Joint distributions associated with patterns,successes and failures in a sequence of multi-state trials

Let {Z _t ,t≥1} be a sequence of trials taking values in a given setA={0, 1, 2,...,m}, where we regard the value 0 as failure and the remainingm values as successes. Let ε be a (single or compound) pattern. In this paper, we provide a unified approach for the study of two joint distributions, i.e., the joint distribution of the numberX _n of occurrences of ε, the numbers of successes and failures inn trials and the joint distribution of the waiting timeT _r until ther-th occurrence of ε, the numbers of successes and failures appeared at that time. We also investigate some distributions as by-products of the two joint distributions. Our methodology is based on two types of the random variablesX _n (a Markov chain imbeddable variable of binomial type and a Markov chain imbeddable variable of returnable type). The present work develops several variations of the Markov chain imbedding method and enables us to deal with the variety of applications in different fields. Finally, we discuss several practical examples of our results. This research was partially supported by the ISM Cooperative Research Program (2002-ISM·CRP-2007).