In this paper, we obtain sufficient and necessary conditions for local asymptotics for the maximum of a Markov modulated random walk with long-tailed increments and negative drifts, where the local asymptotics means asymptotic behaviour of P（ ∈ （x,x ＋ z]） for each z 〉 0, as x→∞ Our results extend and improve the existing ones in the literature.
