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On the Normal Approximation for the Distribution of the Number of Simple or Compound Patterns in a Random Sequence of Multi-state Trials

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  • James C. Fu

    (University of Manitoba)

  • W. Y. Wendy Lou

    (University of Toronto)

Abstract

Distributions of numbers of runs and patterns in a sequence of multi-state trials have been successfully used in various areas of statistics and applied probability. For such distributions, there are many results on Poisson approximations, some results on large deviation approximations, but no general results on normal approximations. In this manuscript, using the finite Markov chain imbedding technique and renewal theory, we show that the number of simple or compound patterns, under overlap or non-overlap counting, in a sequence of multi-state trials follows a normal distribution. Poisson and large deviation approximations are briefly reviewed.

Suggested Citation

  • James C. Fu & W. Y. Wendy Lou, 2007. "On the Normal Approximation for the Distribution of the Number of Simple or Compound Patterns in a Random Sequence of Multi-state Trials," Methodology and Computing in Applied Probability, Springer, vol. 9(2), pages 195-205, June.
    • Handle: RePEc:spr:metcap:v:9:y:2007:i:2:d:10.1007_s11009-007-9019-5
    DOI: 10.1007/s11009-007-9019-5
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    References listed on IDEAS

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    1. Markos V. Koutras & Stavros G. Papastavridis, 1993. "Application of the stein‐chen method for bounds and limit theorems in the reliability of coherent structures," Naval Research Logistics (NRL), John Wiley & Sons, vol. 40(5), pages 617-631, August.
    2. Yosef Rinott & Vladimir Rotar, 2000. "Normal approximations by Stein's method," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 23(1), pages 15-29.
    3. Chang, Yung-Ming, 2005. "Distribution of waiting time until the rth occurrence of a compound pattern," Statistics & Probability Letters, Elsevier, vol. 75(1), pages 29-38, November.
    4. Chen, Jihong & Huo, Xiaoming, 2006. "Distribution of the Length of the Longest Significance Run on a Bernoulli Net and Its Applications," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 321-331, March.
    5. James Fu & W. Lou, 2006. "Waiting Time Distributions of Simple and Compound Patterns in a Sequence of r-th Order Markov Dependent Multi-state Trials," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(2), pages 291-310, June.
    6. M. Koutras, 1997. "Waiting Time Distributions Associated with Runs of Fixed Length in Two-State Markov Chains," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 49(1), pages 123-139, March.
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    Cited by:

    1. Frosso S. Makri & Zaharias M. Psillakis, 2011. "On Success Runs of Length Exceeded a Threshold," Methodology and Computing in Applied Probability, Springer, vol. 13(2), pages 269-305, June.

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