Let A_i, i≥0 be a finite state irreducible aperiodic Markov chain and f a lattice score function such that the average score is negative and such that positive scores are possible. Define S_0 := 0 and S_k := f(A_1) +...+ f(A_k) 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 (S_k − S_j: 0≤j≤k≤n) the local score first defined by Karlin and Altschul (1990). We establish recursive formulae for the exact distribution of S^+ and derive a new approximation for the tail behaviour of Q_1, together with an asymptotic equivalence for the distribution of M_n. Computational methods are explicitly presented in a simple application case. Comparison is performed between the new approximations and the ones proposed by Karlin and Dembo (1992) in order to evaluate improvements, both in the simple application case and on the real data examples considered in Karlin and Altschul (1990}.