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On Generating Functions of Waiting Times and Numbers of Occurrences of Compound Patterns in a Sequence of Multistate Trials

Part of: Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  14 July 2016

Kiyoshi Inoue  and
Sigeo Aki
Kiyoshi Inoue*
Affiliation:
Seikei University
Sigeo Aki*
Affiliation:
Kansai University
*
∗ Postal address: Faculty of Economics, Seikei University, 3-3-1 Kichijoji-Kitamachi, Musasino-shi, Tokyo, 180-8633, Japan. Email address: kinoue@econ.seikei.ac.jp
∗∗ Postal address: Department of Mathematics, Faculty of Engineering, Kansai University, 3-3-35 Yamate-cho, Suita-shi, Osaka, 564-8680, Japan.
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Abstract

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In this paper we study two distributions, namely the distribution of the waiting times until given numbers of occurrences of compound patterns and the distribution of the numbers of occurrences of compound patterns in a fixed number of trials. We elucidate the interrelation between these two distributions in terms of the generating functions. We provide perspectives on the problems related to compound patterns in statistics and probability. As an application, the waiting time problem of counting runs of specified lengths is considered in order to illustrate how the distributions of waiting times can be derived from our theoretical results.

Keywords

Compound patternmultistate trialsooner waiting timelater waiting timerunenumeration schemeprobability functionprobability generating functiondouble generating function

MSC classification

Primary: 60E05: Distributions
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 1 , March 2007 , pp. 71 - 81
Copyright
Copyright © Applied Probability Trust 2007 

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