%  First, use latex to compile the file. Second, use dvi2ps to become .ps file. You can use Gsview to watch it. But
%  I suggest you use ps2pdf to become .pdf file. Because .ps file is for making a final film.
%  There are nine types of backgrounds. You can revise the second parameter. Now I use the darkblue background. All
%  backgrounds are the following:
%  alienglow, autumn, azure, contemporain, darkblue, frames, lignesbleues, nuancegris, troispoints,
% prettybox,rico, winter, corners, whitecross, wj, blends, alcatel,capsules, fyma, gyom, mancini, serpaggi, thomasd.
% Of course, you can make your background if you have the ability and you wish.

\documentclass[pdf,wj,slideColor,colorBG]{prosper}

\usepackage[latin1]{inputenc}
\usepackage{pstricks,pst-node,pst-text,pst-3d}
\usepackage{amsmath}
\usepackage{CJK}
% Definition of new colors
\newrgbcolor{LemonChiffon}{1. 0.98 0.8}
\newrgbcolor{LightBlue}{0.68 0.85 0.9}

% > BEGIN OF OVERLAPPED COLORS
% Code below devised by Pingji Deng (BNU/ - China, dengpj@eyou.com)
\newrgbcolor{LemonChiffon}{1. 0.98 0.8}
\newrgbcolor{LightBlue}{0.68 0.85 0.9}
\makeatletter
\newdimen\pst@dimz

% Draw two overlapped surfaces, with computation of the mixed color for
% the intersection of the surfaces
% #1=first  surface, #2=color of first  surface,
% #3=second surface, #4=color of second surface
\def\ColoredOverlappedSurfaces#1#2#3#4{%
\psset{fillstyle=solid}
% Decode the three components of the first RGB color
\DecodeRGBFirstColor{\csname color@#2\endcsname}%
\psset{fillcolor=#2}
% Draw first surface
#1
% Decode the three components of the second RGB color
\DecodeRGBSecondColor{\csname color@#4\endcsname}%
% Compute the mixed color
\BuildMixedColor
% Draw second surface
\psclip{\psset{fillcolor=#4}#3}
\psset{fillcolor=MixedColor}
% Redraw overlapped surface in the mixed color
#1
\endpsclip}

% Get the three components of the first color
\def\DecodeRGBFirstColor#1{%
\pst@expandafter\pst@getnumiii{#1} {} {} {} {}\@nil
\edef\pst@FirstColorR{\pst@tempg}%
\edef\pst@FirstColorG{\pst@temph}%
\edef\pst@FirstColorB{\pst@tempi}%
%\typeout{Color 1=\pst@tempg,\pst@temph,\pst@tempi}% Debug
}

% Get the three components of the second color
\def\DecodeRGBSecondColor#1{%
\pst@expandafter\pst@getnumiii{#1} {} {} {} {}\@nil
\edef\pst@SecondColorR{\pst@tempg}%
\edef\pst@SecondColorG{\pst@temph}%
\edef\pst@SecondColorB{\pst@tempi}%
%\typeout{Color 2=\pst@tempg,\pst@temph,\pst@tempi}% Debug
}

% Build the mixed RBG color (by means of each three components)
\def\BuildMixedColor{%
% Resulting R component
\pst@dimz=\pst@FirstColorR pt
\advance\pst@dimz\pst@SecondColorR pt
\divide\pst@dimz\tw@
\pst@dimtonum{\pst@dimz}{\pst@MixedColorR}%
% Resulting G component
\pst@dimz=\pst@FirstColorG pt
\advance\pst@dimz\pst@SecondColorG pt
\divide\pst@dimz\tw@
\pst@dimtonum{\pst@dimz}{\pst@MixedColorG}%
% Resulting B component
\pst@dimz=\pst@FirstColorB pt
\advance\pst@dimz\pst@SecondColorB pt
\divide\pst@dimz\tw@
\pst@dimtonum{\pst@dimz}{\pst@MixedColorB}%
% Definition of the mixed color MixedColor
\newrgbcolor{MixedColor}{%
\pst@MixedColorR\space \pst@MixedColorG\space \pst@MixedColorB}
%\typeout{Mixed color=\csname color@MixedColor\endcsname}% Debug
}
\makeatother
% < END OF OVERLAPPED COLORS

\begin{document}
\begin{CJK*}{GBK}{song}

%---------------------------------------------------------------------- SLIDE 0-

\begin{slide}[Box]{\small\blue È«¹úÊýÑ§µÚÊ®¶þ½ìÑÐ¾¿ÉúÊîÆÚÑ§Ð£, 2007}
\vspace{1.5cm}

