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WAITING TIME PROBLEMS FOR A SEQUENCE

OF DISCRETE RANDOM VARIABLES

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SIGEO AKI

*Department of Mathematical Science, Faculty of Engineering Science, Osaka University,*

Machikaneyama-cho, Toyonaka, Osaka 560, Japan
(Received September 20, 1990; revised June 13, 1991)

**Abstract.**
Let *X*_{1}, *X*_{2}, .... be a sequence of nonnegative
integer valued random variables. For each nonnegative integer *i*,
we are given a positive integer *k*_{i}. For every *i* =0,1,2, .... ,
*E*_{i} denotes the event that a run of *i* of length *k*_{i} occurs in
the sequence *X*_{1}, *X*_{2}, ..... For the sequence *X*_{1}, *X*_{2}, ....,
the generalized pgf's of
the distributions of the waiting times until the *r*-th occurrence
among the events { *E*_{i} }_{i=0}^{\infty} are obtained.
Though our situations are
general, the results are very simple. For the special cases that
*X*'s are i.i.d. and {0,1}-valued, the corresponding results are
consistent with previously published results.

*Key words and phrases*:
Sooner and later problems, generalized
probability generating function, discrete distributions, binary
sequence of order *k*.

**Source**
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