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

Abstract

In this article we consider two-state Markov dependent trials and the waiting time T needed for the occurrence of the r ₀-th ℓ₀-overlapping failure run of length k ₀, or the occurrence of the r ₁-th ℓ₁-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.

Keywords:

• Binomial/negative binomial distribution of order k
• Sooner waiting time problem
• Success run
• Markov chain
• ℓ-Overlapping enumeration scheme
• Probability generating function
• Probability mass function
• Moments