Related papers: Convex geometry of max-stable distributions
We discuss isoperimetric inequalities for convex sets. These include the classical isoperimetric inequality and that of Brunn-Minkowski, Blaschke-Santalo, Busemann-Petty and their various extensions. We show that many such inequalities…
The functional characterization of a measure, an essential but delicate aspect of Stein's method, is shown to be accessible for stable probability distributions on convex cones. This notion encompasses the usual stable distributions…
We consider vectors of random variables, obtained by restricting the length of the nodal set of Berry's random wave model to a finite collection of (possibly overlapping) smooth compact subsets of $\mathbb{R}^2$. Our main result shows that,…
Max-stable processes have proved to be useful for the statistical modelling of spatial extremes. Several representations of max-stable random fields have been proposed in the literature. For statistical inference it is often assumed that…
For a sample of absolutely bounded i.i.d. random variables with a continuous density the cumulative distribution function of the sample variance is represented by a univariate integral over a Fourier series. If the density is a polynomial…
The purpose of this note is to present several aspects of concentration phenomena in high dimensional geometry. At the heart of the study is a geometric analysis point of view coming from the theory of high dimensional convex bodies. The…
We study random polytopes of the form $[X_1,\ldots,X_n]$ defined as convex hulls of independent and identically distributed random points $X_1,\ldots,X_n$ in $\mathbb{R}^d$ with one of the following densities: $$ f_{d,\beta} (x) =…
The random polytope $K_n$, defined as the convex hull of $n$ points chosen uniformly at random on the boundary of a smooth convex body, is considered. Proofs for lower and upper variance bounds, strong laws of large numbers and central…
Let K be a convex body in $R^d$. A random polytope is the convex hull $[x_1,...,x_n]$ of finitely many points chosen at random in K. $\Bbb E(K,n)$ is the expectation of the volume of a random polytope of n randomly chosen points. I.…
This paper consists of two halves. In the first half of the paper, we consider real-valued functions $f$ whose domain is the vertex set of a graph $G$ and that are Lipschitz with respect to the graph distance. By placing a uniform…
We establish the essentially optimal form of Donaldson's geodesic stability conjecture regarding existence of constant scalar curvature K\"ahler metrics. We carry this out by exploring in detail the metric geometry of Mabuchi geodesic rays,…
The notion of ball convexity, considered in finite dimensional real Banach spaces, is a natural and useful extension of usual convexity; one replaces intersections of half-spaces by suitable intersections of balls. A subset $S$ of a normed…
In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems that have a singular measure. Using the block maxima approach described in Faranda et al. [2011] we show that,…
It is well known that the ratio of two independent standard Gaussian random variables follows a Cauchy distribution. Any convex combination of independent standard Cauchy random variables also follows a Cauchy distribution. In a recent…
Stochastic approximation is a framework unifying many random iterative algorithms occurring in a diverse range of applications. The stability of the process is often difficult to verify in practical applications and the process may even be…
The paper considers distributed stochastic optimization over randomly switching networks, where agents collaboratively minimize the average of all agents' local expectation-valued convex cost functions. Due to the stochasticity in gradient…
We present a new random walk for uniformly sampling high-dimensional convex bodies. It achieves state-of-the-art runtime complexity with stronger guarantees on the output than previously known, namely in R\'enyi divergence (which implies…
We establish the following max-plus analogue of Minkowski's theorem. Any point of a compact max-plus convex subset of $(R\cup\{-\infty\})^n$ can be written as the max-plus convex combination of at most $n+1$ of the extreme points of this…
We consider an independently identically distributed random dynamical system generated by finitely many, non-uniformly expanding Markov interval maps with a finite number of branches. Assuming a topologically mixing condition and the…
The polygonal distributions are a class of distributions that can be defined via the mixture of triangular distributions over the unit interval. The class includes the uniform and trapezoidal distributions, and is an alternative to the beta…