Geodesic
Joint distribution of the number of vertices and the area of convex hulls generated by a uniform distribution in a convex polygon
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 14 (2021) no. 2, pp. 230-241.

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A convex hull generated by a sample uniformly distributed on the plane is considered in the case when the support of a distribution is a convex polygon. A central limit theorem is proved for the joint distribution of the number of vertices and the area of a convex hull using the Poisson approximation of binomial point processes near the boundary of the support of distribution. Here we apply the results on the joint distribution of the number of vertices and the area of convex hulls generated by the Poisson distribution given in [6]. From the result obtained in the present paper, in particular, follow the results given in [3, 7], when the support is a convex polygon and the convex hull is generated by a homogeneous Poisson point process.
Keywords: convex hull, central limit theorem.
Mots-clés : convex polygon, Poisson point process, binomial point process 
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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. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 14 (2021) no. 2, pp. 230-241. http://geodesic.mathdoc.fr/item/JSFU_2021_14_2_a10/

[1] C. Buchta, “On the distribution of the number of vertices of a random polygon”, Anz. Osterreich. Akad. Wiss. Math. Natur. K1, 139 (2003), 17–19 | MR | Zbl

[2] C. Buchta, “Exact formulae for variances of functionals of convex hulls”, Advances in Applied Probability, 45:4 (2013), 917–924 | DOI | MR | Zbl

[3] A.J. Cabo, P. Groeneboom, “Limit theorems for functiohals of convex hulls”, Probab. Theory Relat. Fields, 100 (1994), 31–55 | DOI | MR | Zbl

[4] H. Carnal, Die konvexe Hulle von n rotations symmetrisch verteilten Punkten, Z. Wahrscheinlichkeits theorie verw. Geb., 1970, no. 15, 168–176 | DOI | MR | Zbl

[5] B. Efron, “The convex hull of a random set of points”, Biometrika, 52 (1965), 331–343 | DOI | MR | Zbl

[6] Sh.K. Formanov, I.M. Khamdamov, “On joint probability distribution of the number of vertices and area of the convex hulls generated by a Poisson point process”, Statistics and Probability Letters, 169 (2021), 108966, 1–7 | DOI | MR

[7] P. Groeneboom, “Limit theorems for convex hulls”, Probab. Theory Related Fields, 79 (1988), 327–368 | DOI | MR | Zbl

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

[9] 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

[10] I.M. Khamdamov, “On Limit Theorem for the Number of Vertices of the Convex Hulls in a Unit Disk”, Journal of Siberian Federal University. Mathematics and Physics, 13:3 (2020), 275–284 | DOI | MR

[11] I.M. Khamdamov, “Properties of convex hull generated by inhomogeneous Poisson point process”, Ufa Mathematical Journal, 12:3 (2020), 81–96 | DOI | MR | Zbl

[12] 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

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

[14] J. Pardon, “Central limit theorems for random polygons in an arbitrary convex set”, The Annals of Probability, 1:3 (2011), 881–903 | MR

[15] J. Pardon, “Central Limit Theorems for Uniform Model Random Polygons”, J. Theor. Probab., 25 (2012), 823–833 | DOI | MR | Zbl

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