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.

中文翻译：

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Markov依赖试验的零一序列中k元组统计量的联合分布

我们考虑n，n≥3，零（0）-一（1）马尔可夫依赖试验的序列。我们关注1s的k元组；即长度为1s的游程至少等于固定整数k，1≤k≤n。定义了表示1s的k元组的数量，其中1s的数量以及序列中1s的第一个和最后一个k元组之间的距离的统计量。假设序列中1s的k元组的数量至少为2，则该工作以封闭形式提供这些统计信息的精确条件联合分布。该研究还涵盖了独立且相同的0-1试验的案例。数值例子进一步说明了理论结果。