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Hoeffding spaces and Specht modules

Published online by Cambridge University Press:  15 October 2010

Giovanni Peccati  and
Jean-Renaud Pycke
Giovanni Peccati
Affiliation:
Équipe , Université Paris Ouest-Nanterre La Dé fense, 200 avenue de la République, 92000 Nanterre, France. giovanni.peccati@gmail.com
Jean-Renaud Pycke
Affiliation:
Département de Mathématiques, Université d'Évry, Évry, France; jrpycke@maths.univ-evry.fr
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Abstract

It is proved that each Hoeffding space associated with a random permutation (or, equivalently, with extractions without replacement from a finite population) carries an irreducible representation of the symmetric group, equivalent to a two-block Specht module.

Keywords

Exchangeabilityfinite population statisticsHoeffding decompositionsirreducible representationsrandom permutationsSpecht modulessymmetric group
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. S58 - S68
Copyright
© EDP Sciences, SMAI, 2011

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References

D.J. Aldous, Exchangeability and related topics. École d'été de Probabilités de Saint-Flour XIII. LNM 1117, Springer, New York (1983).
Bloznelis, M., Orthogonal decomposition of symmetric functions defined on random permutations. Combin. Probab. Comput. 14 (2005) 249–268. CrossRef
Bloznelis, M. and Götze, F., Orthogonal decomposition of finite population statistics and its applications to distributional asymptotics. Ann. Stat. 29 (2001) 353–365.
Bloznelis, M. and Götze, F., Edgeworth, An expansion for finite population statistics. Ann. Probab. 30 (2002) 1238–1265.
P. Diaconis, Group Representations in Probability and Statistics. IMS Lecture Notes – Monograph Series 11, Hayward, California (1988).
J.J. Duistermaat and J.A.C. Kolk, Lie groups. Springer-Verlag, Berlin-Heidelberg-New York (1997).
El-Dakkak, O. and Peccati, G., Hoeffding decompositions and urn sequences. Ann. Probab. 36 (2008) 2280–2310. CrossRef
G.D. James, The representation theory of the symmetric groups. Lecture Notes in Math. 682, Springer-Verlag, Berlin-Heidelberg-New York (1978).
Peccati, G., Hoeffding-ANOVA decompositions for symmetric statistics of exchangeable observations. Ann. Probab. 32 (2004) 1796–1829. CrossRef
Peccati, G. and Pycke, J.-R., Decompositions of stochastic processes based on irreducible group representations. Theory Probab. Appl. 54 (2010) 217–245. CrossRef
B.E. Sagan, The Symmetric Group. Representations, Combinatorial Algorithms and Symmetric Functions, 2nd edition. Springer, New York (2001).
R.J. Serfling, Approximation Theorems of Mathematical Statistics. Wiley, New York (1980).
J.-P. Serre, Linear representations of finite groups, Graduate Texts Math. 42, Springer, New York (1977).
Zhao, L. and Chen, X., Normal approximation for finite-population U-statistics. Acta Math. Appl. Sinica 6 (1990) 263–272. CrossRef