In this paper we study two distributions, namely the distribution of the waiting times until given numbers of occurrences of compound patterns and the distribution of the numbers of occurrences of compound patterns in a fixed number of trials. We elucidate the interrelation between these two distributions in terms of the generating functions. We provide perspectives on the problems related to compound patterns in statistics and probability. As an application, the waiting time problem of counting runs of specified lengths is considered in order to illustrate how the distributions of waiting times can be derived from our theoretical results.

The sooner and later waiting time problems have been extensively studied and applied in various areas of statistics and applied probability. In this paper, we give a comprehensive study of ordered series and later waiting time distributions of a number of simple patterns with respect to nonoverlapping and overlapping counting schemes in a sequence of Markov dependent multistate trials. Exact distributions and probability generating functions are derived by using the finite Markov chain imbedding technique. Examples are given to illustrate our results.

Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

Let be the scan statistic of window size r for a sequence of n bistate trials . The scan statistic Sn(r) has been successfully used in various fields of applied probability and statistics, and its distribution has been studied extensively in the literature. Currently, all existing formulae for the distribution of Sn(r) are rather complex, and they can only be numerically implemented when is a sequence of Bernoulli trials, the window size r is less than 20 and the length of the sequence n is not too large. Hence, these formulae have been limiting the practical applications of the scan statistic. In this article, we derive a simple and effective formula for the distribution of Sn(r) via the finite Markov chain embedding technique to overcome some of the limitations of the existing complex formulae. This new formula can be applied when is either a sequence of Bernoulli trials or a sequence of Markov dependent bistate trials. Selected numerical examples are given to illustrate our results.