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\begin{document}
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\title[单死过程首达时的矩与遍历理论]{单死过程首达时的矩与遍历理论}

\author[张余辉]{\sihao 张余辉}

\date[10.20, 2019]{\wuhao 2019年随机分析及其应用研讨会\\ (复旦大学, 10.17-20, 2019) }

\institute[北京师范大学]{\wuhao 北京师范大学}

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


%----------------------------------------------------------------------------------------2
\frame{ \frametitle{Background}\small
\begin{itemize}
\item $Q$-process: 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
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{Background}\small
\begin{itemize}

\item branching processes:

\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: $\sup_{n\geqslant 1}\mathbb E_n\tau_0$.

\item 2017.6.29-2017.7.1, Nanning, The 6th IMS-China International Conference on Statistics and Probability, Notes on single death processes.

\end{itemize}
}
\section{First hitting times, ergodicity and strong ergodicity}

%----------------------------------------------------------------------------------------6
\frame{\frametitle{First hitting times, 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{First hitting times, 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
$\sigma_H=\inf\{t\geqslant\text{\,the first jump time}: X(t)\in H\}$.

\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{First hitting times, 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{First hitting times, 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{First hitting times, 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{First hitting times, 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 Define $\tau_i=\inf\{t\geqslant 0: X(t)=i\}$. $\mathbb E_n\tau_{n-1}=?$
\end{itemize}

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

}

%----------------------------------------------------------------------------------------9
\frame{\frametitle{First hitting times, 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 side by $m_n$, the right hand side by $h_n$.
\begin{block}{}
$$h_n=\frac 1{q_{n,n-1}}\Bigg(1+\sum_{k\geqslant n+1}q_n^{(k)}h_k\Bigg).$$
\end{block}
}


%----------------------------------------------------------------------------------------10
\frame{\frametitle{First hitting times, 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{First hitting times, 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{First hitting times, 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{First hitting times, ergodicity and strong ergodicity}\small

\begin{block}{Lemma 2. Under the conditions of Theorem 1}
$(\mathbb E_i\tau_{i_0}: i\geqslant i_0)$ is the minimal nonnegative solution to
$$
x_{i_0}=0,\qquad x_i=\sum_{j\ne i,j> i_0}\frac {q_{ij}}{q_i} x_j+\frac 1{q_i},\qquad  i> i_0.
$$
\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{First hitting times, 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{First hitting times, 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:
$$
\lambda y_i=\sum_{j=0}^\infty y_jq_{ji},\qquad y_i\geqslant 0,\ i\geqslant 0\quad\text{ and }\quad \sum_{i=0}^\infty y_i<\infty.
$$
\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{First hitting times, 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{First hitting times, 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}
}

%----------------------------------------------------------------------------------------15
%\frame{\frametitle{Three examples }\small
\frame{\frametitle{First hitting times, ergodicity and strong ergodicity}\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{First hitting times, ergodicity and strong ergodicity}\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{First hitting times, ergodicity and strong ergodicity}\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 $\alpha ip_{k+1}:i\to i+k\,(k\geqslant 1,i\in \mathbb Z_+)$. An extended class of branching processes:

\begin{block}{Example 3.}
$$
q_{ij}=\begin{cases} q_{0j},\quad &j>i=0;\\-q_0,&j=i=0;\\r_ip_0,&j=i-1,\,i\geqslant 1;\\
r_ip_{k+1},&j=i+k,\,i,k\geqslant 1;\\-r_i(1-p_1),&j=i\geqslant 1;\\
0,&\text{else},\quad i,j\in \mathbb Z_+.\end{cases}$$
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{First hitting times, ergodicity and strong ergodicity}\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{First hitting times, ergodicity and strong ergodicity}\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
\frame{\frametitle{First hitting times, ergodicity and strong ergodicity}\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{High-order moments and exponential ergodicity}

%----------------------------------------------------------------------------------------21
\frame{\frametitle{High-order moments and exponential ergodicity}\small

\begin{block}{Theorem 4.} Assume that the single death $Q$ is irreducible and 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}{Corollary 2.} Assume that the single death $Q$-matrix is regular and the process is exponentially ergodic. Then
$$\delta:=\sup_{i\geqslant 1}\sum_{1\leqslant k\leqslant i}\sum_{\ell\geqslant i}\frac {G_k^{(\ell)}}{q_{\ell,\ell-1}}<\infty.$$
\end{block}
}
%-------------------------------------------------------------------------22
\frame{\frametitle{High-order moments and exponential ergodicity}\small
\begin{block}{} $\ell$-ergodicity: $\mathbb E_j\sigma_j^\ell<\infty$ for some $j$. {\red $1$-erg. $=$ posit. recur., $0$-erg. $=$ null recur.}\end{block}

\begin{block}{Corollary 3.} Assume that the single death $Q$-matrix is regular and the process is recurrent. Then the process is $\ell$-ergodic if and only if
$$d_\ell:=\sum_{k\geqslant 1}q_{0}^{(k)}\sum_{j\geqslant k}\frac{G_k^{(j)}}{q_{j,j-1}}\mathbb E_j\tau_0^{\ell-1}<\infty.$$
\end{block}
}
%----------------------------------------------------------------------------------------21
\frame{\frametitle{High-order moments and exponential ergodicity}\small

