18 research outputs found
Negative correlation and log-concavity
We give counterexamples and a few positive results related to several
conjectures of R. Pemantle and D. Wagner concerning negative correlation and
log-concavity properties for probability measures and relations between them.
Most of the negative results have also been obtained, independently but
somewhat earlier, by Borcea et al. We also give short proofs of a pair of
results due to Pemantle and Borcea et al.; prove that "almost exchangeable"
measures satisfy the "Feder-Mihail" property, thus providing a "non-obvious"
example of a class of measures for which this important property can be shown
to hold; and mention some further questions.Comment: 21 pages; only minor changes since previous version; accepted for
publication in Random Structures and Algorithm
Correlation bounds for fields and matroids
Let be a finite connected graph, and let be a spanning tree of
chosen uniformly at random. The work of Kirchhoff on electrical networks can be
used to show that the events and are negatively
correlated for any distinct edges and . What can be said for such
events when the underlying matroid is not necessarily graphic? We use Hodge
theory for matroids to bound the correlation between the events ,
where is a randomly chosen basis of a matroid. As an application, we prove
Mason's conjecture that the number of -element independent sets of a matroid
forms an ultra-log-concave sequence in .Comment: 16 pages. Supersedes arXiv:1804.0307
A BK inequality for randomly drawn subsets of fixed size
The BK inequality (\cite{BK85}) says that,for product measures on
, the probability that two increasing events and `occur
disjointly' is at most the product of the two individual probabilities. The
conjecture in \cite{BK85} that this holds for {\em all} events was proved by
Reimer (cite{R00}).
Several other problems in this area remained open. For instance, although it
is easy to see that non-product measures cannot satisfy the above inequality
for {\em all} events,there are several such measures which, intuitively, should
satisfy the inequality for all{\em increasing} events. One of the most natural
candidates is the measure assigning equal probabilities to all configurations
with exactly 1's (and probability 0 to all other configurations).
The main contribution of this paper is a proof for these measures. We also
point out how our result extends to weighted versions of these measures, and to
products of such measures.Comment: Revised version for PTRF. Equation (13) corrected. Several, mainly
stylistic, changes; more compac
Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures
Sufficient conditions are developed, under which the compound Poisson
distribution has maximal entropy within a natural class of probability measures
on the nonnegative integers. Recently, one of the authors [O. Johnson, {\em
Stoch. Proc. Appl.}, 2007] used a semigroup approach to show that the Poisson
has maximal entropy among all ultra-log-concave distributions with fixed mean.
We show via a non-trivial extension of this semigroup approach that the natural
analog of the Poisson maximum entropy property remains valid if the compound
Poisson distributions under consideration are log-concave, but that it fails in
general. A parallel maximum entropy result is established for the family of
compound binomial measures. Sufficient conditions for compound distributions to
be log-concave are discussed and applications to combinatorics are examined;
new bounds are derived on the entropy of the cardinality of a random
independent set in a claw-free graph, and a connection is drawn to Mason's
conjecture for matroids. The present results are primarily motivated by the
desire to provide an information-theoretic foundation for compound Poisson
approximation and associated limit theorems, analogous to the corresponding
developments for the central limit theorem and for Poisson approximation. Our
results also demonstrate new links between some probabilistic methods and the
combinatorial notions of log-concavity and ultra-log-concavity, and they add to
the growing body of work exploring the applications of maximum entropy
characterizations to problems in discrete mathematics.Comment: 30 pages. This submission supersedes arXiv:0805.4112v1. Changes in
v2: Updated references, typos correcte