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Estimating the Error of a Permutational Central Limit Theorem

Published online by Cambridge University Press:  27 July 2009

Chern-Ching Chao ,
Lincheng Zhao  and
Wen-Qi Liang
Chern-Ching Chao
Affiliation:
Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan, R.O.C.
Lincheng Zhao
Affiliation:
Department of Mathematics, University of Science & Technology Hefei, Anhui 230026, China
Wen-Qi Liang
Affiliation:
Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan, R.O.C.
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Extract

Motivated by two measures of presortedness, number of runs and oscillation of a permutation, related to the sorting problem, we derive an error bound for normal approximation to the distribution of Here, αij's are given real numbers and π is a uniformly distributed random permutation of {l,…, n}. The derivation is based on Stein's method.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 10 , Issue 4 , October 1996 , pp. 533 - 541
Copyright
Copyright © Cambridge University Press 1996

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References

1.Angus, J.E. (1995). A coupling proof of the asymptotic normality of the permutation oscillation. Probability in the Engineering and Informational Sciences 9: 615–621.Google Scholar
2.Barton, D.E. & Mallows, C.L. (1965). Some aspects of the random sequence. Annals of Mathematical Statistics 36: 236–260.CrossRefGoogle Scholar
3.Bender, E.A. (1973). Central and local limit theorems applied to asymptotic enumeration. Journal of Combinatorial Theory, Series A 15: 91–111.Google Scholar
4.Carlitz, L., Kurtz, D.C., Scoville, R., & Stackelberg, O.P. (1972). Asymptotic properties of Eulerian numbers. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Cebiete 23: 47–54.Google Scholar
5.Chao, C.C., Bai, Z., & Liang, W.Q. (1993). Asymptotic normality for oscillation of permutation. Probability in the Engineering and Informational Sciences 7: 227–235.CrossRefGoogle Scholar
6.David, F.N. & Barton, D.E. (1962). Combinatorial chance. London: Griffin.Google Scholar
7.Dwass, M. (1973). The number of increases in a random permutation. Journal of Combinatorial Theory, Series A 15: 192–199.CrossRefGoogle Scholar
8.Estivill-Castro, V., Mannila, H., & Wood, D. (1993). Right invariant metrics and measures of presortedness. Discrete Applied Mathematics 42: 1–16.Google Scholar
9.Harris, B. & Park, C.J. (1994). A generalization of the Eulerian numbers with a probabilistic application. Statistics & Probability Letters 20: 37–47.Google Scholar
10.Knuth, D.E. (1973). The art of computer programming. Vol. 3: Sorting and searching. Reading, MA: Addison-Wesley.Google Scholar
11.Levcopoulos, C. & Petersson, O. (1989). Heapsort-adapted for prosorted files. Proceedings of the 1989 Workshop on Algorithms and Data Structures, Lecture Notes in Computer Science 382: 499–509. Berlin: Springer.Google Scholar
12.Levcopoulos, C. & Petersson, O. (1993). Adaptive heapsort. Journal of Algorithms 14: 395–413.CrossRefGoogle Scholar
13.Mannila, H. (1985). Measures of presortedness and optimal sorting algorithms. IEEE Transactions on Computers C-34(4): 318–325.Google Scholar
14.Stein, C.M. (1986). Approximate computations of expectations, Lecture Notes-Monograph Series, Vol. 7. Hayward, CA: Institute of Mathematical Statistics.CrossRefGoogle Scholar
15.Tanny, S. (1973). A probabilistic interpretation of Eulerian numbers. Duke Mathematical Journal 40: 717–722.Google Scholar