\centerline{\Large\bf µ¥Éú¹ý³ÌµÄ±éÀúÀíÂÛ}

\vskip1cm \centerline{\large ÕÅÓà»Ô }

\vskip0.5cm \centerline{ ±±¾©Ê¦·¶´óÑ§ÊýÑ§¿ÆÑ§Ñ§Ôº }
\vskip0.5cm \centerline{\tiny (Îäºº´óÑ§, 2007Äê8ÔÂ3ÈÕÖÁ4ÈÕ)}
\end{slide}

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

%---------------------------------------------------------------------- SLIDE 1-
\begin{slide}[Split]{Contents}
\vskip0.2cm
\begin{itemize}
\item Background
\item Criteria for uniqueness
\item Criteria for recurrence
\item Criteria for ergodicity
\item Criteria for strong ergodicity
\item Stationary distribution
\item {\blue Criteria for $\ell$-ergodicity}
\item {\blue Exponential ergodicity}
\item {\blue A class of single birth processes}
\item {\blue Applications}
\item {\blue Related topics}
\end{itemize}
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 2-
\begin{slide}[Glitter]{Background}
\vskip0.2cm
Single birth $Q$-matrix $Q=(q_{ij}:i,j\ge 0)$:

$q_{i,i+1}>0$, $q_{ij}=0$ if $j>i+1$ for all $i\ge 0$.

$$\left(%
\begin{array}{ccccc}
  - & + & 0 & 0 & \cdots \\
  * & - & + & 0 & \cdots \\
  * & * & - & + & \cdots\\
  \vdots & \vdots & \vdots & \vdots & \ddots \\
  \end{array}%
\right)$$
\vskip0.2cm
1. {\blue irreversible};

2. {\blue single extremal point, single parameter};

3. {\blue applications in other fields}.

4. {\blue applications in mathematics}.

\vskip0.3cm
So the explicit criteria are expected.
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 3-
\overlays{3}{
\begin{slide}[Split]{Background}
\vskip0.2cm
Upwardly skip-free processes (W.J. Anderson, 1991);
\vskip0.2cm
generalized birth and death processes (J.K. Zhang, 1984);
\vskip0.2cm
\FromSlide{2}
Population processes: birth and death processes with catastrophes; birth, death and catastrophe processes
\OnlySlide{3}
$$\aligned i&\to i+1\\
&\to i-1\\
&\to i-2\\
&\to\cdots\\
&\to 0\endaligned
\quad
\aligned \hbox{at rate}\quad & {\blue b_i}\\
& {\blue a_i}+{ c_i}{\red f_{i,i-1}}\\
& { c_i}{\red f_{i,i-2}}\\
&\cdots\\
& { c_i}{\red f_{i0}}\endaligned$$
where $\sum_{j=0}^{i-1}{\red f_{ij}}=1$.
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 4-
\begin{slide}[Blinds]{Background}
\vskip1cm
Brockwell, Gani, Resnick and Pakes et al (1982-1986) for some special
catastrophes $f_{ij}$ ($b_i=b+\lambda i$, $a_i=0$, $c_i=ci$, $i\ge 0$).

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

\vskip0.3cm
Aim: extinction times and probability of extinction.
\vskip0.3cm
Keys: generating function of $Q$-resolvent.
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 5-
\begin{slide}[Box]{Background}
\vskip1cm
B. Cairns \& P. Pollett (2004).
$$\aligned i&\to i+1\\
&\to i-1\\
&\to\cdots\\
&\to 1\\
&\to 0\endaligned
\quad
\aligned \hbox{at rate}\quad & {\blue g_i} b\\
& {\blue g_i}{\red d_{1}}\\
&\cdots\\
& {\blue g_i}{\red d_{i-1}}\\
& {\blue g_i}\hbox{\red $\sum_{k\ge i}d_k$}\endaligned$$
where $b+\sum_{k\ge 1}d_k=1$.
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 6-
\begin{slide}[Box]{Background}
\vskip0.5cm
single birth process:

Yan, S.-J. and Chen, M.-F.(1986), Multidimensional $Q$-processes, Chinese Ann. Math.,
\textbf{7(B)}(1), 90-110.

