Towards a theory of negative dependence

By Robin Pemantle. 2000

226 citations recorded by the source corpus.

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.

Subfields: Economics and Econometrics, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/3b5debd55ee36baf905e39acbdb9c904b9c68b6f

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Citations in the corpus

Determinantal probability measures (2002, background): “In the language of Pemantle (2000), this corollary says that the measure PH is condi- tionally negatively associated (CNA).”
Hyperdeterminantal relations among symmetric principal minors (2006, background): “al minors give the joint densities of the process. The matrix K is not necessarily Hermitian (or real symmetric), even though very often it is. In the theory of negatively correlated random variables [12, 14], principal minors of a real n×nmatrix give values of a function ω : 2[n] → [0,∞) on the Boo”
The Random-Cluster Model (2002, background): “The intuition is that certain positive correlations should be replaced by negative correlations; however, the theory of negative correlation is more problematic than that of positive correlation (see [114]).”
Hyperbolicity and stable polynomials in combinatorics and probability (2012, background): “This goal was popularized in the article [Pem00] which posed several challenges and stated many conjectures but contained few concrete results.”
Negative Dependence in Sampling (2012, background): “In a pioneering paper, Pemantle (2000), called out for a search for a useful theory of negative dependence.”
Log-concavity and the maximum entropy property of the Poisson distribution (2006, background): “Note that in Pemantle [17], Liggett [14], and Wang and Yeh [23], this property is referred to as ‘ultra log-concavity of order ∞’ – see Equation (7) below for the definition of ultra log-concavity of order n.”
Unimodality, log-concavity, real-rootedness and beyond (2015, background): “ Proposition 9.1, it is not hard to prove that C is strong Rayleigh, see [12]. Negative dependence is an important notion in probability theory, statistics and statistical mechanics, see the survey [75]. In [12] several strong negative dependence properties of strong Rayleigh measures were establis”
Negative dependence and the geometry of polynomials (2007, methodology): “In particular, this solves an open problem of Pemantle [65] and Liggett [55] stating that the distribution of a symmetric exclusion process at time t ≥ 0 with non-random/deterministic initial configuration is negatively associated, and shows that the same is actually true whenever the initial distri”
On the topology of random complexes built over stationary point processes. (2012, background): “unctions. In general, the literature contains fewer examples of negatively associated processes than their positively associated counterparts, a phenomenon that occurs even in simpler situations; cf. [33]. We shall give two examples of negatively associated point processes below (determinantal point”
Demystifying Orthogonal Monte Carlo and Beyond (2020, methodology): “The key tool we apply to obtain our general theoretical results is the notion of negative dependence [ND] [32, 40] that is also used in the theory of Determinantal Point Processes (DPPs) [28]: Definition 1 (Negative Dependence (ND)).”