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ON NUMBER OF OCCURRENCES OF SUCCESS RUNS

OF SPECIFIED LENGTH IN A HIGHER-ORDER

TWO-STATE MARKOV CHAIN

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MASAYUKI UCHIDA

*The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu,*

Minato-ku, Tokyo 106-8569, Japan
(Received October 24, 1996; revised August 14, 1997)

**Abstract.**
Let *X*_{-m+1}, *X*_{-m+2} , ... ,
*X*_{0}, *X*_{1}, *X*_{2}, ... , *X*_{n} be a
time-homogeneous {0,1}-valued *m*-th order Markov chain. The
probability distributions of numbers of runs of ``1'' of
length *k* (*k* __>__ *m*) and of ``1'' of length *k* (*k* < *m*)
in the sequence of a {0,1}-valued *m*-th order Markov
chain are studied. There are some ways of counting numbers
of runs with length *k*. This paper studies the
distributions based on four ways of counting numbers of
runs, i.e., the number of non-overlapping runs of length
*k*, the number of runs with length greater than or equal
to *k*, the number of overlapping runs of length *k* and
the number of runs of length exactly *k*.

*Key words and phrases*:
Probability generating
function, discrete distribution, binomial distribution,
binomial distribution of order *k*, higher order Markov
chain.

**Source**
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