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RUNS, SCANS AND URN MODEL DISTRIBUTIONS:

A UNIFIED MARKOV CHAIN APPROACH

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M. V. KOUTRAS AND V. A. ALEXANDROU

*Department of Mathematics, University of Athens,
Panepistemiopolis, Athens 157 84, Greece*
(Received June 6, 1994; revised December 2, 1994)

**Abstract.**
This paper presents a unified approach for the
study of the exact distribution (probability mass function, mean,
generating functions) of three types of random variables: (a)
variables related to success runs in a sequence of Bernoulli trials
(b) scan statistics, i.e. variables enumerating the moving windows
in a linearly ordered sequence of binary outcomes (success or
failure) which contain prescribed number of successes and (c)
success run statistics related to several well known urn models. Our
approach is based on a Markov chain imbedding which permits the
construction of probability vectors satisfying triangular recurrence
relations. The results presented here cover not only the case of
identical and independently distributed Bernoulli variables, but the
non-identical case as well. An extension to models exhibiting Markov
dependence among the successive trials is also discussed in brief.

*Key words and phrases*:
Success runs, scan statistics,
urn models, Markov chains, triangular multidimensional recurrence
relations, distributions of order *k*.

**Source**
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