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

\begin{block}{}
Define
$$
M_{ik}^{(n-1)}=\sum_{i+1\leqslant \ell\leqslant k}\sum_{1\leqslant s\leqslant n-1}\binom {n}{s}\mathbb E_{\ell}\tau_{\ell-1}^{n-s}\mathbb E_{\ell-1}\tau_{i-1}^s,$$
$$%\begin{equation}\label{e2.2-}
M_{i}^{(n-1)}=\mathbb E_i\tau_{i-1}^{n-1}+\frac 1n\sum_{k\geqslant i+1}q_{ik}M_{ik}^{(n-1)},\qquad n\geqslant 1.
$$%\end{equation}
\end{block}


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

}

%----------------------------------------------------------------------------------------21
\frame{\frametitle{High-order moments and exponential ergodicity}\small

\begin{block}{Proposition. Under the conditions of Theorem 4}
$$\mathbb E_i\tau_{i-1}^n=n\sum_{k\geqslant i}\frac{G_i^{(k)}M_k^{(n-1)}}{q_{k,k-1}},\ i\geqslant 1, n\geqslant 1.$$
\end{block}

\begin{block}{Lemma A. For all $1\leqslant i\leqslant v<u$,}
$$G_i^{(u)}=\sum_{i\leqslant k\leqslant v}\frac{G_i^{(k)}}{q_{k,k-1}}\sum_{\ell=v+1}^u q_{k}^{(\ell)}G_\ell^{(u)}, \qquad 1\leqslant i\leqslant v<u.$$
\end{block}

\begin{block}{Lemma B. Under the conditions of Theorem 4} For any nonnegative sequence $\{a_n\}$ and $n\geqslant 1$,
$$\sum_{k\geqslant i}\frac{G_i^{(k)}}{q_{k,k-1}}\sum_{\ell_1\geqslant k+1}q_{k}^{(\ell_1)}\mathbb E_{\ell_1}\tau_{\ell_1-1}^{n}\sum_{\ell_2=k}^{\ell_1-1}a_{\ell_2}
=n\sum_{u\geqslant i+1}\frac{G_i^{(u)}M_u^{(n-1)}}{q_{u,u-1}}\sum_{\ell_2=i}^{u-1}a_{\ell_2}.$$
\end{block}

}


%----------------------------------------------------------------------------------------21
\frame{\frametitle{High-order moments and exponential ergodicity}\small
\begin{block}{Lemma C. Under the conditions of Theorem 4} For any $1\leqslant i\leqslant k< \ell_1$ and $n\geqslant 2$,
$$M_{k,\ell_1-1}^{(n-1)}+\sum_{s=1}^{n-1}\binom{n}{s}\mathbb E_{\ell_1-1}\tau_{k-1}^{n-s}\mathbb E_{k-1}\tau_{i-1}^{s}=\sum_{\ell_2=k}^{\ell_1-1}\sum_{s=1}^{n-1}\binom{n}{s}\mathbb E_{\ell_2}\tau_{\ell_2-1}^{n-s}\mathbb E_{\ell_2-1}\tau_{i-1}^{s}.$$
\end{block}

\begin{block}{Lemma D. Under the conditions of Theorem 4} For all $1\leqslant s_1< n$,
$$\aligned &\sum_{k\geqslant i}\frac{G_i^{(k)}}{q_{k,k-1}}\sum_{\ell_1\geqslant k+1}q_{k}^{(\ell_1)}\mathbb E_{\ell_1}\tau_{\ell_1-1}^{n-s_1}\Bigg(M_{k,\ell_1-1}^{(s_1-1)}+\sum_{s_2=1}^{s_1-1}\binom{s_1}{s_2}\mathbb E_{\ell_1-1}\tau_{k-1}^{s_1-s_2}\mathbb E_{k-1}\tau_{i-1}^{s_2}\Bigg)\\
&\qquad =(n-s_1)\sum_{u\geqslant i+1}\frac{G_i^{(u)}}{q_{u,u-1}}M_u^{(n-s_1-1)}M_{i,u-1}^{(s_1-1)}.\endaligned$$
\end{block}
}
%----------------------------------------------------------------------------------------21
\frame{\frametitle{High-order moments and exponential ergodicity}\small

\begin{block}{}
$$\mathbb E_i\tau_{i-1}^{n}=n\sum_{k\geqslant i}\frac{G_i^{(k)}\mathbb E_k\tau_{i-1}^{n-1}}{q_{k,k-1}},\qquad i\geqslant 1,n\geqslant 1.$$

$$\mathbb E_i\tau_{k}^{n}=n\sum_{k+1\leqslant j\leqslant i}\sum_{\ell\geqslant j}\frac{G_j^{(\ell)}\mathbb E_\ell\tau_{k}^{n-1}}{q_{\ell,\ell-1}},\qquad 0\leqslant k<i,n\geqslant 1.$$
\end{block}
}
%----------------------------------------------------------------------------------------21
\frame{\frametitle{High-order moments and 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 5.} Let the single death $Q$-matrix be regular and irreducible.
Assume that
$$
\sum_{k\geqslant 1} q_0^{(k)}G_k<\infty.$$
If
$$q:=\inf_{n\geqslant 0}q_n>0\quad\text{and}\quad \sup_{n\geqslant 1}\Bigg[\sum_{k=1}^{n}G_k\Bigg]\Bigg[\sum_{j=n}^\infty\frac 1{q_{j,j-1}G_j}\Bigg]<\infty,$$
then the process is exponentially ergodic.
\end{block}
}

