%  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
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% > BEGIN OF OVERLAPPED COLORS
% Code below devised by Pingji Deng (BNU/ - China, dengpj@eyou.com)
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% Draw two overlapped surfaces, with computation of the mixed color for
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% Draw first surface
#1
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% Compute the mixed color
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% Draw second surface
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% Redraw overlapped surface in the mixed color
#1
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% 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}%
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%\typeout{Color 2=\pst@tempg,\pst@temph,\pst@tempi}% Debug
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% 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}%
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\divide\pst@dimz\tw@
\pst@dimtonum{\pst@dimz}{\pst@MixedColorG}%
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\advance\pst@dimz\pst@SecondColorB pt
\divide\pst@dimz\tw@
\pst@dimtonum{\pst@dimz}{\pst@MixedColorB}%
% Definition of the mixed color MixedColor
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\pst@MixedColorR\space \pst@MixedColorG\space \pst@MixedColorB}
%\typeout{Mixed color=\csname color@MixedColor\endcsname}% Debug
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% < END OF OVERLAPPED COLORS

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

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

\begin{slide}[Box]{\small\blue Workshop on Markov processes and related topics 2006}
\vspace{1cm}

\centerline{\Large Criteria for several types of ergodicity}
\centerline{\Large  for single birth processes}

\vskip1cm \centerline{ Yuhui Zhang (ÕÅÓà»Ô) }

\vskip0.5cm \centerline{ Beijing Normal University} \vskip0.5cm
\centerline{ (Central South University, Changsha, August 21-25)}
\end{slide}

%------------------------------------------------------------------------------
%---------------------------------------------------------------------- SLIDE 1-
\begin{slide}[Split]{Contents}
\vskip1cm
\begin{itemize}
\item Concepts and notations
\item Background
\item Three series of notations
\item Criteria for uniqueness, recurrence and ergodicity
\item Criterion for strong ergodicity
\item Stationary distribution
\item Criterion for $\ell$-ergodicity
\item Exponential ergodicity
\end{itemize}
\end{slide}
%------------------------------------------------------------------------------
%---------------------------------------------------------------------- SLIDE 1-
\overlays{3}{
\begin{slide}[Split]{Concepts and notations}
\vskip1cm
Let $Q=(q_{ij})$ be
regular on countable $E$, i.e.

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

$Q$ determines uniquely $P_t=(p_{ij}(t))$ ($Q$-process or Markov chain). Assume that $Q$ is irreducible. $\pi=(\pi_i)$.
\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
$\Longleftrightarrow\displaystyle\lim_{t\to\infty}{\red e^{\beta t}}\sup_i|p_{ij}(t)-\pi_j|=0$.
\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 2-
\begin{slide}[Blinds]{Concepts and notations}
\vskip1cm
Set $\sigma_j=\inf\{t\ge$ the first jumping time:
$X_t=j\}$ and {\blue $m_{ij}^{(\ell)}=\mathbf E_i\sigma_j^\ell$}.

\vskip0.3cm
For a positive integer $\ell$, the recurrent
chain $P_t$ is said to be $\ell$-ergodic if
{\red $m_{jj}^{(\ell)}<\infty$} for some (and hence for all) $j\in E$.

\vskip0.3cm
{\blue $1$}-ergodic $=$ positive recurrent (ergodic),

{\blue $0$}-ergodic $=$ null recurrent.

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

%---------------------------------------------------------------------- SLIDE 3-
\begin{slide}[Box]{Concepts and notations}
\vskip1cm
Discrete time:
\begin{itemize}
\item J.G. Kemeny, J.L. Snell \& A.W. Knapp (1976)
\item Y.H. Mao (2003)
\item Z.T. Hou \& Y.Y. Liu (2003). Queue Theory
\end{itemize}
\vskip0.5cm
Continuous time:
\begin{itemize}
\item P. Coolen-Schrijner \& E.A. van Doorn (2002)
\item Y.H. Mao (2004)
\end{itemize}
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 4-
\overlays{2}{
\begin{slide}[Wipe]{Concepts and notations}
\vskip1cm
\begin{itemstep}
\item $\ell$-ergodicity provides an {\red algebraic} convergence rate:

