Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-14T04:27:17.091Z Has data issue: false hasContentIssue false
  • English
  • Français

Dependent Random Variables with Independent Subsets - II

Published online by Cambridge University Press:  20 November 2018

Y. H. Wang
Y. H. Wang*
Affiliation:
Concordia University, Montreal, Québec, H3G 1M8
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.

In this paper, we consolidate into one two separate problems - dependent random variables with independent subsets and construction of a joint distribution with given marginals. Let N = {1,2,3,...} and X = {Xn; n ∊ N} be a sequence of random variables with nondegenerate one-dimensional marginal distributions {Fn; n ∊ N}. An example is constructed to show that there exists a sequence of random variables Y = {Yn; n ∊ N} such that the components of a subset of Y are independent if and only if its size is ≦ k, where k ≧ 2 is a prefixed integer. Furthermore, the one-dimensional marginal distributions of Y are those of X.

Keywords

60E05Random variablespairwise independenceindependencejoint distributionmarginal distributions

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 33 , Issue 1 , 01 March 1990, pp. 24 - 28
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Fréchet, M., Sur les tableaux de corrélation dont les marges sont données, Ann. Univ. Lyon, 14 A (1951) 53–77.Google Scholar
2. Geisser, S. and Mantel, N., Pairwise independence of jointly dependent variables, Ann. Math. Statist. 33 (1962) 290.Google Scholar
3. Gumbel, E. J., Distributions à plusieurs variables dont les marges sont données, Comptes Rendus de l'Académie Sa, Paris, 246 (1958) 2717-2720.Google Scholar
4. Hoeffding, W., Masstabinvariante Korrelationstheorie, Schriften des Mathematischen Instituts und des Instituts fur Angewandte Mathematik des Universitat Berlin, 5 (1940) 199–233.Google Scholar
5. Joffe, A., On a sequence of almost deterministic pairwise independent random variables, Proc. Amer. Math. Soc. 29 (1971) 381–382.Google Scholar
6. Joffe, A., On a set of almost deterministic k-independent random variables, Ann. Probab. 2 (1974) 161–162.Google Scholar
7. Morgenstern, D., Eingache Beispiele zweidimensionaler Verteilungen, Mitteilungsbl. Math. Statist. 8 (1956) 234–235.Google Scholar
8. Stoyanov, J. M., Counter Examples in Probability, John Wiley & Sons, England, 1987.Google Scholar
9. Wang, Y. H., Dependent random variables with independent subsets, Amer. Math. Monthly, 86 (1979) 290–292.Google Scholar
10. Whitt, W., Bivariate distribution with given marginals, Ann. Statist. 4 (1976) 1280-1289.Google Scholar