infty$.
If the birth-death process is ergodic (or exponentially ergodic, or strongly ergodic respectively), then
so is the single death process.
$$\aligned &\sum_{i\geqslant 1}\frac 1{q_{i,i-1}G_i}<\infty\Rightarrow \text{ergodic}\\
&M<\infty\Rightarrow \text{exponentially ergodic}\\
&\sum_{n\geqslant 1}G_n\sum_{k\geqslant n}\frac 1{q_{k,k-1}G_k}<\infty\Rightarrow \text{strongly ergodic}
\endaligned$$
\end{block}
}
\section{Recurrence and return probability}

%----------------------------------------------------------------------------------------26
\frame{\frametitle{Recurrence and return probability}\small

Define
$$\widetilde G_n=\varlimsup_{m\to\infty}\frac {G_n^{(m)}}{G_1^{(m)}},\qquad n\geqslant 1.$$
Assume that
$$\sum_{k\geqslant n+1}\frac {q_n^{(k)}}{G_1^{(k)}}<\infty,\qquad n\geqslant 1.\qquad\qquad (**)$$
Then
$$
\widetilde G_n\leqslant\frac 1{q_{n,n-1}}\sum_{k=n+1}^\infty q_n^{(k)}\widetilde G_k,\qquad n\geqslant 1.$$

\begin{block}{Theorem 6.} Let the single death $Q$-matrix be regular and irreducible.
If
$\sum_{n\geqslant 1}G_n=\infty,$
then the process is recurrent. Conversely, under the assumption ($**$),
if  the process is recurrent, then $\sum_{n\geqslant 1}\widetilde G_n=\infty$.
\end{block}
}

%----------------------------------------------------------------------------------------27
\frame{\frametitle{Recurrence and return probability}\small

\begin{itemize}
\item
For the birth-death process $(a_i, b_i)$, we have
$$\sum_{n\geqslant 1}G_n=\sum_{n\geqslant 1}\widetilde G_n=\sum_{n\geqslant 1}\frac {\mu_1a_1}{\mu_na_n}=\sum_{n\geqslant 0}\frac {b_0}{\mu_nb_n}$$
$$\sum_{k\geqslant n+1}\frac {q_n^{(k)}}{G_1^{(k)}}=\frac{b_0}{\mu_n}<\infty,\qquad n\geqslant 1.$$
Hence the birth-death process is recurrent if and only if $\sum_{n\geqslant 0} {1}/(\mu_nb_n)=\infty$.

\item
For Example 2 when $b>1$. Then
$$\sum_{k\geqslant n+1}\frac {q_n^{(k)}}{G_1^{(k)}}=(b-1)^n<\infty,\qquad n\geqslant 1$$
$$\sum_{n\geqslant 1}G_n=\sum_{n\geqslant 1}\widetilde G_n=\sum_{n\geqslant 1}(b-1)^{n-1},$$
which is infinite when $b\geqslant 2$ and finite when $1<b<2$. So the process is recurrent if and only if $b\geqslant 2$.

\end{itemize}
}

%----------------------------------------------------------------------------------------28
\frame{\frametitle{Recurrence and return probability}\small

Define $G_0=\frac 1{q_0}\sum_{n\geqslant 1}q_0^{(n)}G_n$, $G=\sum_{n\geqslant 1}G_n$, $\widetilde G_0=\frac 1{q_0}\sum_{n\geqslant 1}q_0^{(n)}\widetilde G_n$,  $\widetilde G =\sum_{n\geqslant 1}\widetilde G_n.$

\begin{block}{Theorem 7.} Let the single death $Q$-matrix be regular and irreducible. Assume that $(**)$ holds.
Then
$$\mathbb P_0(\sigma_0<\infty)\leqslant 1-\frac{\widetilde G_0}{\widetilde G},\ \mathbb P_n(\sigma_0<\infty)\leqslant 1-\frac 1{\widetilde G}\sum_{1\leqslant k\leqslant n}\widetilde G_k,\  n\geqslant 1.$$
In addition, assume that $\displaystyle
\varliminf_{m\to\infty}\frac 1{G_1^{(m)}}\sum_{1\leqslant j\leqslant k}G_j^{(m)}=\sum_{1\leqslant j\leqslant k}G_j,\ k\geqslant 1.\qquad (***)$

