On the distribution of the number of successes in fourth- or lower-order Markovian trials

This paper presents an algorithm which may be used to compute the distribution of the number of successes in a sequence of binary trials which displays dependency. The case of independent trials is treated even in elementary texts, and these results are frequently extended in the literature to the case of first-order Markovian dependence. However, results are scarce for higher-order Markovian dependence. In this article, an algorithm is developed for computing the distribution of the number of successes in binary sequences, under the assumption that the dependence structure is fourth-order Markovian. The importance of using the best model order for a particular data set is discussed. Examination of the computed distribution estimates for various orders may assist in model determination.