Abstract
We prove a correlation inequality for n increasing functions on a distributive lattice, which for n = 2 reduces to a special case of the FKG inequality. The key new idea is to reformulate the inequalities for all n into a single positivity statement in the ring of formal power series. We also conjecture that our results hold in greater generality.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Ahilswede and D. E. Daykin: An inequality for the weights of two families of sets, their unions and intersections; Z. Wahrsch. Verw. Gebiete 43 (1978), 183–185.
N. Alon and J. H. Spencer: The Probabilistic Method, John Wiley & Sons, Inc., New York, 1992.
C. Fortuin, P. Kasteleyn and J. Ginibre: Correlation inequalities on some partially ordered sets, Comm. Math. Phys. 22 (1971), 89–103.
R. L. Graham: Applications of the FKG inequality and its relatives, in: Mathematical Programming: The State of the Art, Springer, Berlin, 1983, pp. 115–131.
J. Glimm and A. Jaffe: Quantum Physics: A Functional Integral Point of View, 2nd ed., Springer, New York, 1987.
S. Karlin and Y. Rinott: A generalized Cauchy-Binet formula and applications to total positivity and majorization, J. Multivar. Anal. 27 (1988), 284–299.
J. L. Lebowitz: Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems, Comm. Math. Phys. 28 (1972), 313–321.
D. Richards: Algebraic method toward higher-order probability inequalities II, Ann. of Prob. 32 (2004), 1509–1544.
Y. Rinott and M. Saks: Correlation inequalities and a conjecture for permanents, Combinatorica 13(3), (1993), 269–277.
L. A. Shepp: The XYZ conjecture and the FKG inequality, Ann. of Prob. 10 (1982), 824–827.