%  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.
% Of course, you can make your background if you have the ability and you wish.

\documentclass[pdf,darkblue,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

\title{$\ell$-ergodicity for single birth processes}
\author{\Large\green Yuhui Zhang}
\institution{Beijing Normal University\\
May 20, 2004}

\begin{document}
\begin{CJK*}{GBK}{song}
\CJKtilde\sloppy \maketitle

%---------------------------------------------------------------------- SLIDE 1-
\overlays{3}{
\begin{slide}[Split]{Backgroud} 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:
$\displaystyle\lim_{t\to\infty}{\yellow e^{\alpha t}}|p_{ij}(t)-\pi_j|=0$.

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

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

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

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

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

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

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

%---------------------------------------------------------------------- SLIDE 3-
\begin{slide}[Box]{Backgroud}

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}
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]{Backgroud}
\begin{itemstep}
\item $\ell$-ergodicity provides an {\green algebraic} convergence rate:

$p_{ij}(t)-\pi_j=o({\yellow t^{-(\ell-1)}})$ as $t\to\infty$.

\item Criterion: $P_t$ 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.
\end{itemstep}
\hypertarget{Third}{}
\hyperlink{Fourth}{$\triangleright$}

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

%---------------------------------------------------------------------- SLIDE 5-
\begin{slide}[Dissolve]{Backgroud}
Given a regular birth-death $(a_i,b_i)$. Assume that $P_t$ is recurrent.

\centerline{\yellow $P_t$ is $\ell$-ergodic iff $\sum_{i=1}^\infty\mu_i m_{i0}^{(\ell-1)}<\infty$,}

\vskip0.3cm
where $\mu_0=1$, $\mu_i=b_0\cdots b_{i-1}/a_1\cdots a_i (i\ge 1)$,

$\displaystyle
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{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\begin{slide}[Glitter]{Backgroud}
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 $i\ge 0$.

$$\left(%
\begin{array}{ccccc}
  - & + & 0 & 0 & \cdots \\
  * & - & + & 0 & \cdots \\
  * & * & - & + & \cdots\\
  \vdots & \ddots & \ddots & \ddots &  \\
  \end{array}%
\right)$$

1. {\yellow irreversible}; 2. {\yellow single extremal point}.

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

%---------------------------------------------------------------------- SLIDE 6-
\overlays{3}{
\begin{slide}[Split]{Backgroud}
Upwardly skip-free processes;

\FromSlide{2}
Population processes: birth and death processes with catastrophes
\OnlySlide{3}
$$\aligned i&\to i+1\\
&\to i-1\\
&\to i-2\\
&\to\cdots\\
&\to 0\endaligned
\quad
\aligned \hbox{at rate}\quad & {\green b_i}\\
& {\green a_i}+{\green c_i}{\yellow f_{i,i-1}}\\
& {\green c_i}{\yellow f_{i,i-2}}\\
&\cdots\\
& {\green c_i}{\yellow f_{i0}}\endaligned$$
where $\sum_{j=0}^{i-1}{\yellow f_{ij}}=1$.
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 7-
\begin{slide}[Blinds]{Backgroud}
W.J. Anderson (1991)

\vskip0.2cm
Brockwell, Gani, Resnick and Pakes et al (1982-1986) for special
$f_{ij}$, e.g. geometric, uniform, binomial catastrophes.

\vskip0.4cm
Keys: generating function of $Q$-resolvent.
\end{slide}

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

\begin{slide}[Box]{Backgroud}

B. Cairns \& P. Pollett (2004).
$$\aligned i&\to i+1\\
&\to i-1\\
&\to\cdots\\
&\to 1\\
&\to 0\endaligned
\quad
\aligned \hbox{at rate}\quad & {\green g_i b}\\
& {\green g_i}{\yellow d_{1}}\\
&\cdots\\
& {\green g_i}{\yellow d_{i-1}}\\
& {\green g_i}\hbox{\yellow $\sum_{k\ge i}d_k$}\endaligned$$
where $b+\sum_{k\ge 1}d_k=1$.
\end{slide}

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

%---------------------------------------------------------------------- SLIDE 8-
\begin{slide}[Wipe]{Single birth processes}
$q_n^{(k)}=\sum_{j=0}^kq_{nj},\ 0\le k<n\,(k,n\ge 0)$
$$\aligned &m_0=\frac 1{q_{01}},\quad m_n=\frac 1{q_{n,n+1}}
\Big(1+\sum_{k=0}^{n-1}q_n^{(k)}m_k\Big),\quad n\ge 1,\\
&F_n^{(n)}=1,\quad  F_n^{(i)}=\frac 1{q_{n,n+1}}
\sum_{k=i}^{n-1}q_n^{(k)}F_k^{(i)},\quad 0\le i<n, \\
&d_0=0,\quad {\yellow d_n=\frac 1{q_{n,n+1}}\Big(1+\sum_{k=0}^{n-1}q_n^{(k)}d_k\Big)},
\quad n\ge 1.
\endaligned$$
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 9-
\overlays{4}{%
\begin{slide}[Dissolve]{Single birth processes}
M.F. Chen (1992,1999), J.K. Zhang (1984) and Y.H. Zhang (2001).

