Joint distribution of k-tuple statistics in zero-one sequences of Markov-dependent trials

摘要

We consider a sequence of n , n ≥3, zero (0) - one (1) Markov-dependent trials. We focus on k -tuples of 1s; i.e. runs of 1s of length at least equal to a fixed integer number k , 1≤ k ≤ n . The statistics denoting the number of k -tuples of 1s, the number of 1s in them and the distance between the first and the last k -tuple of 1s in the sequence, are defined. The work provides, in a closed form, the exact conditional joint distribution of these statistics given that the number of k -tuples of 1s in the sequence is at least two. The case of independent and identical 0?1 trials is also covered in the study. A numerical example illustrates further the theoretical results.