\vskip0.2cm
Chen, M.-F.(2004), From Markov Chains to Non-Equilibrium Particle Systems, Second Edition,
World Scientific, Singapore.

\vskip0.2cm
³ÂÄ¾·¨, Ã«ÓÀ»ª(2007), Ëæ»ú¹ý³Ìµ¼ÂÛ, ¸ßµÈ½ÌÓý³ö°æÉç, ±±¾©.

\vskip0.4cm
Population processes:

Anderson, W. J.(1991), Continuous-Time Markov Chains, Springer-Verlag, New York.
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 7-
\overlays{1}{%
\begin{slide}[Dissolve]{Criteria for uniqueness}
\vskip0.5cm

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

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

\vskip0.2cm
$Q$-process: $P'(t)|_{t=0}=Q$.

$Q$-process exist? unique?

\vskip0.2cm
Let $Q=(q_{ij})$ be totally stable and conservative on countable $E$, i.e.

$q_{ij}\ge 0 (i\ne j)$, $q_i:=-q_{ii}=\sum_{j\ne i}q_{ij}<\infty(i\in E)$.

\vskip0.3cm
\begin{itemize}
\item Every $Q$-process satisfies the backward Kolmogorov equation.

\item $Q$-process always exists. The minimal process.
\end{itemize}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 8-
\overlays{1}{%
\begin{slide}[Dissolve]{Criteria for uniqueness}
\vskip1cm

$P_t$ is determined uniquely if and only if for some (equivalently, all) $\lambda>0$,
$$u_i=\sum_{j\ne i}q_{ij}u_j/(\lambda+q_i),\quad 0\le u_i\le 1,\,
i\ge 0$$
has only a trivial solution.

\vskip0.4cm

´ËÅÐ±ð×¼ÔòµÄÓÅÃÀÖ®´¦ÊÇ~$Q$~¾ØÕó, ¶øÇÒ·½³ÌµÄ×î´ó½âÒ²´æÔÚµü´úËã·¨.
µ«¶Ôµ¥Éú¹ý³Ì, Ò»¸ö¸üºÃµÄÅÐ±ð×¼Ôò---ÏÔÊ½µÄ¡¢ÍêÈ«¿É¼ÆËãµÄ, ÊÇÆÚÅÎµÄ.
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 9-
\overlays{2}{%
\begin{slide}[Dissolve]{Criteria for uniqueness}
\vskip0.8cm
\begin{itemstep}
\item Zhang, J.-K.(1984), Yan, S.-J. and Chen, M.-F.(1986).

\vskip0.2cm
For $0\leq k<n$, define $q_n^{(k)}=\sum_{j=0}^k q_{nj}$ and
$$\aligned
&m_0=\frac 1{q_{01}},\qquad m_n=\frac 1{q_{n,n+1}}\bigg(1+\sum_{k=0}^{n-1}q_n^{(k)}m_k\bigg),\quad n\ge 1,
\\
&F_n^{(n)}=1, \qquad F_n^{(i)}=\frac 1{q_{n,n+1}}
\sum_{k=i}^{n-1}q_n^{(k)}F_k^{(i)},\quad 0\le i<n.\
\endaligned$$

\item Uniqueness (regurlarity) $\Longleftrightarrow$ $R:=\sum_{n=0}^\infty m_n=\infty$.
\end{itemstep}

\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 10-
\overlays{1}{%
\begin{slide}[Dissolve]{Criteria for uniqueness}
\vskip0.5cm ¶ÔÉúÃð~$Q$~¾ØÕó~$(a_i,b_i)$, ÓÐ¼òµ¥µÄÐÎÊ½:
$$m_n=\frac 1{\mu_nb_n}\mu[0,n],\qquad n\ge 0,$$
ÆäÖÐ
$$\mu_0=1, \qquad \mu_i=\frac{b_0b_1\cdots b_{i-1}}{a_1a_2\cdots a_i},\quad i\ge 1,$$
ÇÒ~$\mu[i,k]=\sum_{j=i}^k\mu_j$.

