On the concentration of runs of ones of length exceeding a threshold in a Markov chain

ABSTRACT

Consider a homogeneous two state (failure-success or zero-one) Markov chain of first order. The paper deals with the position and the length of the shortest segment of the first n, $n\ge 5$, trials of the chain in which all runs of ones of length greater than or equal to a fixed number are concentrated. Accordingly, we define random variables denoting the starting/ending position of the first/last such runs in the chain as well as the implied distance between them. The paper provides exact closed form expressions for the probability mass function of these random variables given that the number of the considered runs in the chain is at least two. An application concerning DNA sequences is discussed. It is accompanied by numerics which exemplify further the theoretical results.

KEYWORDS:

• Exact distributions
• runs
• Markov chain
• binary trials
• DNA

AMS CLASSIFICATIONS:

• 60E05
• 62E15
• 60J10
• 60C05
• 92D20

Acknowledgements

The authors wish to thank the anonymous referees for the thorough reading, useful comments and suggestions which helped to improve the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.