%----------------------------------------------------------------------------------------22
\frame{\frametitle{High-order moments and 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}
}


%----------------------------------------------------------------------------------------23
\frame{\frametitle{High-order moments and 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
$$M=\sup_{n\geqslant 1}\Bigg[\sum_{k=1}^{n}\frac 1{\mu_ka_k}\Bigg]\Bigg[\sum_{j=n}^\infty\mu_j\Bigg]=\sup_{n\geqslant 1}\Bigg[\sum_{k=0}^{n-1}\frac 1{\mu_kb_k}\Bigg]\Bigg[\sum_{j=n}^\infty\mu_j\Bigg].
$$
The birth-death process is exponentially ergodic if and only if $M<\infty$.
}

%----------------------------------------------------------------------------------------24
\frame{\frametitle{High-order moments and 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.
}

%----------------------------------------------------------------------------------------24
\frame{\frametitle{High-order moments and exponential ergodicity }\small
\begin{block}{Theorem 6.} Let the single death $Q$-matrix be regular and irreducible.
Assume that
$$
\sum_{k\geqslant 1} \frac{q_0^{(k)}}{G_1^{(k)}}<\infty.$$
If
$$q:=\inf_{n\geqslant 0}q_n>0\quad\text{and}\quad M:=\sup_{n\geqslant 1}\Bigg(\sum_{k=1}^{n}\frac 1{G_1^{(k)}}\Bigg)\Bigg(\sum_{j=n}^\infty\frac {G_1^{(j)}}{q_{j,j-1}}\Bigg)<\infty,$$
then the process is exponentially ergodic.
\end{block}

\begin{block}{} $$M\geqslant \delta.$$\end{block}
}

%----------------------------------------------------------------------------------------22
\frame{\frametitle{High-order moments and exponential ergodicity}\small
\begin{block}{} Define a positive matrix $H=(h_{ij})_{i,j\geqslant 1}$:
$$h_{ij}=\frac 1{q_{j,j-1}}\sum_{k=1}^{i\wedge j}G_{k}^{(j)},\quad H=((-Q)|_{i\geqslant 1})^{-1}=\bigg(\int_{0}^{\infty}p_{ij}^{\text{min}}(t)\text{d}t\bigg)_{i,j\geqslant 1}.$$
\end{block}

$$\mathbb E_k\text{e}^{\alpha\tau_0}=\sum_{n=0}^\infty((\alpha H)^n\mathbbold 1)(k).$$

\begin{block}{}
$$\frac 1{\alpha_{\text{max}}}=\inf_{y\in\mathscr F}\sup_{k\geqslant 1}\frac{(Hy+H\mathbbold 1)(k)}{y_k}<\infty \Leftrightarrow \text{exp. erg.}\Leftrightarrow \lambda_{\text{max}}<\infty.$$
where
$$\mathscr F=\{y: 0<y\upuparrows, \sum_{k\geqslant 1}q_{0k}y_k<\infty\}.$$
\end{block}

}

%----------------------------------------------------------------------------------------22
\frame{\frametitle{High-order moments and exponential ergodicity}\small

{\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(王梓坤).

\vskip0.2cm
定义

$$\xi_i=\int_{0}^{\tau_i} V(X(t))\text{ d} t, V\geqslant 0, V\text{不恒为}0.$$

\begin{block}{}
$$\mathbb{E}_k \xi_i^{n}=n \sum_{i+1 \leqslant j \leqslant k} \sum_{\ell \geqslant j} \frac{G_j^{(\ell)}V(\ell) \mathbb{E}_{\ell} \xi_i^{n-1}}{q_{\ell, \ell-1}},\qquad k\geqslant i\geqslant 0, n\geqslant 1.$$
\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 7.} 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 8.} 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{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. \& Yan. 单死过程一些数字特征的概率意义, 北京师范大学学报, 2018, 54(6): 695-700.
\item Z. 单死过程研究的一些最新进展, 中国科学: 数学, 2020.
\item Z. Moments of the first hitting times for birth-death processes on trees, Frontiers of Mathematics in China, 2019, 14(4): 833-854.
\item Z. \& Zhou. High-order moments of the first hitting times for single death processes, Frontiers of Mathematics in China, 2019. https://rdcu.be/bPTMU
\item 王婧, Z. 单死过程的向下积分型泛函, 2019.
\item Wang \& Z. High-order moments of integral-type functional downward of single death processes, 2019.
\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}
