Waiting times for patterns in a sequence of multistate trials

Abstract

Let {X_n, n ≥ 1} be a sequence of trials taking values in a given set A, let 𝓔 be a pattern (simple or compound), and let X_r,𝓔 be a random variable denoting the waiting time for the rth occurrence of 𝓔. In the present article a finite Markov chain imbedding method is developed for the study of X_r,𝓔 in the case of the non-overlapping and overlapping way of counting runs and patterns. Several extensions and generalizations are also discussed.