Then
$$\mathbb P_0(\sigma_0<\infty)\geqslant 1-\frac {G_0}{G},\ \mathbb P_n(\sigma_0<\infty)\geqslant 1-\frac 1G\sum_{1\leqslant k\leqslant n}G_k,\  n\geqslant 1.$$
\end{block}
}

%----------------------------------------------------------------------------------------28
\frame{\frametitle{Recurrence and return probability}\small
\begin{block}{Lemma 3.}
$$x_0=1-\frac {\widetilde G_0}{\widetilde G},\qquad x_k=1-\frac 1{\widetilde G}\sum_{1\leqslant n\leqslant k}\widetilde G_n$$
satisfies the inequality
$x_i\geqslant\sum_{k\ne 0,i}\frac{q_{ik}}{q_i}x_k+\frac{q_{i0}}{q_i}(1-\delta_{i0}),\ \ i\geqslant 0.$
\end{block}

$(\mathbb P_i(\sigma_0<\infty))$ is the minimal nonnegative solution to the equation above. Thus
$$\mathbb P_i(\sigma_0<\infty)\leqslant x_i,\qquad i\geqslant 0.$$

\begin{block}{Lemma 4.}
$$\mathbb P_k(\tau_0<\tau_m)=\frac{\sum_{k+1\leqslant n\leqslant m}G_n^{(m)}}{ \sum_{1\leqslant n\leqslant m}G_n^{(m)}},\qquad k\geqslant 0.$$
\end{block}

$$\mathbb P_k(\sigma_0<\infty)\geqslant 1-\frac 1G\sum_{1\leqslant n\leqslant k}G_n,\qquad k\geqslant 1.$$

}

%----------------------------------------------------------------------------------------28
\frame{\frametitle{Recurrence and return probability}\small

\begin{block}{Remark 1.} If there exist the limits
$$\lim_{m\to\infty}\frac{G_n^{(m)}}{G_1^{(m)}}$$
and the assumption ($**$) holds, then $G_n=\widetilde G_n\,(n\geqslant 0)$ and ($***$) holds. Hence
$$\mathbb P_0(\sigma_0<\infty)= 1-\frac{G_0}{G},\qquad \mathbb P_n(\sigma_0<\infty)= 1-\frac 1{G}\sum_{1\leqslant n\leqslant k}G_n,\qquad  n\geqslant 1.$$
Thus, $\mathbb P_i(\sigma_0<\infty)=1$ for all $i\geqslant 1$ if and only if $\mathbb P_0(\sigma_0<\infty)=1$, equivalently, if and only if $G=\infty$\,($G_0<\infty$). See Example 1 and 2.
\end{block}

}

\section{Birth-death processes on trees}
%----------------------------------------------------------------------------------------28
\frame{\frametitle{Birth-death processes 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 processes 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 $j\in J(i)$.

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

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 processes on trees}\small

\begin{block}{Theorem 8.} Assume that the process corresponding to the $Q$-matrix on the tree $T$ is recurrent. Then
$$\mathbb E_i\tau_k^n=n\sum_{j\in\mathscr P(i)\setminus\mathscr P(k)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell\mathbb E_\ell\tau_k^{n-1},
\qquad k\in \mathscr P(i),$$
$$\mathbb E_i\tau_o^n=n\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell\mathbb E_\ell\tau_o^{n-1},\qquad i\ne o;$$
in particular,
$$\mathbb E_i\sigma_o=\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell,\qquad i\ne o,$$
$$\mathbb E_o\sigma_o=\frac \mu{q_o};$$
\end{block}
}
%----------------------------------------------------------------------------------------28
\frame{\frametitle{Birth-death processes on trees}\small
\begin{block}{Theorem 8.(continued)}
Furthermore, the process is ergodic if and only if $\mu<\infty$;
the process is strongly ergodic if and only
$$S:=\sup_{k\ne o}\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell<\infty.$$
Actually, for the last conclusion, the recurrence assumption can be replaced by the uniqueness one.
\end{block}
}

