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

  • James C. Fu
  • W. Y. Wendy LouEmail author


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.


Runs and patterns Finite Markov chain imbedding Waiting time distribution 

AMS 2000 Subject Classification

Primary 60E05 Secondary 60J10 


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  1. R. Arratia, L. Goldstein, and L. Gordon, “Poisson approximation and the Chen–Stein method,” Statistical Science vol. 5 pp. 403–434, 1990.zbMATHMathSciNetGoogle Scholar
  2. N. Balakrishnan and M. V. Koutras, Runs and Scans with Applications, Wiley: New York, 2002.zbMATHGoogle Scholar
  3. R. R. Bahadur,“Some limit theorems in statistics,” Regional Conference Series in Applied Mathematics, SIAM, 1971.Google Scholar
  4. A. D. Barbour and O. Chryssaphinou, “Compound Poisson approximation: A user’s guide,” Annals of Applied Probability vol. 11 pp. 964–1002, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  5. A. D. Barbour, O. Chryssaphinou, and M. Roos, “Compound Poisson approximation in reliability theory,” IEEE Transactions on Reliability vol. 44 pp. 393–402, 1995.CrossRefGoogle Scholar
  6. A. D. Barbour, L. Holst, and S. Janson, Poisson Approximation, Oxford Studies in Probability, Oxford University Press: New York, 1992.zbMATHGoogle Scholar
  7. P. Billingsley, Convergence of Probability Measures, Wiley: New York, 1968.zbMATHGoogle Scholar
  8. J. Cai, “Reliability of a large consecutive-k-out-of-n:F system with unequal component reliability,” IEEE Transactions on Reliability vol. 43 pp. 107–111, 1994.CrossRefGoogle Scholar
  9. Y. M. Chang, “Distribution of waiting time until the r-th occurrence of a compound pattern,” Statistics & Probability Letters vol. 75 pp. 29–38, 2005.zbMATHCrossRefMathSciNetGoogle Scholar
  10. L. H. Y. Chen, “Poisson approximation for dependent trials,” Annals of Probability vol. 3 pp. 534–545, 1975.zbMATHGoogle Scholar
  11. J. Chen and X. Huo, “Distribution of the length of the longest significance run on a Bernoulli net and its applications,” Journal of the American Statistical Association vol. 473 pp. 321–331, 2006.CrossRefMathSciNetGoogle Scholar
  12. V. M. Dwyer, “The influence of microstructure on the probability of early failure in aluminum-based interconnects,” Journal of Applied Physics vol. 96 pp. 2914–2922, 2004.CrossRefGoogle Scholar
  13. W. Feller, An Introduction to Probability Theory and its Applications (Vol. I, 3rd ed.) Wiley: New York, 1968.zbMATHGoogle Scholar
  14. J. C. Fu, “Bounds for reliability of large consecutive-k-out-of-n:F systems with unequal component reliability,” IEEE Transactions on Reliability vol. 35 pp. 316–319, 1986.zbMATHCrossRefGoogle Scholar
  15. J. C. Fu and W. Y. W. Lou, Distribution Theory of Runs and Patterns and Its Applications: A Finite Markov Chain Imbedding Approach, World Scientific: New Jersey, 2003.zbMATHGoogle Scholar
  16. J. C. Fu and W. Y. W. Lou, “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 vol. 28 pp. 291–310, 2006CrossRefMathSciNetGoogle Scholar
  17. W. Hoeffding and H. Robbins, “The central limit theorem for dependent random variables,” Duke Mathematical Journal vol. 15 pp. 773–780, 1948.zbMATHCrossRefMathSciNetGoogle Scholar
  18. M. V. Koutras, “On a waiting time distribution in a sequence of Bernoulli trials,” Annals of the Institute of Statistical Mathematics vol. 48 pp. 789–806, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  19. M. V. Koutras and S. G. Papastavridis, “Application of the Stein–Chen method for bounds and limit theorems in the reliability of coherent structures,” Naval Research Logistic vol. 40 pp. 617–631, 1993.zbMATHMathSciNetGoogle Scholar
  20. M. V. Koutras, “Waiting time distributions associated with runs of fixed length in two-state Markov chains,” Annals of the Institute of Statistical Mathematics vol. 49 pp. 123–139, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  21. P. A. MacMahon, Combinatory Analysis, Cambridge University Press: London, 1915.Google Scholar
  22. D. L. McLeish, “Dependent central limit theorems and invariance principles,” Annals of Probability vol. 2 pp. 620–628, 1974.zbMATHMathSciNetGoogle Scholar
  23. M. Muselli, “New improved bounds for reliability of consecutive-k-out-of-n:F systems,” Journal of Applied Probability vol. 37 pp. 1164–1170, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  24. G. Nuel, Fast p-value Computations Using Finite Markov Chain Imbedding: Application to Local Score and Pattern Statistics, TR223 Université d’Evry Val d’Essonne, 2005.Google Scholar
  25. G. Nuel, “LD-SPatt: Large deviations statistics for patterns on Markov chains,” Journal of Computational Biology vol. 11 pp. 1023-1033, 2004.CrossRefGoogle Scholar
  26. Y. Rinott and V. Rotar, “Normal approximations by Stein’s method,” Decisions in Economics and Finance vol. 23 pp. 15–29, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  27. J. Riordan, An Introduction to Combinatorial Analysis, Wiley: New York, 1958.zbMATHGoogle Scholar
  28. S. Robin and J. J. Daudin, “Exact distribution of word occurrences in a random sequence of letters,” Journal of Applied Probability vol. 36 pp. 179–193, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  29. C. Stein, “Approximate computation of expectations,” Institute of Mathematical Statistics Lecture Notes—Monograph Series 7, Institute of Mathematical Statistics, Hayward, CA, 1986.Google Scholar
  30. M. S. Waterman, Introduction to Computational Biology, Chapman and Hall: New York, 1995.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of ManitobaWinnipegCanada
  2. 2.Department of Public Health Sciences and Department of StatisticsUniversity of TorontoTorontoCanada

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