Related papers: Sylvester's problem for beta-type distributions

We study the problem of binary classification from the point of view of learning convex polyhedra in Hilbert spaces, to which one can reduce any binary classification problem. The problem of learning convex polyhedra in finite-dimensional…

In the d dimensional Euclidean space, any set of n+1 independent random points, uniformly distributed in the interior of a unit ball of center O, determines almost surely a circumsphere of center C and of radius R, with n positive and less…

The simplicial depth, like other relevant multivariate statistical data depth functions, vanishes right outside the convex hull of the support of the distribution with respect to which the depth is computed. This is problematic when it is…

To complete a previous paper, the probability density functions of the center-of-gravity as positioning algorithm are derived with classical methods. These methods, as suggested by the textbook of Probability, require the preliminary…

This paper deals with the typical cell in a Poisson line tessellation in the plane whose directional distribution is concentrated on three equally spread values with possibly different weights. Such a random polygon can only be a triangle,…

We study the weak convergence of $\beta$- and $\beta'$-Delaunay tessellations in $\mathbb{R}^{d-1}$ that were introduced in part I of this paper, as $\beta\to\infty$. The limiting stationary simplicial random tessellation, which is called…

Let $X_1,\ldots, X_n$ be independent random points in the unit ball of $\mathbb R^d$ such that $X_i$ follows a beta distribution with the density proportional to $(1-\|x\|^2)^{\beta_i}1_{\{\|x\| <1\}}$. Here, $\beta_1,\ldots, \beta_n> -1$…

The stationary distribution of a sample taken from a Wright-Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree to…

This paper proposes a unified approach to enable the study of diverse distributions in the real, complex, quaternion and octonion cases, simultaneously. In particular, the central, nonsingular matricvariate and matrix multivariate Pearson…

Learning joint probability distributions on n random variables requires exponential sample size in the generic case. Here we consider the case that a temporal (or causal) order of the variables is known and that the (unknown) graph of…

We consider Gibbs distributions, which are families of probability distributions over a discrete space $\Omega$ with probability mass function of the form $\mu^\Omega_\beta(\omega) \propto e^{\beta H(\omega)}$ for $\beta$ in an interval…

We prove the central limit theorem for the volume and the $f$-vector of the Poisson random polytope $\Pi_{\eta}$ in a fixed convex polytope $P\subset\mathbb{R}^d$. Here, $\Pi_{\eta}$ is the convex hull of the intersection of a Poisson…

Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave…

Let $S_k$ be a random walk in $R^d$ such that its distribution of increments does not assign mass to hyperplanes. We study the probability $p_n$ that the convex hull $conv (S_1, \ldots , S_n)$ of the first $n$ steps of the walk does not…

For $d\in\mathbb{N}$, let $S$ be a set of points in $\mathbb{R}^d$ in general position. A set $I$ of $k$ points from $S$ is a $k$-island in $S$ if the convex hull $\mathrm{conv}(I)$ of $I$ satisfies $\mathrm{conv}(I) \cap S = I$. A…

We provide a streamlined proof and improved estimates for the weak multivariate Gnedenko law of large numbers on concentration of random polytopes within the space of convex bodies (in a fixed or a high dimensional setting), as well as a…

Consider a random set of points on the unit sphere in $\mathbb{R}^d$, which can be either uniformly sampled or a Poisson point process. Its convex hull is a random inscribed polytope, whose boundary approximates the sphere. We focus on the…

Robust Bayesian inference is the calculation of posterior probability bounds given perturbations in a probabilistic model. This paper focuses on perturbations that can be expressed locally in Bayesian networks through convex sets of…

The convex hull generated by the restriction to the unit ball of a stationary Poisson point process in the $d$-dimensional Euclidean space is considered. By establishing sharp bounds on cumulants, exponential estimates for large deviation…

Let $M$ be an arbitrary subset in $\mathbb R^n$ with a conic (or positive) hull $C$. Consider its Gaussian image $AM$, where $A$ is a $k\times n$-matrix whose entries are independent standard Gaussian random variables. We show that the…