Factorization Methods and Boundary Problems for Sums of Random Variables Given on Markov Chains

This work concerns asymptotic study of the joint distribution of the maximum of a sequence of sums and the sums themselves on finite regular Markov chains. To this end, a series of lemmas are demonstrated on factorization of elements of general algebraic rings, as well as factorization identities having interest in themselves which provide a connection between various distributions pertaining to the random processes in question.