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  1. arXiv:1108.5011 (Published 2011-08-25, updated 2015-09-13)

    Gaussian behavior on hyperplanes

    Daniel Fresen

    We observe that the function $\Lambda _{n}:\mathbb{R}^{n}\rightarrow (0,\infty )$ defined by \begin{equation*} \Lambda _{n}(x)=\exp \left( x_{1}-\pi \sum_{i=2}^{n}x_{i}^{2}\right) \end{equation*} appears in the tails of a large class of functions, with properties involving log-concavity, independence, and homogeneity, as well as the gamma function.

  2. arXiv:1102.2570 (Published 2011-02-13, updated 2011-02-20)

    Comments on the floating body and the hyperplane conjecture

    Daniel Fresen

    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 $K_{\delta}$ inside $K$.

  3. arXiv:1102.1128 (Published 2011-02-06, updated 2011-02-20)

    Simultaneous concentration of order statistics

    Daniel Fresen

    Let $\mu$ be a probability measure on $\mathbb{R}$ with cumulative distribution function $F$, $(x_{i})_{1}^{n}$ a large i.i.d. sample from $\mu$, 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.

  4. arXiv:1101.4887 (Published 2011-01-25, updated 2013-10-21)

    A multivariate Gnedenko law of large numbers

    Daniel Fresen
    Comments: Published in at http://dx.doi.org/10.1214/12-AOP804 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
    Journal: Annals of Probability 2013, Vol. 41, No. 5, 3051-3080
    Categories: math.PR, math.FA

    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.