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Zapiski Nauchnykh Seminarov POMI, 2012, Volume 408, Pages 154–174
(Mi znsl5498)
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This article is cited in 2 scientific papers (total in 2 papers)
Convex hulls of regularly varying processes
Yu. Davydova, C. Dombryb a Université des sciences et technologies de Lille, Laboratoire Paul Painlevé, UMR CNRS 8524, U.F.R. de Mathematiques, Villeneuve d'Ascq Cedex, France
b Université de Poitiers, Laboratoire LMA, UMR CNRS 7348, Futuroscope-Chasseneuil cedex, France
Abstract:
We consider the asymptotic behaviour of the compact convex subset $\widetilde W_n$ of $\mathbb R^d$ defined as the closed convex hull of the ranges of independent and identically distributed (i.i.d.) random processes $(X_i)_{1\leq i\leq n}$. Under a condition of regular variations on the law of $X_i$'s, we prove the weak convergence of the rescaled convex hulls $\widetilde W_n$ as $n\to\infty$ and analyse the structure and properties of the limit shape. We illustrate our results on several examples of regularly varying processes and show that, in contrast with Gaussian setting, in many cases the limit shape is a random polytope of $\mathbb R^d$.
Key words and phrases:
convex hull, regular variations, limit theorem, stability property.
Received: 15.10.2012
Citation:
Yu. Davydov, C. Dombry, “Convex hulls of regularly varying processes”, Probability and statistics. Part 18, Zap. Nauchn. Sem. POMI, 408, POMI, St. Petersburg, 2012, 154–174; J. Math. Sci. (N. Y.), 199:2 (2014), 150–161
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https://www.mathnet.ru/eng/znsl5498 https://www.mathnet.ru/eng/znsl/v408/p154
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Abstract page: | 132 | Full-text PDF : | 27 | References: | 30 |
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