Variance asymptotics and scaling limits for Gaussian polytopes

• 成果类型: Article
• 署名作者: Calka, Pierre; Yukich, J. E.
• 署名单位: Universite de Rouen Normandie; Lehigh University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0592-6
• 发表日期: 2015
• 页码: 259-301
• 关键词: convex hulls; points; THEOREMS

摘要

Let K-n be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on . We establish variance asymptotics as for the re-scaled intrinsic volumes and -face functionals of , , resolving an open problem (Weil and Wieacker, Handbook of Convex Geometry, vol. B, pp. 1391-1438. North-Holland/Elsevier, Amsterdam, 1993). Variance asymptotics are given in terms of functionals of germ-grain models having parabolic grains with apices at a Poisson point process on with intensity . The scaling limit of the boundary of as 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.