arXiv subjects

Daniel Fresen

Publications and source records attributed to Daniel Fresen.

5 recordsLinked to original sources

arXiv2019

Exponential and Gaussian behavior in the tails of multivariate functions

We observe that approximate copies of the function $Λ_{n}:\mathbb{R}^{n}\rightarrow (0,\infty )$ defined by \begin{equation*} Λ_{n}(x)=\exp \left( -x_{1}-π\sum_{i=2}^{n}x_{i}^{2}\right) \end{equation*} appear in the tails of a large class of functions, with properties related to coordinate independence, convexity, homotheticity, and homogeneity. The function $Λ_{n}$ is an entropy maximizer (on a half-space) that is uniquely determined by a homogeneity condition together with rotational invariance about the $x_{1}$ direction and its behavior near the origin. These results are connected to the limiting Poisson point processes found near the edges of large random samples, as well as the conditioning of random vectors on certain rare events, and can be thought of as variations of Laplace's method for estimating integrals.

math.PR↗

arXiv2013

Explicit Euclidean Embeddings in Permutation Invariant Normed Spaces

Let $(X,\left\Vert \cdot \right\Vert )$ be a real normed space of dimension $N\in \mathbb{N}$ with a basis $(e_{i})_{1}^{N}$ such that the norm is invariant under coordinate permutations. Assume for simplicity that the basis constant is at most $2$. Consider any $n\in \mathbb{N}$ and $0<\varepsilon <1/4$ such that $n\leq c(\log \varepsilon ^{-1})^{-1}\log N$. We provide an explicit construction of a matrix that generates a $(1+\varepsilon )$ embedding of $\ell _{2}^{n}$ into $X$.

math.FA↗

arXiv2013

A multivariate Gnedenko law of large numbers

We show that the convex hull of a large i.i.d. sample from an absolutely continuous log-concave distribution approximates a predetermined convex body in the logarithmic Hausdorff distance and in the Banach-Mazur distance. For log-concave distributions that decay super-exponentially, we also have approximation in the Hausdorff distance. These results are multivariate versions of the Gnedenko law of large numbers, which guarantees concentration of the maximum and minimum in the one-dimensional case. We provide quantitative bounds in terms of the number of points and the dimension of the ambient space.

math.PR↗

arXiv2011

Simultaneous concentration of order statistics

Let $μ$ be a probability measure on $\mathbb{R}$ with cumulative distribution function $F$, $(x_{i})_{1}^{n}$ a large i.i.d. sample from $μ$, and $F_{n}$ the associated empirical distribution function. The Glivenko-Cantelli theorem states that with probability 1, $F_{n}$ converges uniformly to $F$. In so doing it describes the macroscopic structure of $\{x_{i}\}_{1}^{n}$, however it is insensitive to the position of individual points. Indeed any subset of $o(n)$ points can be perturbed at will without disturbing the convergence. We provide several refinements of the Glivenko-Cantelli theorem which are sensitive not only to the global structure of the sample but also to individual points. Our main result provides conditions that guarantee simultaneous concentration of all order statistics. The example of main interest is the normal distribution.

math.PR↗

arXiv2011

Comments on the floating body and the hyperplane conjecture

We provide a reformulation of the hyperplane conjecture (the slicing problem) in terms of the floating body and give upper and lower bounds on the logarithmic Hausdorff distance between an arbitrary convex body $K\subset \mathbb{R}^{d}$\ and the convex floating body $K_δ$ inside $K$.

math.FA↗