Normalized information-based divergences

This paper is devoted to the mathematical study of some divergences based on the mutual information well-suited to categorical random vectors. These divergences are generalizations of the "entropy distance" and "information distance". Their main characteristic is that they combine a complexity term and the mutual information. We then introduce the notion of (normalized) information-based divergence, propose several examples and discuss their mathematical properties in particular in some prediction framework.Comment: 36 page

Representing three-dimensional cross fields using 4th order tensors

This paper presents a new way of describing cross fields based on fourth order tensors. We prove that the new formulation is forming a linear space in $\mathbb{R}^9$. The algebraic structure of the tensors and their projections on \mbox{SO}(3) are presented. The relationship of the new formulation with spherical harmonics is exposed. This paper is quite theoretical. Due to pages limitation, few practical aspects related to the computations of cross fields are exposed. Nevetheless, a global smoothing algorithm is briefly presented and computation of cross fields are finally depicted

A Random Difference Equation with Dufresne Variables revisited

The Dufresne laws (laws of product of independent random variables with gamma and beta distributions) occur as stationary distribution of certain Markov chains $X_n$ on $R$ defined by: \begin{equation} X_n = A_n ( X_{n-1} + B_n ) \end{equation} where $X_0 , (A_1,B_1),...,(A_n,B_n)$ are independent and the $(A_i,B_i)'$s are identically distributed. This paper generalizes an explicit example where $A$ is the product of two independent $\beta_{a,1} , \beta_{b,1}$ and $B \sim \gamma_1$ or $\gamma_2$. Keywords: beta, gamma and Dufresne distributions,Markov chains, stationary distributions, hypergeometric differential equations, Poisson process.Comment: 11 pages, 2 tables, 1 figur

Some properties of the range of super-Brownian motion

We consider a super-Brownian motion $X$. Its canonical measures can be studied through the path-valued process called the Brownian snake. We obtain the limiting behavior of the volume of the $\epsilon$-neighborhood for the range of the Brownian snake, and as a consequence we derive the analogous result for the range of super-Brownian motion and for the support of the integrated super-Brownian excursion. Then we prove the support of $X_t$ is capacity-equivalent to $[0,1]^2$ in $\R^d$, $d\geq 3$, and the range of $X$, as well as the support of the integrated super-Brownian excursion are capacity-equivalent to $[0,1]^4$ in $\R^d$, $d\geq 5$

The lineage process in Galton--Watson trees and globally centered discrete snakes

We consider branching random walks built on Galton--Watson trees with offspring distribution having a bounded support, conditioned to have $n$ nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of ``globally centered discrete snake'' that extends the usual settings in which the displacements are supposed centered. We show that under some additional moment conditions, when $n$ goes to $+\infty$, ``globally centered discrete snakes'' converge to the Brownian snake. The proof relies on a precise study of the lineage of the nodes in a Galton--Watson tree conditioned by the size, and their links with a multinomial process [the lineage of a node $u$ is the vector indexed by $(k,j)$ giving the number of ancestors of $u$ having $k$ children and for which $u$ is a descendant of the $j$th one]. Some consequences concerning Galton--Watson trees conditioned by the size are also derived.Comment: Published in at http://dx.doi.org/10.1214/07-AAP450 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org