\vskip0.5cm ¸ø¶¨ÉúÃð~$Q$~¾ØÕó~$(a_i, b_i)$. ÔòÉúÃð¹ý³ÌÎ¨Ò»µ±ÇÒ½öµ±
~$R:=\sum_{n=0}^\infty \frac 1{\mu_nb_n}\mu[0,n]=\infty$.
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 11-
\overlays{3}{%
\begin{slide}[Dissolve]{Criteria for recurrence}
%\vskip0.5cm
\begin{itemstep}
\item $P(t)$ is recurrent: discrete chain $P(h)$ is recurrent for all $h>0$ $\Longleftrightarrow$
$\int_0^\infty p_{ii}(t)\textrm{d}t=\infty$.

\vskip0.2cm
\item For a regular and irreducible $Q=(q_{ij})$,  $P(t)$ ie recurrent if and only if for some
(equivalently, all) $j_0$,
$$
x_i=\sum_{j\ne j_0,i}q_{ij}x_j/q_i,\qquad 0\le x_i\le 1,\,i\ge 0
$$
has only a trivial solution.

regular: totally stable, conservative and $Q$-process is unique.
\vskip0.2cm
\item
Suppose that the $Q$-matrix is regular and irreducible, then

Recurrence $\Longleftrightarrow$ $\sum_{n=0}^\infty F_n^{(0)}=\infty$.

\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 12-
\overlays{1}{%
\begin{slide}[Dissolve]{Criteria for recurrence}
\vskip1cm
¶ÔÉúÃð~$Q$~¾ØÕó~$(a_i,b_i)$, ÓÐ¼òµ¥µÄÐÎÊ½:
$$F_n^{(0)}=\frac{b_0}{\mu_nb_n},\qquad n\ge 0.$$

\vskip0.5cm
¸ø¶¨ÉúÃð~$Q$~¾ØÕó~$(a_i, b_i)$. ÔòÉúÃð¹ý³Ì³£·µµ±ÇÒ½öµ±
~$\sum_{n=0}^\infty \frac{1}{\mu_nb_n}=\infty$.
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 13-
\overlays{4}{
\begin{slide}[Split]{Criteria for ergodicity}
\vskip0.5cm
Assume that $Q$ is irreducible. $\pi=(\pi_i)$ is stationary distribution.
\begin{itemstep}\small
\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{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 14-
\overlays{2}{%
\begin{slide}[Dissolve]{Criteria for ergodicity}
\vskip0.5cm
\begin{itemstep}
\item
Let $H\ne\emptyset$ be a finite subset of $E$. $P(t)$ is ergodic if and only if the equation
$$\begin{cases}\sum_j q_{ij}y_j\le -1,\quad i\notin H&\\
\sum_{i\in H}\sum_{j\ne i}q_{ij}y_j<\infty&
\end{cases}$$
has a {\red finite} nonnegative solution.

\vskip0.4cm\item
Take $H=\{0\}$. For single birth $Q$-matrix,
$$
\sum_{j\ne i} q_{ij}(y_j-y_i)+1\le 0,\qquad i\ge 1.
$$
\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 15-
\overlays{3}{%
\begin{slide}[Dissolve]{Criteria for ergodicity}
%\vskip1cm
\begin{itemstep}
\item Define
$$d_0=0,\quad d_n=\frac 1{q_{n,n+1}}\Big(1+\sum_{k=0}^{n-1}q_n^{(k)}d_k\Big),
\quad n\ge 1.
$$

Êµ¼ÊÉÏ, Èý×é¼ÇºÅ¿ÉÒÔÓÃÍ³Ò»±íÊ¾, ÒòÎª

\item
$$m_n=\sum_{k=0}^n\frac{F_n^{(k)}}{q_{k,k+1}},\quad d_n=\sum_{k=1}^n\frac{F_n^{(k)}}{q_{k,k+1}},
\qquad n\ge 0.$$

