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Berry-Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing

Part of: Combinatorial probability Limit theorems

Published online by Cambridge University Press:  14 July 2016

Larry Goldstein
Larry Goldstein*
Affiliation:
University of Southern California
*
∗Postal address: Department of Mathematics, University of Southern California, 3620 Vermont Avenue, KAP 108, Los Angeles, CA 90089-2532, USA. Email address: larry@math.usc.edu
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Abstract

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Berry-Esseen-type bounds to the normal, based on zero- and size-bias couplings, are derived using Stein's method. The zero biasing bounds are illustrated in an application to combinatorial central limit theorems in which the random permutation has either the uniform distribution or one that is constant over permutations with the same cycle type, with no fixed points. The size biasing bounds are applied to the occurrences of fixed, relatively ordered subsequences (such as rising sequences) in a random permutation, and to the occurrences of patterns, extreme values, and subgraphs in finite graphs.

Keywords

Smoothing inequalityStein's methodpermutationgraph

MSC classification

Primary: 60F05: Central limit and other weak theorems 60C05: Combinatorial probability
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 3 , September 2005 , pp. 661 - 683
Copyright
© Applied Probability Trust 2005 

References

Baldi, P. and Rinott, Y. (1989). Asymptotic normality of some graph-related statistics. J. Appl. Prob. 26, 171–175.Google Scholar
Baldi, P., Rinott, Y. and Stein, C. (1989). A normal approximation for the number of local maxima of a random function on a graph. In Probability, Statistics, and Mathematics, Academic Press, Boston, MA, pp. 59–81.Google Scholar
Barbour, A. D., Janson, S., Karoński, M. and Ruciński, A. (1990). Small cliques in random graphs. Random Structures Algorithms 1, 403–434.Google Scholar
Bayer, D. and Diaconis, P. (1992). Trailing the dovetail shuffle to its lair. Ann. Appl. Prob. 2, 294–313.Google Scholar
Bhattacharya, R. N. and Ranga Rao, R. (1986). Normal Approximation and Asymptotic Expansions. Krieger, Melbourne, FL.Google Scholar
Biggs, N. (1993). Algebraic Graph Theory. Cambridge University Press.Google Scholar
Bolthausen, E. (1984). An estimate of the remainder in a combinatorial central limit theorem. Z. Wahrscheinlichkeitsth. 66, 379–386.Google Scholar
Bolthausen, E. and Götze, F. (1993). The rate of convergence for multivariate sampling statistics. Ann. Statist. 21, 1692–1710.Google Scholar
Brouwer, A. E., Cohen, A. M. and Neumaier, A. (1989). Distance-Regular Graphs. Springer, Berlin.Google Scholar
Chen, L. H. Y. and Shao, Q.-M. (2004). Normal approximation under local dependence. Ann. Prob. 32, 1985–1985.Google Scholar
Darling, R. W. R. and Waterman, M. S. (1986). Extreme value distribution for the largest cube in a random lattice. SIAM J. Appl. Math. 46, 118–132.Google Scholar
Darling, R. W. R. and Waterman, M. S. (1985). Matching rectangles in d dimensions: algorithms and laws of large numbers. Adv. Math. 55, 1–12.Google Scholar
Glaz, J., Naus, J. and Wallenstein, S. (2001). Scan Statistics. Springer, New York.Google Scholar
Goldstein, L. (2004). Normal approximation for hierarchical sequences. Ann. Appl. Prob. 14, 1950–1950.Google Scholar
Goldstein, L. and Reinert, G. (1997). Stein's method and the zero bias transformation with application to simple random sampling. Ann. Appl. Prob. 7, 935–952.Google Scholar
Goldstein, L. and Reinert, G. (2005). Distributional transformations, orthogonal polynomials, and Stein characterizations. J. Theoret. Prob. 18, 185–208.Google Scholar
Goldstein, L. and Rinott, Y. (1996). On multivariate normal approximations by Stein's method and size bias couplings. J. Appl. Prob. 33, 1–17.Google Scholar
Goldstein, L. and Rinott, Y. (2004). A permutation test for matching and its asymptotic distribution. Metron 61, 375–388.Google Scholar
Götze, F. (1991). On the rate of convergence in the multivariate CLT. Ann. Prob. 19, 724–739.Google Scholar
Ho, S. T. and Chen, L. H. Y. (1978). An L_{p} bound for the remainder in a combinatorial central limit theorem. Ann. Prob. 6, 231–249.Google Scholar
Huang, H. (2002). Error bounds on multivariate normal approximations for word count statistics. Adv. Appl. Prob. 34, 559–586.CrossRefGoogle Scholar
Kolchin, V. F. and Chistyakov, V. P. (1973). On a combinatorial limit theorem. Theory Prob. Appl. 18, 728–739.Google Scholar
Naus, J. I. (1982). Approximations for distributions of scan statistics. J. Amer. Statist. Assoc. 77, 177–183.CrossRefGoogle Scholar
Rinott, Y. and Rotar, V. (1996). A multivariate CLT for local dependence with n{-1/2}log n rate and applications to multivariate graph related statistics. J. Multivariate Anal. 56, 333–350.Google Scholar
Rinott, Y. and Rotar, V. (1997). On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted U-statistics. Ann. Appl. Prob. 7, 1080–1105.Google Scholar
Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Statist. Prob. (Berkeley, CA, 1970/1971), Vol. 2, University of California Press, pp. 583–602.Google Scholar
Stein, C. (1972). Estimation of the mean of a multivariate normal distribution. Ann. Prob. 9, 1135–1151.Google Scholar
Stein, C. (1986). Approximate Computation of Expectations (Inst. Math. Statist. Lecture Notes Monogr. Ser. 7). IMS, Hayward, CA.Google Scholar
Von Bahr, B. (1976). Remainder term estimate in a combinatorial limit theorem. Z. Wahrscheinlichkeitsth. 35, 131–139.Google Scholar