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

\vskip0.2cm
Next, suppose that the $Q$-matrix is irreducible, then

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

\vskip0.2cm In the regular case,

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

\item Strong ergodicity $\Longleftrightarrow$
$\sup_{k\ge 0}\sum_{n=0}^k(F^{(0)}_nd-d_n)<\infty$.
\end{itemstep}
\hypertarget{First}{}
\hyperlink{Second}{$\triangleright$}
\end{slide}
}
%------------------------------------------------------------------------------

%-----------------------------------------------------------------------slide 10-
\overlays{4}{%
\begin{slide}[Wipe]{Single birth processes}
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. {\yellow $d=\mathbf E_1\sigma_0$.}

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

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

%---------------------------------------------------------------------- SLIDE 11-
\overlays{2}{%
\begin{slide}[Replace]{Single birth processes}
\small
Exp. ergo. (Y.H. Mao \& Y.H. Zhang (2004)): $\Longleftarrow$
$$\inf_i q_i>0\ \hbox{and}\ \sup_{i> 0}\sum_{j=0}^{i-1}F_j^{(0)}\sum_{j=i}^\infty \frac 1{q_{j,j+1}F_j^{(0)}}<\infty.$$

\OnlySlide{2}
$P_t$ is exponential ergodic iff for some $\lambda>0$ with $\lambda <q_i$ for all $i$,
$$\sum_j q_{ij}y_j\le -\lambda y_i-1,\quad i\ge 1$$
has a nonnegative finite solution $(y_i)$.
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 12-
\overlays{2}{%
\begin{slide}[Glitter]{Single birth processes}

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

\OnlySlide{2}
\centerline{\large\yellow $P_t$ is $\ell$-ergodic iff $\sum_{i=1}^\infty\pi_i m_{i0}^{(\ell-1)}<\infty$.}

\hyperlink{First}{$\triangleleft$}
\hypertarget{Second}{}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE -
\overlays{3}{%
\begin{slide}[Box]{Single birth processes}
$$d_0=0,\quad d_i=\frac 1{q_{i,i+1}}\Big(
\fromSlide*{2}{\rnode{NA}{\pscirclebox[linecolor=red]{1}}}
\onlySlide*{1}{\rnode{NA}{\pscirclebox[linecolor=red,linestyle=none]{1}}}
+\sum_{k=0}^{i-1}q_i^{(k)}d_k\Big),\quad i\ge 1.$$

\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 13-
\overlays{3}{%
\begin{slide}[Dissolve]{$\ell$-ergodicity}
\centerline{\large\yellow For $\ell\ge 1$, $P_t$ is $\ell$-ergodic iff $d^{(\ell)}<\infty$,}
where
$$d^{(\ell)}:=\sup_{i\ge 0}\frac{\sum_{j=0}^i
\fromSlide*{2}{\rnode{NA}{\pscirclebox[linecolor=red]{d^{(\ell)}_j}}}
\onlySlide*{1}{\rnode{NA}{\pscirclebox[linecolor=red,linestyle=none]{d^{(\ell)}_j}}}
}{\sum_{j=0}^iF_j^{(0)}}.$$

\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}}

\OnlySlide{3}

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

%---------------------------------------------------------------------- SLIDE 14-
\begin{slide}[Split]{$\ell$-ergodicity}
{\large\green Key:}

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

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

%---------------------------------------------------------------------- SLIDE 15-
\overlays{4}{
\begin{slide}[Glitter]{Exponential ergodicity}
{\yellow $P_t$ is exponentially ergodic iff

\FromSlide{2}\vskip0.5cm
$d^{(\ell)}/(\ell-1)!\le \gamma^\ell$ for some $\gamma>0$ and
$\ell\gg 1$,

\FromSlide{3}\vskip0.5cm
equivalently, $\displaystyle\varlimsup_{\ell\to\infty}\root{\ell}\of{d^{(\ell)}}/\ell<\infty$.}

\vskip1cm
\OnlySlide{4}
 \rput(9,0.2){
      \onlySlide{4}{
        \psellipse[fillstyle=solid,fillcolor=LemonChiffon](0.3,-0.7)(1.5,1)}}
 \rput(9.5,-0.5){
      \onlySlide{4}{\blue --Thanks--}}
\end{slide}
}
%------------------------------------------------------------------------------
\clearpage
\end{CJK*}
\end{document}

%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End:
