On waiting time problems associated with runs in Markov dependent trials
The aim of the paper is to develop a general technique useful to the study of waiting time distribution for the \(k\)th occurrence of a success run of length \(k\) in a sequence of Markov dependent trials. Let \(W^{(a)}_{r,k}\) be a random variable denoting the number of Markov dependent trials until the occurrence of the \(r\)th success run of length \(k\). The superscript a denotes the counting scheme employed and takes the following values: \(a=\text{I}\) for the non-overlapping scheme; \(a=\text{II}\) for the at least \(k\) scheme; \(a=\text{III}\) for the overlapping scheme; \(a=\text{IV}\) for the exactly \(k\) scheme. Also, let \(W_S\) \((W_L)\) be a random variable denoting the waiting time for a run of \(k\) successes or (and) a run of \(s\) failures whichever comes sooner (later). The proposed technique for the study of \(W^{(a)}_{r,k}\), \(W_S\), and \(W_L\) is equivalent to the method of generalized probability generating functions. Section 2 provides the necessary tools for the study of the waiting time distributions. In Section 3, by exploiting the Markov chain embedding technique, the probability mass function and the probability generating function are derived for \(W^{(a)}_{r,k}\), \(W_S\), and \(W_L\).
scientific article; zbMATH DE number 1368793