4,013 research outputs found
Determinantal probability measures
Determinantal point processes have arisen in diverse settings in recent years
and have been investigated intensively. We study basic combinatorial and
probabilistic aspects in the discrete case. Our main results concern
relationships with matroids, stochastic domination, negative association,
completeness for infinite matroids, tail triviality, and a method for extension
of results from orthogonal projections to positive contractions. We also
present several new avenues for further investigation, involving Hilbert
spaces, combinatorics, homology, and group representations, among other areas.Comment: 50 pp; added reference to revision. Revised introduction and made
other small change
Distance covariance in metric spaces
We extend the theory of distance (Brownian) covariance from Euclidean spaces,
where it was introduced by Sz\'{e}kely, Rizzo and Bakirov, to general metric
spaces. We show that for testing independence, it is necessary and sufficient
that the metric space be of strong negative type. In particular, we show that
this holds for separable Hilbert spaces, which answers a question of Kosorok.
Instead of the manipulations of Fourier transforms used in the original work,
we use elementary inequalities for metric spaces and embeddings in Hilbert
spaces.Comment: Published in at http://dx.doi.org/10.1214/12-AOP803 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Hyperbolic Space Has Strong Negative Type
It is known that hyperbolic spaces have strict negative type, a condition on
the distances of any finite subset of points. We show that they have strong
negative type, a condition on every probability distribution of points (with
integrable distance to a fixed point). This implies that the function of
expected distances to points determines the probability measure uniquely. It
also implies that the distance covariance test for stochastic independence,
introduced by Sz\'ekely, Rizzo and Bakirov, is consistent against all
alternatives in hyperbolic spaces. We prove this by showing an analogue of the
Cram\'er-Wold device.Comment: 6 p
Factors of IID on Trees
Classical ergodic theory for integer-group actions uses entropy as a complete
invariant for isomorphism of IID (independent, identically distributed)
processes (a.k.a. product measures). This theory holds for amenable groups as
well. Despite recent spectacular progress of Bowen, the situation for
non-amenable groups, including free groups, is still largely mysterious. We
present some illustrative results and open questions on free groups, which are
particularly interesting in combinatorics, statistical physics, and
probability. Our results include bounds on minimum and maximum bisection for
random cubic graphs that improve on all past bounds.Comment: 18 pages, 1 figur
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