Geodesic
On limit theorem for the number of vertices of the convex hulls in a unit disk
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 3, pp. 275-284.

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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, Markovian jump process, Central limit theorem.
Mots-clés : Poisson point process, martingales 
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Isakjan M. Khamdamov. On limit theorem for the number of vertices of the convex hulls in a unit disk. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 3, pp. 275-284. http://geodesic.mathdoc.fr/item/JSFU_2020_13_3_a1/

[1] A.J. Cabo, P. Groeneboom, Probab. Theory Relat. Fields, 100 (1994), 31–55 | DOI | MR | Zbl

[2] H. Carnal, Z. Wahrscheinlichkeits theorie verw. Geb., 15 (1970), 168–176 | DOI | MR | Zbl

[3] B. Efron, Biometrika, 52 (1965), 331–343 | DOI | MR | Zbl

[4] P. Groeneboom, Probab. Theory Related Fields, 79 (1988), 327–368 | DOI | MR | Zbl

[5] P. Groeneboom, “Convex hulls of uniform samples from a convex polygon”, Adv. Appl. Prob. (SGSA), 44 (2012), 330–342 | DOI | MR | Zbl

[6] T. Hsing, “On the asymptotic distribution of the area outside a random convex hull in a disk”, The Annals of Applied Probability, 4:2 (1994), 478–493 | DOI | MR | Zbl

[7] I. Hueter, “The convex hull of a normal sample”, Adv. Appl. Prob., 26 (1994), 855–875 | DOI | MR | Zbl

[8] I.A. Ibragimov, Yu.U. Linnik, Independent and stationary sequences of random variables, Izdatelstvo Nauka, M., 1965 (in Russian) | MR

[9] I.M. Khamdamov, T. Kh.Adirov, “Martingale properties of vertex functionals generated by Poisson point processes”, Reports of the Academy of Sciences of Uzbekistan, 2015, no. 1, 9–11 (in Russian)

[10] I.M. Khamdamov, T. Kh.Adirov, “One of the properties of the convex hull generated by a Poisson point process”, Uzbek Mathematical Journal, 2019, no. 3, 60–63 | DOI | MR

[11] A.V. Nagaev, “Some properties of convex hulls generated by homogeneous Poisson point processes in an unbounded convex domain”, Ann. Inst. Statist. Math., 47:1 (1995), 21–29 | DOI | MR | Zbl

[12] A.V. Nagaev, I.M. Khamdamov, Limit theorems for functionals of random convex hulls, Preprint of Institute of Mathematics, Academy of Sciences of Uzbekistan, Tashkent, 1991 (in Russian)

[13] A.V. Nagaev, I.M. Khamdamov, “On the Role of Extreme Summands in Sums of Independent Random Variables”, Theory of Probability and Its Applications, 47:3 (2003), 533–541 | DOI | MR

[14] H. Raynaud, “Sur l'enveloppe convexe des nuages de points aleatoires dans”, J. Appl. Prob., 7 (1970), 35–48 | MR | Zbl

[15] A. Reny, R. Sulanke, “Uber diekovexe Hulle von zufalling gewahlten Punkten”, Z. Wahrscheinlichkeits theorie verw. Geb., 2 (1963), 75–84 | DOI | MR | Zbl