AISM 52, 767-777

## Numbers of success-runs of specified length until certain stopping time rules and generalized binomial distributions of order k

### Sigeo Aki1 and Katuomi Hirano2

1Department of Informatics and Mathematical Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, 560-8531, Japan
2The Institute of Statistical Mathematics, 4-6-7 Minani-Azabu, Minato-ku, Tokyo, 106-8569, Japan

(Received April 21, 1998; revised February 22, 1999)

Abstract.    A new distribution called a generalized binomial distribution of order $k$ is defined and some properties are investigated. A class of enumeration schemes for success-runs of a specified length including non-overlapping and overlapping enumeration schemes is rigorously studied. For each nonnegative integer $\mu$ less than the specified length of the runs, an enumeration scheme called $\mu$-overlapping way of counting is defined. Let $k$ and $\ell$ be positive integers satisfying $\ell < k$. Based on independent Bernoulli trials, it is shown that the number of $(\ell -1)$-overlapping occurrences of success-run of length $k$ until the $n$-th overlapping occurrence of success-run of length $\ell$ follows the generalized binomial distribution of order $(k- \ell)$. In particular, the number of non-overlapping occurrences of success-run of length $k$ until the $n$-th success follows the generalized binomial distribution of order $(k-1)$. The distribution remains unchanged essentially even if the underlying sequence is changed from the sequence of independent Bernoulli trials to a dependent sequence such as higher order Markov dependent trials. A practical example of the generalized binomial distribution of order $k$ is also given.

Key words and phrases:    Binomial distribution of order $k$, Markov chain, probability generating function, stopping time, success-run, waiting time.

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