A multivariate Gnedenko law of large numbers

• Autores: Daniel Fresen
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 5, 2013, págs. 3051-3080
• Idioma: inglés
• DOI: 10.1214/12-AOP804
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• Resumen

  • 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.