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Pré-Publication, Document De Travail Année : 2014

Variance asymptotics and scaling limits for Gaussian Polytopes

Pierre Calka
J. E. Yukich
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Résumé

Let $K_n$ be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on $\R^d$. We establish variance asymptotics as $n \to \infty$ for the re-scaled intrinsic volumes and $k$-face functionals of $K_n$, $k \in \{0,1,...,d-1\}$, resolving an open problem. Variance asymptotics are given in terms of functionals of germ-grain models having parabolic grains with apices at a Poisson point process on $\R^{d-1} \times \R$ with intensity $e^h dh dv$. The scaling limit of the boundary of $K_n$ as $n \to \infty$ converges to a festoon of parabolic surfaces, coinciding with that featuring in the geometric construction of the zero viscosity solution to Burgers' equation with random input.
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Dates et versions

hal-00955664 , version 1 (05-03-2014)
hal-00955664 , version 2 (27-09-2014)

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Pierre Calka, J. E. Yukich. Variance asymptotics and scaling limits for Gaussian Polytopes. 2014. ⟨hal-00955664v1⟩
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