\item $$m_n=\frac{F_n^{(0)}}{q_{01}}+d_n,\qquad n\ge 0.$$
\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 16-
\overlays{2}{%
\begin{slide}[Dissolve]{Criteria for ergodicity}
\vskip0.5cm
\begin{itemstep}
\item
$$F_n^{(i)}=\sum_{k=i+1}^n\frac{F_n^{(k)}q_k^{(i)}}{q_{k,k+1}},\quad n>i\ge 0.$$

\vskip0.5cm\item
Suppose that the $Q$-matrix is regular and irreducible, then

Ergodicity $\Longleftrightarrow$
$$d:=\sup_{k\ge 0}\frac{\sum_{n=0}^k d_n}{\sum_{n=0}^k F_n^{(0)}}<\infty.$$
\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 17-
\overlays{5}{%
\begin{slide}[Dissolve]{Criteria for ergodicity}
Sketch of the proof:
\begin{itemstep}
\item
Éè~$(u_i)$~ÊÇ·½³ÌµÄÓÐÏÞ·Ç¸º½â. È¡~$u_0=0$. ÓÃ¹éÄÉ·¨Ö¤µÃ
$$u_{n+1}-u_n\leq F_n^{(0)}u_1-d_n,\qquad n\ge 0.$$
\item
$$0\le u_{n+1}\le u_1\sum_{k=0}^nF_k^{(0)}-\sum_{k=0}^nd_k,\qquad n\ge 0.$$
\item
~$d\le u_1<\infty$.

\item
·´Ö®, ¼ÙÉè~$d<\infty$. Áî
$$u_0=0,\quad u_1=d,\quad u_{n+1}=u_n+F_n^{(0)}u_1-d_n,\qquad n\ge 0.$$

\item ÑéÖ¤~$(u_n)$~ÊÇ·½³ÌµÄÒ»¸öÓÐÏÞ·Ç¸º½â.

\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 18-
\begin{slide}[Dissolve]{Criteria for ergodicity}
\vskip1cm
Àý: ¸ø¶¨ÕýÔò²»¿ÉÔ¼µ¥Éú~$Q$~¾ØÕó~$Q=(q_{ij})$. Èô~$\inf_{k\ge 1}q_{k0}>0$, Ôòµ¥Éú¹ý³Ì±éÀú.
ÒòÎª
ÓÉºÏ·Ö±ÈÐÔÖÊ¼°¿ÉµÃ
$$d\le\sup_{i\ge 1}\frac{d_i}{F_i^{(0)}}\le\sup_{k\ge 1}\frac 1{q_k^{(0)}}=\frac 1{\inf_{k\ge 1}q_{k0}}.$$
ËùÒÔ~$d<\infty$.

\vskip0.5cm
Êµ¼ÊÉÏ, ´Ë¹ý³ÌÊÇÇ¿±éÀúµÄ!
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 19-
\begin{slide}[Dissolve]{Criteria for ergodicity}
\vskip1cm
¶ÔÉúÃð~$Q$~¾ØÕó~$(a_i,b_i)$, ÓÐ¼òµ¥µÄÐÎÊ½:
$$d_n=\frac 1{\mu_nb_n}\mu[1,n],\qquad n\ge 0.$$

\vskip0.5cm ¸ø¶¨ÕýÔòÉúÃð~$Q$~¾ØÕó~$(a_i, b_i)$. ÔòÉúÃð¹ý³Ì±éÀúµ±ÇÒ½öµ±
~$\mu[0,\infty)<\infty$.
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 20-
\overlays{1}{%
\begin{slide}[Dissolve]{Criteria for strong ergodicity}
\vskip0.5cm

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

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

\vskip0.5cm
Let $H\ne\emptyset$ be a finite subset of $E$. Then $P(t)$ is strongly ergodic if and only if the equation
$$\begin{cases}\sum_j q_{ij}y_j\le -1,\quad i\notin H&\\
\sum_{i\in H}\sum_{j\ne i}q_{ij}y_j<\infty&
\end{cases}$$
has a {\red bounded} nonnegative solution.
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 21-
\overlays{3}{%
\begin{slide}[Dissolve]{Criteria for strong ergodicity}
\vskip0.4cm
\begin{itemstep}

