Limit distributions of maximal segmental score among Markov-dependent partial sums

Let s1, ., sn be generated governed by an r-state irreducible aperiodic Markov chain. The partial sum process is determined by a realization of states with s0 = . and the real-valued i.i.d. bounded variables X.ß associated with the transitions si = ., si+1 = .. Assume . .. has negative stationary mean. The explicit limit distribution of the maximal segmental sum is derived. Computational methods with potential applications to the analysis of random Markov-dependent letter sequences (e.g. DNA and protein sequences) are presented.