###
ON GENERATING FUNCTIONS OF WAITING TIME PROBLEMS

FOR SEQUENCE PATTERNS OF DISCRETE RANDOM

VARIABLES

###
MASAYUKI UCHIDA

*The Institute of Statistical Mathematics,*

4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan
(Received July 31, 1996; revised August 28, 1997)

**Abstract.**
Let *X*_{1}, *X*_{2},... be
a sequence of independent and identically
distributed random variables, which take values in a countable set
*S* = { 0,1,2, ... }.
By a pattern we mean a finite sequence of elements in *S*.
For every *i* =0,1,2,..., we denote by
*P*_{i} = ``*a*_{i,1}*a*_{i,2} ··· *a*_{i},_{ki}''
the pattern of some length *k*_{i}, and
*E*_{i} denotes the event that the pattern
*P*_{i} occurs in the sequence *X*_{1}, *X*_{2}, ....
In this paper,
we have derived the generalized probability generating functions of
the distributions of the waiting times until the
*r*-th occurrence
among the events {*E*_{i}}_{i=0}^{\infty}.
We also have derived the probability generating functions
of the distributions of the number of occurrences of sub-patterns of
length *l* (*l* < *k*)
until the first occurrence of the pattern of length *k*
in the higher order Markov chain.

*Key words and phrases*:
Sequence patterns, runs,
sooner and later problems, *r*-th occurrence problem,
discrete distribution of order *k*,
generalized probability generating function, higher order Markov chain.

**Source**
( TeX ,
DVI ,
PS )