Exchangeable Hoeffding decompositions over finite sets: A combinatorial characterization and counterexamples

Journal of Multivariate Analysis

Volumn 131, Issue , 2014, Pages 51-64

  El Dakkak, Omar ^a   Peccati, Giovanni ^b   Prünster, Igor ^c  

^a UNIVERSITÉ PARIS OUEST NANTERRE LA DÉFENSE   (France)

^b UNIVERSITY OF LUXEMBOURG   (Luxembourg)

^c UNIVERSITY OF TURIN   (Italy)

Author keywords

Exchangeability; Hoeffding decomposition; P lya urn; Urn sequence; Weak independence

Indexed keywords

EID: 84903885567     PISSN: 0047259X     EISSN: 10957243     Source Type: Journal    
DOI: 10.1016/j.jmva.2014.04.012     Document Type: Article

References (25)

1. Aldous D.J. Exchangeability and related topics. LNM 1983, vol. 1117. Springer, New York.
2. Arcones M.A., Giné E. Limit theorems for U-processes. Ann. Probab. 1993, 21:1494-1542.
3. Avram F., Bertsimas D. On central limit theorems in geometric probabilities. Ann. Appl. Probab. 1993, 3:1033-1046.
4. Bentkus V., Götze F., van Zwet W.R. An Edgeworth expansion for symmetric statistics. Ann. Statist. 1997, 25:851-896.
5. Blackwell D., MacQueen J. Ferguson distribution via Pólya urn schemes. Ann. Statist. 1973, 1(2):353-355.
6. Bloznelis M. Orthogonal decomposition of symmetric functions defined on random permutations. Combin. Probab. Comput. 2005, 14:249-268.
7. Bloznelis M., Götze F. Orthogonal decomposition of finite population statistics and its applications to distributional asymptotics. Ann. Statist. 2001, 29:353-365.
8. Bloznelis M., Götze F. An Edgeworth expansion for finite population statistics. Ann. Probab. 2002, 30:1238-1265.
9. El-Dakkak O., Peccati G. Hoeffding decompositions and urn processes. Ann. Probab. 2008, 36(6):2280-2310.
10. Ferguson T.S. A Bayesian analysis of some nonparametric problems. Ann. Statist. 1973, 1:209-230.
11. Hajek J. Asymptotic normality of simple linear rank statistics under alternatives. Ann. Math. Stat. 1968, 39:325-346.
12. Hill B.H., Lane D., Sudderth W. Exchangeable urn processes. Ann. Probab. 1987, 15(4):1586-1592.
13. Hoeffding W. A class of statistics with asymptotically normal distribution. Ann. Math. Stat. 1948, 19:293-325.
14. James L.F., Lijoi A., Prünster I. Conjugacy as a distinctive feature of the Dirichlet process. Scand. J. Statist. 2006, 33:105-120.
15. Karlin S., Rinott Y. Applications of ANOVA type decompositions for comparisons of conditional variance statistics including jackknife estimates. Ann. Statist. 1982, 10(2):485-501.
16. Koroljuk V.S., Borovskich Yu.V. Theory of U-Statistics 1994, Kluwer Academic Publishers, London.
17. Lijoi A., Prünster I. Models beyond the Dirichlet process. Bayesian Nonparametrics 2010, 80-136. Cambridge University Press, Cambridge. N.L. Hjort, C.C. Holmes, P. Müller, S.G. Walker (Eds.).
18. Major P. Tail behaviour of multiple random integrals and U-statistics. Probab. Surv. 2005, 2:448-505.
19. Peccati G. Hoeffding decompositions for exchangeable sequences and chaotic representation for functionals of Dirichlet processes. C. R. Acad. Sci. Math. 2003, 336(10):845-850.
20. Peccati G. Hoeffding-ANOVA decompositions for symmetric statistics of exchangeable observations. Ann. Probab. 2004, 32(3A):1796-1829.
21. Peccati G. Multiple integral representation for functionals of Dirichlet processes. Bernoulli 2008, 14(1):91-124.
22. Pitman J. Some developments of the Blackwell-MacQueen urn scheme. Lecture Notes-Monograph Series 1996, vol. 30. Institute of Mathematical Statistics, Hayward, California.
23. Regazzini E., Lijoi A., Prünster I. Distributional results for means of random measures with independent increments. Ann. Statist. 2003, 31:560-585.
24. Vitale R.A. Covariances of symmetric statistics. J. Multivariate Anal. 1992, 41:14-26.
25. Zhao L., Chen X. Normal approximation for finite-population U-statistics. Acta Math. Appl. Sin. 1990, 6(3):263-272.