Abstract:
We consider the asymptotic behaviour of the compact convex subset ˜Wn of Rd defined as the closed convex hull of the ranges of independent and identically distributed (i.i.d.) random processes (Xi)1≤i≤n. Under a condition of regular variations on the law of Xi's, we prove the weak convergence of the rescaled convex hulls ˜Wn as n→∞ 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 Rd.
Key words and phrases:convex hull, regular variations, limit theorem, stability property.
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
\Bibitem{DavDom12}
\by Yu.~Davydov, C.~Dombry
\paper Convex hulls of regularly varying processes
\inbook Probability and statistics. Part~18
\serial Zap. Nauchn. Sem. POMI
\yr 2012
\vol 408
\pages 154--174
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl5498}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3032214}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2014
\vol 199
\issue 2
\pages 150--161
\crossref{https://doi.org/10.1007/s10958-014-1842-y}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84902244659}
Linking options:
https://www.mathnet.ru/eng/znsl5498
https://www.mathnet.ru/eng/znsl/v408/p154
This publication is cited in the following 2 articles:
Wojciech Cygan, Nikola Sandrić, Stjepan Šebek, “Convex hulls of stable random walks”, Electron. J. Probab., 27:none (2022)
Borodin A.N. Davydov Yu.A. Nevzorov V.B., “On the History of the St. Petersburg School of Probability and Statistics. III. Distributions of Functionals of Processes, Stochastic Geometry, and Extrema”, Vestn. St Petersb. Univ.-Math., 51:4 (2018), 343–359