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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. (English) Zbl 1122.60023
Summary: 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.

60F05 Central limit and other weak theorems
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60F10 Large deviations
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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. · Zbl 0955.62542
[2] N. Balakrishnan and M. V. Koutras, Runs and Scans with Applications, Wiley: New York, 2002. · Zbl 0991.62087
[3] R. R. Bahadur,”Some limit theorems in statistics,” Regional Conference Series in Applied Mathematics, SIAM, 1971. · Zbl 0257.62015
[4] A. D. Barbour and O. Chryssaphinou, ”Compound Poisson approximation: A user’s guide,” Annals of Applied Probability vol. 11 pp. 964–1002, 2001. · Zbl 1018.60051 · doi:10.1214/aoap/1015345355
[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. · Zbl 04527633 · doi:10.1109/24.406572
[6] A. D. Barbour, L. Holst, and S. Janson, Poisson Approximation, Oxford Studies in Probability, Oxford University Press: New York, 1992. · Zbl 0746.60002
[7] P. Billingsley, Convergence of Probability Measures, Wiley: New York, 1968. · Zbl 0172.21201
[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. · Zbl 04519991 · doi:10.1109/24.285122
[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. · Zbl 1081.60060 · doi:10.1016/j.spl.2005.05.007
[10] L. H. Y. Chen, ”Poisson approximation for dependent trials,” Annals of Probability vol. 3 pp. 534–545, 1975. · Zbl 0335.60016 · doi:10.1214/aop/1176996359
[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. · Zbl 1118.62308 · doi:10.1198/016214505000000574
[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. · doi:10.1063/1.1771825
[13] W. Feller, An Introduction to Probability Theory and its Applications (Vol. I, 3rd ed.) Wiley: New York, 1968. · Zbl 0155.23101
[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. · Zbl 0597.90036 · doi:10.1109/TR.1986.4335442
[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. · Zbl 1030.60063
[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, 2006 · Zbl 1099.60050 · doi:10.1007/s10463-006-0038-8
[17] W. Hoeffding and H. Robbins, ”The central limit theorem for dependent random variables,” Duke Mathematical Journal vol. 15 pp. 773–780, 1948. · Zbl 0031.36701 · doi:10.1215/S0012-7094-48-01568-3
[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. · Zbl 1002.60517 · doi:10.1007/BF00052333
[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. · Zbl 0795.60084 · doi:10.1002/1520-6750(199308)40:5<617::AID-NAV3220400506>3.0.CO;2-4
[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. · Zbl 0913.60019 · doi:10.1023/A:1003118807148
[21] P. A. MacMahon, Combinatory Analysis, Cambridge University Press: London, 1915. · JFM 45.1271.01
[22] D. L. McLeish, ”Dependent central limit theorems and invariance principles,” Annals of Probability vol. 2 pp. 620–628, 1974. · Zbl 0287.60025 · doi:10.1214/aop/1176996608
[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. · Zbl 0985.60080 · doi:10.1239/jap/1014843097
[24] G. Nuel, Fast p-value Computations Using Finite Markov Chain Imbedding: Application to Local Score and Pattern Statistics, TR223 Universit√© d’Evry Val d’Essonne, 2005.
[25] G. Nuel, ”LD-SPatt: Large deviations statistics for patterns on Markov chains,” Journal of Computational Biology vol. 11 pp. 1023-1033, 2004. · doi:10.1089/cmb.2004.11.1023
[26] Y. Rinott and V. Rotar, ”Normal approximations by Stein’s method,” Decisions in Economics and Finance vol. 23 pp. 15–29, 2000. · Zbl 0985.60024 · doi:10.1007/s102030050003
[27] J. Riordan, An Introduction to Combinatorial Analysis, Wiley: New York, 1958. · Zbl 0078.00805
[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. · Zbl 0945.60008 · doi:10.1239/jap/1032374240
[29] C. Stein, ”Approximate computation of expectations,” Institute of Mathematical Statistics Lecture Notes–Monograph Series 7, Institute of Mathematical Statistics, Hayward, CA, 1986.
[30] M. S. Waterman, Introduction to Computational Biology, Chapman and Hall: New York, 1995. · Zbl 0831.92011
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