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\begin{center}\tcr{\rmH Markov Chains as Discrete\vspace{5pt}
\\ Linear Systems} 
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\newcommand{\cS}{{\cal S}}
\newcommand{\cA}{{\cal A}}


\begin{description}  
\item[\tcr{Given:}] \tcr{1)} A \tcg{system} \mc{\cS} which can be in 
\tcr{precisely one} of \mc{m} \tcg{states} \mc{s_{\scr1},\ldots,s_m}.

\tcr{2)} A \tcg{procedure} \mc{\cA} which is applied to the system at regular time 
intervals \mc{t = 0, 1, \ldots} in such a way that its  \tcg{transition probability 
(matrix)} \mc{A = (a_{ij})} does not depend on \mc{t}. Here
$$
\tcb{a_{ij} =\;} \parbox[t]{15cm}{the \tcg{probability} that the procedure takes a 
system in state \mc{s_j} to state \mc{s_i}.}
$$
Such a pair \mc{(\cS, \cA)} is called a \tcr{Markov chain}, and the \mc{m\times m} 
matrix \mc{A} is called the \tcr{transition matrix} 
of the Markov chain.  

\item[\tcr{Then:}] If we put \mc{\vec{v}_n = (v_{n,1}, v_{n,2}, \ldots, 
v_{n,m})^{\scr t}}, where
$$
\tcb{v_{n,i} =\;}\parbox[t]{15cm}{the probability that the system \mc{\cS} is in 
state \mc{s_i} at time \mc{t = n},}
$$
then we have the associated \tcr{discrete linear system} 
%(with \tcg{constant coefficients})
$$
\tcb{\vec{v}_{n+\scr1} = A \vec{v}_n.}
$$

\item[\tcr{Note:}] The transition matrix \mc{A = (a_{ij})} is a 
\tcr{stochastic matrix}: \mc{a_{ij} \ge 0} and the \tcg{sum of each column} is 
\mc{1}.
\end{description} 
\end{titlepage}
\end{document}                          
