79 research outputs found
General moments of the inverse real Wishart distribution and orthogonal Weingarten functions
Let be a random positive definite symmetric matrix distributed according
to a real Wishart distribution and let be its inverse
matrix. We compute general moments explicitly. To do so, we employ the orthogonal Weingarten
function, which was recently introduced in the study for Haar-distributed
orthogonal matrices. As applications, we give formulas for moments of traces of
a Wishart matrix and its inverse.Comment: 29 pages. The last version differs from the published version, but it
includes Appendi
One-sided Cauchy-Stieltjes Kernel Families
This paper continues the study of a kernel family which uses the
Cauchy-Stieltjes kernel in place of the celebrated exponential kernel of the
exponential families theory. We extend the theory to cover generating measures
with support that is unbounded on one side. We illustrate the need for such an
extension by showing that cubic pseudo-variance functions correspond to
free-infinitely divisible laws without the first moment. We also determine the
domain of means, advancing the understanding of Cauchy-Stieltjes kernel
families also for compactly supported generating measures
Combinatorial Markov chains on linear extensions
We consider generalizations of Schuetzenberger's promotion operator on the
set L of linear extensions of a finite poset of size n. This gives rise to a
strongly connected graph on L. By assigning weights to the edges of the graph
in two different ways, we study two Markov chains, both of which are
irreducible. The stationary state of one gives rise to the uniform
distribution, whereas the weights of the stationary state of the other has a
nice product formula. This generalizes results by Hendricks on the Tsetlin
library, which corresponds to the case when the poset is the anti-chain and
hence L=S_n is the full symmetric group. We also provide explicit eigenvalues
of the transition matrix in general when the poset is a rooted forest. This is
shown by proving that the associated monoid is R-trivial and then using
Steinberg's extension of Brown's theory for Markov chains on left regular bands
to R-trivial monoids.Comment: 35 pages, more examples of promotion, rephrased the main theorems in
terms of discrete time Markov chain
Nonparametric Information Geometry
The differential-geometric structure of the set of positive densities on a
given measure space has raised the interest of many mathematicians after the
discovery by C.R. Rao of the geometric meaning of the Fisher information. Most
of the research is focused on parametric statistical models. In series of
papers by author and coworkers a particular version of the nonparametric case
has been discussed. It consists of a minimalistic structure modeled according
the theory of exponential families: given a reference density other densities
are represented by the centered log likelihood which is an element of an Orlicz
space. This mappings give a system of charts of a Banach manifold. It has been
observed that, while the construction is natural, the practical applicability
is limited by the technical difficulty to deal with such a class of Banach
spaces. It has been suggested recently to replace the exponential function with
other functions with similar behavior but polynomial growth at infinity in
order to obtain more tractable Banach spaces, e.g. Hilbert spaces. We give
first a review of our theory with special emphasis on the specific issues of
the infinite dimensional setting. In a second part we discuss two specific
topics, differential equations and the metric connection. The position of this
line of research with respect to other approaches is briefly discussed.Comment: Submitted for publication in the Proceedings od GSI2013 Aug 28-30
2013 Pari
Random Convex Hulls and Extreme Value Statistics
In this paper we study the statistical properties of convex hulls of
random points in a plane chosen according to a given distribution. The points
may be chosen independently or they may be correlated. After a non-exhaustive
survey of the somewhat sporadic literature and diverse methods used in the
random convex hull problem, we present a unifying approach, based on the notion
of support function of a closed curve and the associated Cauchy's formulae,
that allows us to compute exactly the mean perimeter and the mean area enclosed
by the convex polygon both in case of independent as well as correlated points.
Our method demonstrates a beautiful link between the random convex hull problem
and the subject of extreme value statistics. As an example of correlated
points, we study here in detail the case when the points represent the vertices
of independent random walks. In the continuum time limit this reduces to
independent planar Brownian trajectories for which we compute exactly, for
all , the mean perimeter and the mean area of their global convex hull. Our
results have relevant applications in ecology in estimating the home range of a
herd of animals. Some of these results were announced recently in a short
communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special
issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting
Aortic valvuloplasty of calcific aortic stenosis with monofoil and trefoil balloon catheters: practical considerations
In order to evaluate the relation between balloon design (monofoil, trefoil) and valvular configuration, experimental aortic valvuloplasty was performed in four post-mortem hearts with calcific aortic sten
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