Generalized binomial and negative binomial distributions of order \(k\) by the \(\ell\)-overlapping enumeration scheme
The purpose of this paper is to investigate the exact distribution of the waiting time for the \(r\)\,th \(\ell\)-overlapping occurrence of success-runs of specified length in a sequence of two-state Markov dependent trials. The present work extends several properties of distributions of order \(k\) and involves a new type of geneses of discrete distributions. The main technical tool to derive the results within this paper is the Markov chain embedding method [see \textit{J. C. Fu}, IEEE Trans. Reliab. 35, 316--319 (1986; Zbl 0597.90036)], which has a great potential to be extended to other problems. The paper is organized as follows: Section 2 studies the waiting time distribution for the \(r\)\,th \(\ell\)-overlapping occurrence of success-run of length \(k\) in a sequence of \(\{0,1\}\)-valued Markov dependent trials, and derives the probability generating functions. The corresponding variable is expressed as a sum of \(r\) independent waiting time random variables. Section 3 investigates the asymptotic behaviour of the distributions analyzed in Section 2 as \(r\to\infty\), and discusses the relationship between binomial distributions of the number of \(\ell\)-overlapping occurrences of success-runs of length \(k\) in a sequence of a fixed number of trials, and obtains the probability generating functions. The \(\ell\)-overlapping enumeration scheme is based on a new type of geneses of order-\(k\) distributions.
\(\ell\)-overlapping occurrence of success-runs
scientific article; zbMATH DE number 2106187