On the joint distribution of runs in the sequence of Markov-dependent multi-state trials (Q2493877)

The authors consider the problem of deriving the joint distribution of the numbers of runs of various lengths for five counting schemes (exact, non-overlapping, overlapping, at least, \(\ell\) overlapping \((0\leq\ell\leq k-1))\) in Markov-dependent multi-state trials. Using the method of conditional probability generating functions (cpgf's), they first derive a system of recursive relations for cpgf's and then the exact pgf in terms of a matrix polynomial of degree \(n\). Expanding this polynomial, they obtain a recursive formula for evaluating the exact joint probability distribution function (pdf). The feasibility and compatibility of this formula is illustrated numerically.