Abstract:
This paper is devoted to further investigation of the property of a number of vertices of convex hulls generated by independent observations of a two-dimensional random vector with regular distributions near the boundary of support when it is a unit disk. Following P. Groeneboom [4], the Binomial point process is approximated by the Poisson point process near the boundary of support and vertex processes of convex hulls are constructed. The properties of strong mixing and martingality of vertex processes are investigated. Using these properties, asymptotic expressions are obtained for the expectations and variance of the vertex processes that correspond to the results previously obtained by H. Carnal [2]. Further, using the properties of strong mixing of vertex processes, the central limit theorem for a number of vertices of a convex hull is proved.
Keywords:
convex hull, Poisson point process, Markovian jump process, martingales, Central limit theorem.
Received: 12.02.2020 Received in revised form: 06.03.2020 Accepted: 03.04.2020
Bibliographic databases:
Document Type:
Article
UDC:519.21
Language: English
Citation:
Isakjan M. Khamdamov, “On limit theorem for the number of vertices of the convex hulls in a unit disk”, J. Sib. Fed. Univ. Math. Phys., 13:3 (2020), 275–284
\Bibitem{Kha20}
\by Isakjan~M.~Khamdamov
\paper On limit theorem for the number of vertices of the convex hulls in a unit disk
\jour J. Sib. Fed. Univ. Math. Phys.
\yr 2020
\vol 13
\issue 3
\pages 275--284
\mathnet{http://mi.mathnet.ru/jsfu837}
\crossref{https://doi.org/10.17516/1997-1397-2020-13-3-275-284}
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Linking options:
https://www.mathnet.ru/eng/jsfu837
https://www.mathnet.ru/eng/jsfu/v13/i3/p275
This publication is cited in the following 3 articles:
Isakjan Khamdamov, Lola Sharipova, S. Yekimov, V. Tsipko, “Poisson approximation of binomial point processes”, E3S Web Conf., 587 (2024), 01021
Isakjan M. Khamdamov, Zoya S. Chay, “Joint distribution of the number of vertices and the area of convex hulls generated by a uniform distribution in a convex polygon”, Zhurn. SFU. Ser. Matem. i fiz., 14:2 (2021), 230–241
Shakir Formanov, Isakjan Khamdamov, INTERNATIONAL UZBEKISTAN-MALAYSIA CONFERENCE ON “COMPUTATIONAL MODELS AND TECHNOLOGIES (CMT2020)”: CMT2020, 2365, INTERNATIONAL UZBEKISTAN-MALAYSIA CONFERENCE ON “COMPUTATIONAL MODELS AND TECHNOLOGIES (CMT2020)”: CMT2020, 2021, 060012
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