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JOINT DISTRIBUTIONS OF NUMBERS OF SUCCESS-RUNS

UNTIL THE FIRST CONSECUTIVE *k* SUCCESSES 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 March 13, 1996; revised May 8, 1997)

**Abstract.**
Let *X*_{-m+1}, *X*_{-m+2}, ...., *X*_{0},
*X*_{1}, *X*_{2}, .... be a time-homogeneous {0,1}-valued
*m*-th order Markov chain. Joint distributions of the
numbers of trials, failures and successes, of the numbers
of trials and success-runs of length *l* (*m* __<__ *l* __<__ *k*)
and of the numbers of trials and success-runs of length *l*
(*l* __<__ *m* __<__ *k*) until the first consecutive *k* successes
are obtained in the sequence *X*_{1}, *X*_{2}, ..... There
are some ways of counting numbers of runs of length *l*.
This paper studies the joint distributions based on four
ways of counting numbers of runs, i.e., the number of
non-overlapping runs of length *l*, the number of runs of
length greater than or equal to *l*, the number of
overlapping runs of length *l* and the number of runs of
length exactly *l*. Marginal distributions of them can be
obtained immediately, and surprisingly their distributions
are very simple.

*Key words and phrases*:
Probability generating
function, discrete distribution, success and failure runs,
geometric distribution, geometric distribution of order
*k*, higher order Markov chain.

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