Waiting Times and Number of Appearances of Runs: A Unified Approach

• Published in: Communications in statistics. Theory and methods 2003-01, Vol.32 (7), p.1289-1315
• Main Author: Antzoulakos, Demetrios L.
• Format: Article
• Language: English
• Subjects: Algebra; Algebraic geometry; Binomial/negative binomial distribution of order k; Distribution theory; Exact sciences and technology; Markov chain; Markov processes; Mathematics; Moments; Probability and statistics; Probability generating function; Probability mass function; Probability theory and stochastic processes; Sciences and techniques of general use; Sooner waiting time problem; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Statistics; Success run; ℓ-Overlapping enumeration scheme
• ISSN: 0361-0926; 1532-415X
• DOI: 10.1081/STA-120021560

In this article we consider two-state Markov dependent trials and the waiting time T needed for the occurrence of the r 0 -th ℓ 0 -overlapping failure run of length k 0 , or the occurrence of the r 1 -th ℓ 1 -overlapping success runs of length k 1 , whichever comes sooner. We derive the joint pgf of a certain random vector associated with the waiting time T, and as byproduct we derive the pgf of the random vector (T r,ℓ , N f , N s ), where T r,ℓ denotes the waiting time for the r-th appearance of a ℓ-overlapping success run of length k, and N f and N s denote the number of failures and successes, respectively, observed at that time. Also we consider the random variable N n,ℓ which denotes the number of ℓ-overlapping success run of length k in the first n Markov dependent trials. Several formulae are provided for the evaluation of the pgf, pmf, and moments of T r,ℓ and N n,ℓ . For the special cases ℓ = 0 and ℓ = k−1 our results reduce to respective ones regarding the non-overlapping and overlapping counting scheme for runs, respectively. Furthermore, our results can be instantly adjusted to cover the framework of i.i.d. trials.