%----------------------------------------------------------------------------------------28
\frame{\frametitle{Birth-death processes on trees}\small
Define $q_i^{(+)}=\sum_{j\in J(i)}q_{ij}$ for all $i\in T$ and
$$h_i=\frac n{\mu_iq_{ii^*}}\sum_{k\in T_i}\mu_kq_{kk^*}f_k,\qquad i\ne o,$$
where
$$f_i=\frac 1{q_{ii^*}}\Bigg(\mathbb E_i\tau_{i^*}^{n-1}+\sum_{j\in J(i)}q_{ij}\sum_{m=0}^{n-2}\binom {n-1} m\frac 1{m+1}\mathbb E_j\tau_{j^*}^{m+1}\mathbb E_{i}\tau_{i^*}^{n-1-m}\Bigg), \qquad i\ne o.$$

\begin{block}{Lemma 5.} $(h_i,i\ne o)$ is the minimal nonnegative solution of the following equations
$$
y_i=\frac{q_{i}^{(+)}}{q_i}y_i+\sum_{j\in J(i)}\frac{q_{ij}}{q_i}y_j+\frac n{q_i}\mathbb E_i\tau_{i^*}^{n-1},\qquad i\ne o
$$
and
$$h_i=\frac n{\mu_iq_{ii^*}}\sum_{k\in T_i}\mu_k\mathbb E_k\tau_{i^*}^{n-1},\qquad i\ne o.$$
\end{block}
}

%----------------------------------------------------------------------------------------28
\frame{\frametitle{Birth-death processes on trees}\small
\begin{block}{Lemma 6.} Assume that the process corresponding to the $Q$-matrix on the tree $T$ is recurrent.  Then
$$\mathbb E_i\tau_k^n\leqslant n\sum_{j\in\mathscr P(i)\setminus\mathscr P(k)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell\mathbb E_\ell\tau_k^{n-1},
\qquad k\in \mathscr P(i)$$
and
$$\mathbb E_i\tau_o^n\leqslant n\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell\mathbb E_\ell\tau_o^{n-1}. $$
In particular,
$$\mathbb E_i\tau_{i^*}^n\leqslant \frac n{\mu_i q_{ii^*}}\sum_{k\in T_i}\mu_k\mathbb E_k\tau_{i^*}^{n-1},\qquad i\ne o.$$
\end{block}
}
%----------------------------------------------------------------------------------------28
\frame{\frametitle{Birth-death processes on trees}\small

Under the assumption that the process is recurrent. Then
$$\mathbb E_i\tau_{i^*}^n=\frac n{\mu_iq_{ii^*}}\sum_{k\in T_i}\mu_k\mathbb E_k\tau_{i^*}^{n-1},\qquad i\ne o.$$
$$\mathbb E_i\tau_{k}^n=n\sum_{j\in\mathscr P(i)\setminus\mathscr P(k)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell\mathbb E_\ell\tau_k^{n-1},
\qquad k\in \mathscr P(i).$$

{\blue S. Martinez, B. Ycart(2001). Decay rates and cutoff for convergence and hitting times of Markov chains with countably infinite state space. Adv. Appl. Prob., 33(1): 188-205}
}
%----------------------------------------------------------------------------------------28
\frame{\frametitle{Birth-death processes on trees}\small

\begin{block}{Corollary 2.} Assume that the process is exponential ergodic. Then
$$\delta:=\sup_{i\in T\setminus\{o\}}\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_i}\mu_\ell<\infty.$$
\end{block}

$$\mathbb E_i\tau_o^n=n\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_j}\mu_\ell \mathbb E_\ell\tau_o^{n-1},\qquad i\in T, n\geqslant 1.
$$

$$\aligned \mathbb E_i\tau_o^n&\geqslant n\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_i}\mu_\ell\mathbb E_\ell\tau_o^{n-1}
\geqslant n\Bigg(\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_i}\mu_\ell\Bigg)\mathbb E_i\tau_o^{n-1}\\
&\geqslant\cdots\geqslant n!\Bigg(\sum_{j\in\mathscr P(i)}\frac 1{\mu_jq_{jj^*}}\sum_{\ell\in T_i}\mu_\ell\Bigg)^n,
\qquad n\geqslant 1.\endaligned$$
}


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