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 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 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.