$p_{ij}(t)-\pi_j=o({\blue t^{-(\ell-1)}})$ as $t\to\infty$.
\vskip0.3cm
\item Given a regular birth-death $(a_i,b_i)$. Assume that $P_t$ is recurrent.
\vskip0.2cm
\centerline{$P_t$ is $\ell$-ergodic iff $\sum_{i=1}^\infty\mu_i m_{i0}^{(\ell-1)}<\infty$,}

\vskip0.2cm
where $\mu_0=1$, $\mu_i=b_0\cdots b_{i-1}/a_1\cdots a_i (i\ge 1)$,

$$
m_{i0}^{(n)}=n\sum_{j=0}^{i-1}\frac1{\mu_jb_j}\sum_{k=j+1}^\infty\mu_km_{k0}^{(n-1)},\quad i\ge 1,\,n\ge 1.
$$
\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 5-
\begin{slide}[Glitter]{Background}
\vskip1cm
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}; 3.{\blue application}.
\vskip0.3cm
So the explicit criteria are expected.
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 6-
\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 7-
\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 8-

\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 9-
\begin{slide}[Wipe]{Three series of notations}
\vskip0.4cm
$$q_n^{(k)}=\sum_{j=0}^kq_{nj},\ 0\le k<n\,(k,n\ge 0).$$
$$\aligned &m_0=\frac 1{q_{01}},\quad m_n=\frac 1{q_{n,n+1}}
\Big(1+\sum_{k=0}^{n-1}q_n^{(k)}m_k\Big),\quad n\ge 1,\\
&F_n^{(n)}=1,\quad  F_n^{(i)}=\frac 1{q_{n,n+1}}
\sum_{k=i}^{n-1}q_n^{(k)}F_k^{(i)},\quad 0\le i<n, \\
&d_0=0,\quad d_n=\frac 1{q_{n,n+1}}\Big(1+\sum_{k=0}^{n-1}q_n^{(k)}d_k\Big),
\quad n\ge 1.
\endaligned$$
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 10-
\overlays{3}{%
\begin{slide}[Dissolve]{Criteria for uniq., recur. and erg.}
\vskip1cm
J.K. Zhang (1984), M.F. Chen (1992).
\vskip0.2cm
\begin{itemstep}
\item Uniqueness (regurlarity) $\Longleftrightarrow$ $R:=\sum_{n=0}^\infty m_n=\infty$.

\vskip0.4cm
Suppose that the $Q$-matrix is irreducible, then

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

\vskip0.4cm In the regular case,

\item Ergodicity $\Longleftrightarrow
d:=\sup_{k\ge 0}(\sum_{n=0}^k d_n)/(\sum_{n=0}^k F_n^{(0)})<\infty$.

\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------
%---------------------------------------------------------------------- SLIDE 11-
\overlays{3}{%
\begin{slide}[Dissolve]{Criterion for strong erg.}
\vskip0.4cm
\begin{itemstep}

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

Strong ergodicity $\Longleftrightarrow$
$\sup_{k\ge 0}\sum_{n=0}^k(F^{(0)}_nd-d_n)<\infty$.
\vskip0.2cm
\item H.J. Zhang, X. Lin and Z.T. Hou (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 M.F. Chen (1992).
$$S<\infty\Longrightarrow\hbox{exponential ergodic}.$$
\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 12-
\overlays{1}{%
\begin{slide}[Dissolve]{Uniqueness}
\vskip1.5cm
For a totally stable and conservative $Q=(q_{ij})$,
$P_t$ is determined uniquely iff 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.
\end{slide}
}
%------------------------------------------------------------------------------
%---------------------------------------------------------------------- SLIDE 13-
\overlays{1}{%
\begin{slide}[Dissolve]{Recurrence}
\vskip1.5cm
For a regular and irreducible $Q=(q_{ij})$,  $P(t)$ ie recurrent iff 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.
\end{slide}
}
%------------------------------------------------------------------------------
%---------------------------------------------------------------------- SLIDE 14-
\overlays{1}{%
\begin{slide}[Dissolve]{Erg. and Strong Erg.}
\vskip1.5cm
Let $H\ne\emptyset$ be a finite subset of $E$. $P_t$ is ergodic (resp. strong ergodic) iff 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 finite (resp. bounded) nonnegative solution.
\end{slide}
}
%------------------------------------------------------------------------------

