Towards a theory of negative dependence

The FKG theorem says that the positive lattice condition, an easily checkable hypothesis which holds for many natural families of events, implies positive association, a very useful property. Thus there is a natural and useful theory of positively dependent events. There is, as yet, no corresponding theory of negatively dependent events. There is, however, a need for such a theory. This paper, unfortunately, contains no substantial theorems. Its purpose is to present examples that motivate a need for such a theory, give plausibility arguments for the existence of such a theory, outline a few possible directions such a theory might take, and state a number of specific conjectures which pertain to the examples and to a wish list of theorems.

[1]  J. A. Fill,et al.  Stochastic monotonicity and realizable monotonicity , 2000, math/0005267.

[2]  David Reimer,et al.  Proof of the Van den Berg–Kesten Conjecture , 2000, Combinatorics, Probability and Computing.

[3]  Desh Ranjan,et al.  Balls and bins: A study in negative dependence , 1996, Random Struct. Algorithms.

[4]  Thomas M. Liggett.  Ultra Logconcave Sequences and Negative Dependence , 1997, J. Comb. Theory, Ser. A.

[5]  Olle Häggström,et al.  Random-cluster measures and uniform spanning trees , 1995 .

[6]  Tomás Feder,et al.  Balanced matroids , 1992, STOC '92.

[7]  R. Pemantle,et al.  Choosing a Spanning Tree for the Integer Lattice Uniformly , 1991, math/0404043.

[8]  Gian-Carlo Rota.  Inequalities in statistics and probability: Y. L. Tong (Ed.), Institute of Mathematical Statistics, 1984, 253 pp. , 1988 .

[9]  V. D. Berg,et al.  On a Combinatorial Conjecture Concerning Disjoint Occurrences of Events , 1987 .

[10]  Van den Berg,et al.  Inequalities with applications to percolation and reliability , 1985, Journal of Applied Probability.

[11]  C. Newman.  Asymptotic independence and limit theorems for positively and negatively dependent random variables , 1984 .

[12]  K. Joag-dev,et al.  Negative Association of Random Variables with Applications , 1983 .

[13]  Moshe Shaked,et al.  Some Concepts of Negative Dependence , 1982 .

[14]  Richard P. Stanley,et al.  Two Combinatorial Applications of the Aleksandrov-Fenchel Inequalities , 1981, J. Comb. Theory, Ser. A.

[15]  S. Karlin,et al.  Classes of orderings of measures and related correlation inequalities II. Multivariate reverse rule distributions , 1980 .

[16]  S. Karlin,et al.  Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions , 1980 .

[17]  Charles M. Newman,et al.  Normal fluctuations and the FKG inequalities , 1980 .

[18]  D. E. Daykin,et al.  Inequalities for a pair of mapsS×S→S withS a finite set , 1979 .

[19]  T. Liggett,et al.  The stochastic evolution of infinite systems of interacting particles , 1977 .

[20]  D. Welsh,et al.  Combinatorial applications of an inequality from statistical mechanics , 1975, Mathematical Proceedings of the Cambridge Philosophical Society.

[21]  C. Fortuin,et al.  Correlation inequalities on some partially ordered sets , 1971 .

[22]  C. L. Mallows.  An inequality involving multinomial probabilities , 1968 .

[23]  D. Walkup,et al.  Association of Random Variables, with Applications , 1967 .

[24]  E. Lehmann.  Some Concepts of Dependence , 1966 .

[25]  B. Efron.  Increasing Properties of Polya Frequency Function , 1965 .

[26]  T. E. Harris.  A lower bound for the critical probability in a certain percolation process , 1960, Mathematical Proceedings of the Cambridge Philosophical Society.