Electronic Communications in Probability

Electronic Communications in Probability > Vol. 8 (2003) > Paper 11

Positive correlation for increasing events with disjoint dependencies does not imply positive correlation for all increasing events

Nicholas Weininger, Rutgers University

Abstract
Abstract. A probability measure mu on the lattice 2^[n] is said to be positively associated if any two increasing functions on the lattice are positively correlated with respect to mu. Pemantle asked whether, in order to establish positive association for a given mu, it might be sufficient to show positive correlation only for pairs of functions which depend on disjoint subsets of the ground set [n]. We answer Pemantle's question in the negative, by exhibiting a measure which gives positive correlation for pairs satisfying Pemantle's condition but not for general pairs of increasing functions.

Full text: PDF

Pages: 99-101

Published on: July 18, 2003

Bibliography

Doyle, P., Fishburn, P., and Shepp, L. (1988). "The Match Set of a Random Permutation has the FKG Property." Annals of Probability 16, 1194-1214. MR 89i:60035
Fortuin, C., Kasteleyn, P., and Ginibre, J. (1971). "Correlation inequalities for some partially ordered sets." Comm. Math. Phys. 22, 89-103. MR 46 #8607
Pemantle, R. (2000). "Toward a theory of negative dependence." J. Math. Phys. 41, 1371-1390. MR 2001g:62039

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Electronic Communications in Probability. ISSN: 1083-589X