\item Zhang, Y.-H.(2001). Suppose that the $Q$-matrix is irreducible and regular. Then

Strong ergodicity $\Longleftrightarrow$
$S:=\sup_{k\ge 0}\sum_{n=0}^k(F^{(0)}_nd-d_n)<\infty$.
\vskip0.2cm
\item Zhang, H.-J.,  Lin, X.  and Hou, Z.-T. (1998, 2000).
\vskip0.2cm
For regular birth-death $(a_i,b_i)$,
$$\hbox{strong ergodicity}\Longleftrightarrow
S:=\sum_{n=0}^\infty\frac 1{\mu_n b_n}\sum_{k=n+1}^\infty\mu_k
<\infty.$$

\item Chen, M.-F.(1992).
$$S<\infty\Longrightarrow\hbox{exponential ergodic}.$$
\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------
%---------------------------------------------------------------------- SLIDE 22-
\overlays{1}{%
\begin{slide}[Dissolve]{Criteria for strong  ergodicity}
Sketch of the proof:

Ê×ÏÈ, Ö¤Ã÷·½³Ì
$$
y_i=\sum_{j\ne i}\frac{q_{ij}}{q_i}y_j+\frac{1}{q_i}, \quad i\ge1;\quad y_0=0 \quad (*)
$$
ÓÐÒ»ÓÐ½ç·Ç¸º½âµ±ÇÒ½öµ±~$S<\infty$. ´ËÊ±,
$$d=\lim_{n\to\infty}\sum_{k=0}^nd_k\bigg/\sum_{k=0}^nF_k^{(0)},$$
¶øÇÒ·½³Ì($*$)µÄÎ¨Ò»½âÓÐÏÂÃæµÄÐÎÊ½:
$$
y_0=0,\quad y_1=d,\quad y_{n+1}=y_n+F_n^{(0)}y_1-d_n,\quad n\ge1. \quad (**)
$$
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 23-
\overlays{2}{%
\begin{slide}[Dissolve]{Criteria for strong  ergodicity}
\vskip0.2cm
Sketch of the proof:
\begin{itemstep}
\item
¼ÙÉèµ¥Éú¹ý³ÌÇ¿±éÀú, Ôò·½³Ì´æÔÚÒ»ÓÐ½ç·Ç¸º½â~$(u_i)$, ¼´
$$u_i\ge\sum_{j\ne i}\frac{q_{ij}}{q_i}u_j+\frac{1}{q_i},\quad i\ge1;\quad u_0\ge 0.$$
¼Ç~$(u_i^*)$~ÊÇ·½³Ì($*$)µÄ×îÐ¡·Ç¸º½â. ÔòÓÉ±È½Ï¶¨ÀíµÃµ½~$u_i\ge u_i^*\,(i\ge 0)$.
Òò´Ë, $(u_i^*)$~ÓÐ½çÇÒ·½³Ì($*$)ÓÐÒ»ÓÐ½ç·Ç¸º½â~$\Rightarrow S<\infty$.

\item
·´Ö®, Éè~$S<\infty$. ÓÉ($**$)¶¨Òå~$(y_i)$. Ôò~$(y_i)$~ÊÇ·½³Ì($*$)µÄÓÐ½ç·Ç¸º½â.
ÏÔÈ»~$(y_i)$~Ò²ÊÇ·½³ÌµÄÓÐ½ç·Ç¸º½â~$\Rightarrow$~µ¥Éú¹ý³ÌµÄÇ¿±éÀúÐÔ.
\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 24-
\begin{slide}[Dissolve]{Criteria for strong  ergodicity}

\vskip1cm

ÃüÌâ: ¸ø¶¨ÕýÔò²»¿ÉÔ¼µÄ~$Q$~¾ØÕó$Q=(q_{ij})$. Èô´æÔÚ~$k\ge 0$~Âú×ã
~$c:=\inf_{i\ne k}q_{ik}>0$, ÔòÏàÓ¦µÄ~$Q$~¹ý³ÌÊÇÇ¿±éÀúµÄ.


