Joint Distributions of Numbers of Runs of Specified Lengths in a Sequence of Markov Dependent Multistate Trials

Let Z0, Z1,...,Zn be a sequence of Markov dependent trials with state space Ω = {F1,...,Fλ, S1,...,Sν}, where we regard F1,...,Fλ as failures and S1,...,Sν as successes. In this paper, we study the joint distribution of the numbers of Si-runs of lengths kij (i = 1,2,...,ν, j = 1,2,...,ri) based on four different enumeration schemes. We present formulae for the evaluation of the probability generating functions and the higher order moments of this distribution. In addition, when the underlying sequence is i.i.d. trials, the conditional distribution of the same run statistics, given the numbers of success and failure is investigated. We give further insights into the multivariate run-related problems arising from a sequence of the multistate trials. Besides, our results have potential applications to problems of various research areas and will come to prominence in the future.