Towards a theory of negative dependence

  title={Towards a theory of negative dependence},
  author={Robin Pemantle},
  journal={Journal of Mathematical Physics},
  • R. Pemantle
  • Published 3 March 2000
  • Mathematics
  • Journal of Mathematical Physics
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… 

Figures from this paper

Negative Dependence in Sampling
Abstract.  The strong Rayleigh property is a new and robust negative dependence property that implies negative association; in fact it implies conditional negative association closed under external
Negative dependence and the geometry of polynomials
We introduce the class of strongly Rayleigh probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers
Positive Influence and Negative Dependence
A simple proof that the distribution satisfies negative association and it is shown that under a linear match schedule it satisfies the stronger condition of conditional negative association via a non-trivial application of the Feder–Mihail theorem.
A strong log-concavity property for measures on Boolean algebras
BK-type inequalities and generalized random-cluster representations
Recently, van den Berg and Jonasson gave the first substantial extension of the BK inequality for non-product measures: they proved that, for $$k$$-out-of-$$n$$ measures, the probability that two
Negative dependence and stochastic orderings
We explore negative dependence and stochastic orderings, showing that if an integer-valued random variable $W$ satisfies a certain negative dependence assumption, then $W$ is smaller (in the convex
The cavity method for counting spanning subgraphs subject to local constraints
Using the theory of negative association for measures and the notion of random weak limits of sparse graphs, we establish the validity of the cavity method for counting spanning subgraphs subject to
Some implications of local weak convergence for sparse random graphs
In the so-called sparse regime where the numbers of edges and vertices tend to infinity in a comparable way, the asymptotic behavior of many graph invariants is expected to depend only upon local
Monotonicity, Thinning, and Discrete Versions of the
A stronger version of concavity of entropy is proved, which implies a strengthened form of Shannon's Entropy Power Inequality, which is based on the results established by Artstein, Ball, Barthe, and Naor.
Monotonicity of conditional distributions and growth models on trees
We consider a sequence of probability measures v n obtained by conditioning a random vector X = (X 1 ,...,X d ) with nonnegative integer valued components on X 1 + … + X d = n - 1 and give several


On a Combinatorial Conjecture Concerning Disjoint Occurrences of Events
Recently van den Berg and Kesten have obtained a correlation-like inequality for Bernoulli sequences. This inequality, which goes in the opposite direction of the FKG inequality, states that the
Stochastic monotonicity and realizable monotonicity
We explore and relate two notions of monotonicity, stochastic and realizable, for a system of probability measures on a common finite partially ordered set (poset) f when the measures are indexed by
Some Concepts of Negative Dependence
Abstract : The theory of positive dependence notions cannot yield useful results for some widely used distributions such as the multinomial, Dirichlet and the multivariate hypergeometric. Some
Some Concepts of Dependence
Problems involving dependent pairs of variables (X, Y) have been studied most intensively in the case of bivariate normal distributions and of 2 × 2 tables. This is due primarily to the importance of
A lower bound for the critical probability in a certain percolation process
Consider a lattice L in the Cartesian plane consisting of all points (x, y) such that either x or y is an integer. Points with integer coordinates (positive, negative, or zero) are called vertices
Combinatorial applications of an inequality from statistical mechanics
The main part of this paper shows how an inequality of statistical mechanics has several applications in combinatorial theory. The inequality (known as the FKG inequality) was derived by Fortuin,
Choosing a Spanning Tree for the Integer Lattice Uniformly
Consider the nearest neighbor graph for the integer lattice Zd in d dimensions. For a large finite piece of it, consider choosing a spanning tree for that piece uniformly among all possible subgraphs
Normal fluctuations and the FKG inequalities
In a translation invariant pure phase of a ferromagnet, finite susceptibility and the FKG inequalities together imply convergence of the block spin scaling limit to the infinite temperature Gaussian