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Asymptotic Normality for Oscillation of Permutation

Published online by Cambridge University Press:  27 July 2009

Chern-Ching Chao ,
Zhidong Bai  and
Wen-Qi Liang
Chern-Ching Chao
Affiliation:
Institute of Statistical Science, Academia SinicaTaipei 11529, Taiwan, R.O.C.
Zhidong Bai
Affiliation:
Department of Statistics, Temple University, Philadelphia, Pennsylvania 19112
Wen-Qi Liang
Affiliation:
Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan, R.O.C.
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Abstract

Suppose each permutation (πl,…,πn) of ( 1, …, n) has probability 1/n!. The oscillation of (πl; …, πn) is defined as Tn = | πk − πk+1|, where πn+1 = π1. It is proved that (Tn − ETn)/(var Tn)1/2 converges in distribution to N(0,1). The connection between the oscillation and the presortedness measure is also discussed.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 7 , Issue 2 , April 1993 , pp. 227 - 235
Copyright
Copyright © Cambridge University Press 1993

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References

1.Diaconis, P. & Graham, R. (1977). Spearman's footrule as a measure of disarray. Journal of the Royal Statistical Society B 39: 262–268.Google Scholar
2.Hoeffding, W. (1951). A combinatorial central limit theorem. Annals of Mathematical Statistics 22: 558–566.Google Scholar
3.Levcopoulos, C. & Petersson, O. (1989). Heapsort-adapted for presorted files. Proceedings of the 1989 Workshop on Algorithms and Data Structures, Lecture Notes in Computer Science 382: 499–509. Berlin: Springer.Google Scholar
4.Mannila, H. (1985). Measures of presortedness and optimal sorting algorithms. IEEE-Transactionson Computers C-34(4): 318–325.Google Scholar
5.Spearman, C. (1906). A footrule for measuring correlation. British Journal of Psychology 2: 89.Google Scholar