Abstract
In this paper, we investigate the exact distribution of the waiting time for ther-th ℓ-overlapping occurrence of success-runs of a specified length in a sequence of two state Markov dependent trials. The probability generating functions are derived explicitly, and as asymptotic results, relationships of a negative binomial distribution of orderk and an extended Poisson distribution of orderk are discussed. We provide further insights into the run-related problems from the viewpoint of the ℓ-overlapping enumeration scheme. We also study the exact distribution of the number of ℓ-overlapping occurrences of success-runs in a fixed number of trials and derive the probability generating functions. The present work extends several properties of distributions of orderk and leads us a new type of geneses of the discrete distributions.
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Inoue, K., Aki, S. Generalized binomial and negative binomial distributions of orderk by thel-overlapping enumeration scheme. Ann Inst Stat Math 55, 153–167 (2003). https://doi.org/10.1007/BF02530491
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DOI: https://doi.org/10.1007/BF02530491