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Joint Distributions of Numbers of Runs of Specified Lengths in a Sequence of Markov Dependent Multistate Trials

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Abstract

Let Z 0, Z 1,...,Z n be a sequence of Markov dependent trials with state space Ω = {F 1,...,F λ, S 1,...,S ν}, where we regard F 1,...,F λ as failures and S 1,...,S ν as successes. In this paper, we study the joint distribution of the numbers of S i -runs of lengths k ij (i = 1,2,...,ν, j = 1,2,...,r i ) based on four different enumeration schemes. We present formulae for the evaluation of the probability generating functions and the higher order moments of this distribution. In addition, when the underlying sequence is i.i.d. trials, the conditional distribution of the same run statistics, given the numbers of success and failure is investigated. We give further insights into the multivariate run-related problems arising from a sequence of the multistate trials. Besides, our results have potential applications to problems of various research areas and will come to prominence in the future.

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Authors and Affiliations

  1. Faculty of Economics, Seikei University, 3-3-1 Kichijoji-Kitamachi, Musasino-shi, Tokyo, 180-8633, Japan

    Kiyoshi Inoue

  2. Department of Mathematics, Faculty of Engineering, Kansai University, 3-3-35 Yamate-cho, Suita-shi, Osaka, 564-8680, Japan

    Sigeo Aki

Authors
  1. Kiyoshi Inoue
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  2. Sigeo Aki
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Correspondence to Kiyoshi Inoue.

Additional information

This research was partially supported by the ISM Cooperative Research Program (2004-ISM·CRP-2007).

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Inoue, K., Aki, S. Joint Distributions of Numbers of Runs of Specified Lengths in a Sequence of Markov Dependent Multistate Trials. AISM 59, 577–595 (2007). https://doi.org/10.1007/s10463-006-0056-6

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  • DOI: https://doi.org/10.1007/s10463-006-0056-6

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