  Let $(A_i)_{i \geq 0}$ be a finite state irreducible aperiodic Markov chain
and $f$ a lattice score function such that the average score is negative and
positive scores are possible. Define $S_0:=0$ and $S_k:=\sum_{i=1}^k f(A_i)$
the successive partial sums, $S^+$ the maximal non-negative partial sum, $Q_1$
the maximal segmental score of the first non-negative excursion and
$M_n:=\max_{0\leq k\leq\ell\leq n} (S_{\ell}-S_k)$ the local score first
defined by Karlin and Altschul (1990). We establish recursive formulae for the
exact distribution of $S^+$ and derive new approximations for the distributions
of $Q_1$ and $M_n$. Computational methods are presented in a simple application
case and comparison is performed between these new approximations and the ones
proposed by Karlin and Dembo (1992) in order to evaluate improvements.
