Let $\{\xi_{n}\}_{n\geq 0}$ denote
an ergodic Markov chain
with a finite state space
$\mbox{\boldmath$\Xi$}=\{1,2,\cdots, s\}$.
%with a transition matrix
%$Q=(q_{jk})_{j,k\in\mbox{\boldmath$\Xi$}}$.
For each $j,k\in\mbox{\boldmath$\Xi$}$,
let $\{Y^{jk}_{n}\}_{n\geq 1}$
be a sequence of i.i.d. $\{-1,1\}$-valued
random variables which are independent of
$\{\xi_{n}\}$.
%We assume that
%$\{Y^{jk}_{n}\}_{n\geq 1}$ is the
%sequence of identically distributed
%random variables for each $j,k\in\mbox{\boldmath$\Xi$}$.
We define the process $\{S_{n}\}_{n\geq 0}$
by $S_{0}=0$ and
$S_{n}=S_{n-1}+Y^{\xi_{n-1}\xi_{n}}_{n}$ for $n\geq 1$.
%The process $\{S_{n}\}_{n\geq 0}$ is known as
%{\it a random walk defined on a finite
%Markov chain}.
Let $a$ be a positive integer.
% and denote by
%$[0,a]$ the interval consisting of the integers
%$0,1,2,\cdots, a$.
We denote by $T_{x}$ the first exit time of
the process from the interval $[-x,a-x]$
for each $x=0,1,\cdots,a$.
% $T_{x}=\inf\{n>0|x+S_{n}\notin [0,a]\}$
%for each $x\in [0,a]$.
%Set
We give an asymptotic behavior of
the transition functions
%of random walk defined on
%a finite Markov chain with absorbing barriers
$P_{jk}^{(n)}(x,y)
=\mbox{\boldmath$P$}\{x+S_{n}=y; T_{x}>n; \xi_{n}=k
|\xi_{0}=j\}
% \label{asymptotic}
$ as $n\rightarrow\infty$
for each $x,y\in [0,a]$
and all $j,k\in\mbox{\boldmath$\Xi$}$.