\vskip0.5cm
Àý: ¸ø¶¨ÕýÔò²»¿ÉÔ¼µ¥Éú~$Q$~¾ØÕó~$Q=(q_{ij})$. Èô~$\inf_{k\ge 1}q_{k0}>0$, Ôòµ¥Éú¹ý³Ì
Ç¿±éÀú.

\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 25-
\overlays{2}{%
\begin{slide}[Glitter]{Stationary distribution}
\vskip1cm
Can we get the stationary distribution?

How to use $F_n^{(i)}$?

$$
F_n^{(n)}=1,\quad  F_n^{(i)}=\frac 1{q_{n,n+1}}
\sum_{k=i}^{n-1}q_n^{(k)}F_k^{(i)},\quad 0\le i<n. $$

\OnlySlide{2} Stationary distribution (Zhang, Y.-H.(2004)):
$$c_k:=\sup_{i\ge k}\frac{\sum_{j=k}^i m_j}{\sum_{j=k}^iF_j^{(k)}},\quad
\pi_k=\frac 1{q_{k,k+1}c_k},\quad k\ge 0.$$
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 26-
\overlays{2}{%
\begin{slide}[Glitter]{Stationary distribution}
\begin{itemstep}
\item ¿¼ÂÇÉúÃð¹ý³Ì: ÉúËÙ~$b_i=q_{i,i+1}>0\,(i\ge 0)$, ËÀËÙ~$a_i=q_{i,i-1}>0\,(i\ge 1)$.
¶¨Òå~$\mu_0=1$, $\mu_i=b_0\cdots b_{i-1}/a_1\cdots a_i\,(i\ge 1)$. ÔÙÁî$\mu=\sum_{i=0}^\infty\mu_i$.

µÃµ½~$c_i=\mu/(\mu_ib_i)$, Æ½ÎÈ·Ö²¼Îª~$\pi_i=\mu_i/\mu\,(i\ge 0)$.

\item Áîµ¥Éú~$Q$~¾ØÕóÎª: $q_{i,i+1}=1\,(i\ge 0)$, $q_{10}=1$~ÇÒ
~$q_{i,i-2}=1\,(i\ge 2)$, ÆäËüµÄ~$q_{ij}=0\,(i\ne j)$. Ôò~$\{F_j^{(k)}\}_{j=k}^\infty$~ÊÇFibonacciÊýÁÐ:
$$F_{k+n}^{(k)}=\frac 1{\sqrt 5}[A^{n+1}-(-B)^{n+1}],\qquad n\ge 0, k\ge 0,$$
ÆäÖÐ~$A=(\sqrt 5 +1)/2, B=(\sqrt 5 -1)/2$.

µÃµ½~$c_k=A^{k+2}$, Æ½ÎÈ·Ö²¼Îª~$\pi_k=B^{k+2}\,(k\ge 0)$.
\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 27-
\overlays{2}{%
\begin{slide}[Glitter]{Stationary distribution}
\begin{itemstep}
\item Áîµ¥Éú~$Q$~¾ØÕóÎª: $q_{i,i+1}=1\,(i\ge 0)$~ÇÒ~
$q_{i0}=1\,(i\ge 1)$, ÆäËüµÄ~$q_{ij}=0\,(i\ne j)$. ¸Ã¹ý³ÌÊÇÇ¿±éÀúµÄ.
µÃµ½~$c_k=2^{k+1}$, Æ½ÎÈ·Ö²¼Îª~$\pi_k=2^{-k-1}\,(k\ge 0)$.

