Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T19:57:40.119Z Has data issue: false hasContentIssue false
  • English
  • Français

Positive Dependence of Exchangeable Sequences

Published online by Cambridge University Press:  20 November 2018

R. M. Burton  and
A. R. Dabrowski
R. M. Burton
Affiliation:
Department of Mathematics, Oregon State University, Corvallis, Oregon 97331 U.S.A.
A. R. Dabrowski
Affiliation:
Department of Mathematics, University of Ottawa, Ottawa, Ontario, K1N6N5
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Infinite sequences of exchangeable binary random variables have strong positive dependence properties; in particular, we show they are strong FKG. If the infinite exchangeable sequence is allowed to have multiple values this is no longer true. Positive dependence conditions such as association still have natural application in this context. We establish necessary and sufficient conditions for an infinite exchangeable sequence to be associated. This result shows that exchangeable Polyà urn processes are associated. We also establish necessary and sufficient conditions for finite exchangeable sequences to be weakly associated. The match set distribution of a random permutation has recently been shown to be associated by an extensive analysis of cases. Our result easily yields the weak association of such distributions.

Keywords

60E1560K99associationexchangeableFKGmatch set distributionsPolyă urn models
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 4 , 01 August 1992 , pp. 774 - 783
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Barlow, R.E. and Proschan, F., Statistical theory of reliability and life testing probability models, Holt, Rinehart and Wilson, New York, 1975.Google Scholar
2. van, J. den Berg and Burton, R.M., FKG and equivalent conditions for binary random variables, 1989. preprint.Google Scholar
3. Blum, J.R., Chernoff, H., Rosenblatt, M. and Teicher, H., Central limit theorem for interchangeable random variables, Canad. J. Math. 10 (1958), 222–229.Google Scholar
4. Burton, R.M., Dabrowski, A.R. and Dehling, H., An invariance principle for weakly associated random vectors, Stoch. Proc. Applic. 23 (1986), 301–306.Google Scholar
5. deFinetti, B., La prévision, ses lois logiques, ses sources subjectives, Annales de l'Institut Henri Poincaré 7 (1937), 1–68.Google Scholar
6. Fishburn, P.C., Doyle, P.G. and Shepp, L.A., The match set of a randompermutation has the FKG property, Annals of Probability 16 (1988), 1194–1214.Google Scholar
7. Joag-Dev, K., Shepp, L.A. and Vitali, R.A., Remarks and open problems in the area of the FKG inequality. In: Inequalities in Statistics and Probability, (éd. Tong, Y.L.), IMS Lecture Notes, Monograph Series 5 (1984), 121–126.Google Scholar
8. Hill, B.M., Lane, D. and Sudderth, W., Exchangeable urn processes, Annals of Probability 15 (1987), 1586.mdash; 1592.Google Scholar
9. B, J.H.. Kemperman, On the FKG inequality for measures on a partially ordered space, Indag. Math. 39 (1977), 313–331.Google Scholar
10. Newman, C.M., Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Inequalities in Statistics and Probability, (éd. Tong, Y.L.), IMS Lecture Notes, Monograph Series 5 (1984), 127–140.Google Scholar
11. Tong, Y.L., Inequalities in Statistics and Probability, IMS Lecture Notes, Monograph 5 (1984).Google Scholar