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

Let s 1 , …, s n be generated governed by an r-state irreducible aperiodic Markov chain. The partial sum process is determined by a realization of states with s 0 = α and the real-valued i.i.d. bounded variables X αß associated with the transitions s i = α , s i+ 1 = β. Assume Χ αβ 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.