\item ¿¼ÂÇÖÖÈºÀíÂÛÖÐµÄ¾ùÔÈ´óÔÖÄÑÄ£ÐÍ, ¼´~$q_{i,i+1}=\lambda_i:=a+\lambda i\,(i\ge 0)$,
$q_{ij}=\beta\,(0\le j<i)$, ÆäËüµÄ~$q_{ij}=0\,(i\ne j)$,
ÆäÖÐµÄ~$a$, $\lambda$, $\beta$~¾ùÎªÕý³£Êý. Éè¸Ã~$Q$~¾ØÕóÕýÔò. ÔòÇ¿±éÀúÇÒÆ½ÎÈ·Ö²¼Îª
$$\pi_0=\frac{\beta}{a+\beta},\ \pi_k= \frac{(k+1)\beta}{a+\beta}\prod_{j=0}^{k-1}
\frac{\lambda_j}{\lambda_{j+1}+(j+2)\beta},\ k\ge1.$$
ÌØ±ðµ±~$a=\lambda$~Ê±, Ôò~$\pi_k=\beta a^k/(a+\beta)^{k+1}\,(k\ge0)$.
½øÒ»²½, µ±~$a=\lambda=\beta=1$~Ê±, Ôò~$\pi_k=2^{-k-1}\,(k\ge0)$.

\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 28-
\begin{slide}[Blinds]{Stationary distribution}
\vskip0.5cm
Set $\sigma_j=\inf\{t\ge$ the first jumping time:
$X(t)=j\}$ and
$\tau_j=\inf\{t\ge 0: X(t)=j\}$.

\vskip0.5cm
ÈÎÒâ¸ø¶¨~$i_0\ge 0$. µ¥Éú¹ý³ÌµÄ~$i_0$~Ê×ÖÐÊ±µÄÒ»½×¾Ø¿ÉÓÉÈçÏÂ±í´ïÊ½¸ø³ö:
$$
\mathbb E_i\tau_{i_0}=\sum_{j=i}^{i_0-1}m_j,\quad i<i_0;$$
$$
\mathbb E_i\tau_{i_0}=\sum_{j=i_0}^{i-1}(F_j^{(i_0)}c_{i_0}-m_j),\quad i\ge i_0+1.
$$
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 29-
\begin{slide}[Blinds]{Stationary distribution}
\vskip0.5cm
Áî~$u_i=\mathbb E_i\tau_{i_0}$. $(u_i)$~ÊÇÏÂÁÐ·½³ÌµÄ×îÐ¡·Ç¸º½â:
$$
x_{i_0}=0, \qquad x_i=\sum_{j\ne i}\frac{q_{ij}}{q_i}x_j+\frac 1{q_i},\quad i\ne i_0.
$$

\vskip0.2cm
$i$~»÷ÖÐÊ±Ò»½×¾Ø
$$\mathbb E_i\sigma_i=\frac{q_{i,i+1}c_i}{q_i},\qquad i\ge 0.$$

\vskip0.2cm
$$\pi_i q_i\mathbb E_i\sigma_i=1\Rightarrow\pi_i=\frac 1{q_{i,i+1}c_i}.$$

\end{slide}
%------------------------------------------------------------------------------

%-----------------------------------------------------------------------slide 30-
\overlays{4}{%
\begin{slide}[Wipe]{Stationary distribution}
$$
\mathbb E_1\tau_0=d,\quad \mathbb E_i\tau_0=\sum_{j=0}^{i-1}(F_j^{(0)}d-d_j),
\quad i\ge 1.$$
$$
\mathbb E_0\sigma_0=q_{01}^{-1}+\mathbb E_1\sigma_0=q_{01}^{-1}+d.
$$

\vskip0.2cm
Probabilistic meaning.

\begin{itemstep}

\item Uniqueness. $R=\mathbb E_0\sigma_\infty=\infty$.

\item Recurrence. $\mathbb P_0(\sigma_0<\infty)=1-(\sum_{n=0}^\infty F_n^{(0)})^{-1}=1$.

\item Ergodicity. $d=\mathbb E_1\sigma_0<\infty\Leftrightarrow\mathbb E_0\sigma_0<\infty$.

\item Strong ergodicity. $S=\sup_{i\ge 1}\mathbb E_i\sigma_0=\mathbb E_\infty\sigma_0<\infty$.
\end{itemstep}

\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 31-
\overlays{2}{
\begin{slide}[Glitter]{}

\vskip3cm
\centerline{ \Huge\blue Thanks}
\OnlySlide{2}
\vskip0.5cm
\centerline{http://math.bnu.edu.cn/${}^\sim$zhangyh/}
\end{slide}
}
%------------------------------------------------------------------------------

\clearpage
\end{CJK*}
\end{document}
%------------------------------------------------------------------------------