%-----------------------------------------------------------------------slide 15-
\overlays{4}{%
\begin{slide}[Wipe]{}
Probabilistic meaning.

\begin{itemstep}

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

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

\item Ergodicity. {$d=\mathbf E_1\sigma_0$.}

\item Strong ergodicity. $\sup_{i\ge 1}\mathbf E_i\sigma_0$.
\end{itemstep}

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

%---------------------------------------------------------------------- SLIDE 16-
\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 (Y.H. Zhang (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 17-
\overlays{3}{%
\begin{slide}[Box]{Criterion for $\ell$-erg.}
\vskip1cm
\centerline{ $P_t$ is $\ell$-ergodic iff $\sum_{i=1}^\infty\pi_i m_{i0}^{(\ell-1)}<\infty$.}

$$d_0=0,\quad d_i=\frac 1{q_{i,i+1}}\Big(
\fromSlide*{2}{\rnode{NA}{\pscirclebox[linecolor=red]{1}}}
\onlySlide*{1}{\rnode{NA}{\pscirclebox[linecolor=red,linestyle=none]{1}}}
+\sum_{k=0}^{i-1}q_i^{(k)}d_k\Big),\quad i\ge 1.$$

\FromSlide{2}
$$d^{(\ell)}_0=0,\quad d^{(\ell)}_i=\frac 1{q_{i,i+1}}\Big(
\fromSlide*{3}{\rnode{NB}{\pscirclebox[linecolor=red]{m_{i0}^{(\ell-1)}}}}
\onlySlide*{2}{\rnode{NB}{\pscirclebox[linecolor=red,linestyle=none]{m_{i0}^{(\ell-1)}}}}
+\sum_{k=0}^{i-1}q_i^{(k)}d^{(\ell)}_k\Big),
\quad i\ge 1.$$
\fromSlide*{3}{\nccurve[linecolor=red,angleA=270,angleB=90]{->}{NA}{NB}}
\OnlySlide{3}

\end{slide}
}
%------------------------------------------------------------------------------
%---------------------------------------------------------------------- SLIDE 18-
\overlays{4}{%
\begin{slide}[Dissolve]{Criterion for $\ell$-erg.}

\vskip0.5cm
$$d^{(\ell)}:=\sup_{i\ge 0}\frac{\sum_{j=0}^i
\fromSlide*{2}{\rnode{NA}{\pscirclebox[linecolor=red]{d^{(\ell)}_j}}}
\onlySlide*{1}{\rnode{NA}{\pscirclebox[linecolor=red,linestyle=none]{d^{(\ell)}_j}}}
}{\sum_{j=0}^iF_j^{(0)}}.$$

\FromSlide{2}
$$d:=\sup_{i\ge 0}\frac{\sum_{j=0}^i
\fromSlide*{3}{\rnode{NB}{\pscirclebox[linecolor=red]{d_j}}}
\onlySlide*{2}{\rnode{NB}{\pscirclebox[linecolor=red,linestyle=none]{d_j}}}
}{\sum_{j=0}^i F_j^{(0)}}.$$
\fromSlide*{3}{\nccurve[linecolor=red,angleA=90,angleB=270]{->}{NB}{NA}}

\vskip0.5cm
\OnlySlide{4}\centerline{ For $\ell\ge 1$, $P_t$ is $\ell$-ergodic iff $d^{(\ell)}<\infty$,}

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

%---------------------------------------------------------------------- SLIDE 19-
\begin{slide}[Split]{$\ell$-erg.}
\vskip0.4cm
$P_t$ is $\ell$-regodic iff for some (and then for all) $j\in E$,
the system of inequalities
$$\begin{cases}
\sum_k q_{ik}y_k\le-\ell m_{ij}^{(\ell-1)},&i\ne j,\\
\sum_{k\ne j}q_{jk}y_k<\infty
\end{cases}$$
has a nonnegative finite solution.

