Abstract
Let {Z n , n ≥ 1} be a time-homogeneous {0, 1}-valued Markov chain, and let N n be a random variable denoting the number of runs of "1" of length k in the first n trials. In this article we conduct a systematic study of N n by establishing formulae for the evaluation of its probability generating function, probability mass function and moments. This is done in three different enumeration schemes for counting runs of length k, the "non-overlapping", the "overlapping" and the "at least" scheme. In the special case of i.i.d. trials several new results are established.
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Antzoulakos, D.L., Chadjiconstantinidis, S. Distributions of Numbers of Success Runs of Fixed Length in Markov Dependent Trials. Annals of the Institute of Statistical Mathematics 53, 599–619 (2001). https://doi.org/10.1023/A:1014689617413
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DOI: https://doi.org/10.1023/A:1014689617413