\vskip0.4cm
Key:

$$
m_{10}^{(\ell)}=\ell d^{(\ell)},\quad m_{i0}^{(\ell)}=\ell\sum_{j=0}^{i-1}(F_j^{(0)}d^{(\ell)}- d^{(\ell)}_j).
$$
\end{slide}

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

%---------------------------------------------------------------------- SLIDE 20-
\overlays{2}{%
\begin{slide}[Replace]{Exp. erg.}
\vskip0.4cm
Y.H. Mao \& Y.H. Zhang (2004).
$$\inf_i q_i>0\ \hbox{and}\ \delta':=\sup_{i> 0}\sum_{j=0}^{i-1}F_j^{(0)}\sum_{j=i}^\infty \frac 1{q_{j,j+1}F_j^{(0)}}
<\infty.$$
$\Longrightarrow$ exp. erg.
\vskip0.4cm
\OnlySlide{2}
$P_t$ is exp. erg. iff for some $\lambda>0$ with $\lambda <q_i$ for all $i$,
$$\begin{cases}&\sum_j q_{ij}y_j\le -\lambda y_i-1,\quad i\notin H\\
&\sum_{i\in H}\sum_{j\ne i}q_{ij}y_j<\infty
\end{cases}$$
has a nonnegative finite solution $(y_i)$.
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 21-
\overlays{1}{%
\begin{slide}[Replace]{Exp. erg.}
\vskip1cm
M.F. Chen (2000). For birth-death processes,

$$\hbox{exp. erg. } \Longleftrightarrow
\delta:=\sup_{i> 0}\sum_{j=0}^{i-1}\frac 1{\mu_jb_j}\sum_{j=i}^\infty \mu_j<\infty.$$

$$\lambda_1>0\Longleftrightarrow \delta<\infty.$$

$$\lambda_1=\ \hbox{ exp. conv. rate.}$$
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 22-
\overlays{1}{%
\begin{slide}[Replace]{Exp. erg.}
\vskip0.5cm
Mao \& Zhang (2004) gives another proof.
\vskip0.2cm
Keys:
\vskip0.2cm
1. Construct a test function with double summations for sufficiency.
\vskip0.2cm
2. Necessity. Note that

$$\aligned
m_{i0}^{(n)}&\ge n\bigg(\sum_{j=0}^{i-1}\frac1{\mu_jb_j}\sum_{k=i}^\infty\mu_k\bigg)m_{i0}^{(n-1)}\\
&\ge\cdots\ge n!\bigg(\sum_{j=0}^{i-1}\frac1{\mu_jb_j}\sum_{k=i}^\infty\mu_k\bigg)^n.
\endaligned$$
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 23-
\overlays{2}{%
\begin{slide}[Replace]{Exp. erg.}
\vskip0.5cm

Unfortunately, "$\delta'<\infty$" is not necessary for exp. erg.
\vskip0.2cm
Conjecture that the criterion should be

$$\delta:=\sup_{i> 0}\sum_{j=0}^{i-1}F_j^{(0)}\bigg(d-\frac {d_i}{F_i^{(0)}}\bigg)<\infty.$$

\OnlySlide{2}
{\red New result. }Suppose that
$$q_{i+1,j}=p_iq_{ij},\quad 0\le j\le i-1,$$
where $0\le p_i\le c<1\,(i\ge 1)$. Then

$$\hbox{ exp.erg.}\Longleftrightarrow\delta<\infty.$$
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 24-
\overlays{1}{
\begin{slide}[Glitter]{}

\vskip3cm
\centerline{ \Huge\blue Thanks}
\end{slide}
}

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