Properties of convex hull generated by inhomogeneous Poisson point process

I. M. Khamdamov

Ufa mathematical journal, Tome 12 (2020) no. 3, pp. 81-96

Résumé

The paper is devoted to the limit distribution study of the exterior of a convex hull generated by independent observations of two-dimensional random points having Poisson distributions above the parabola. Following P. Groeneboom [1], we note that near the boundary of support, the Binomial point process is almost indistinguishable from the Poisson point process. Therefore, the approximation of a Binomial point process to a Poisson process is not considered here; it is believed that it is sufficient to study the functionals of the convex hull generated by the Poisson point process. Using the modified P. Groeneboom technique, the so-called strong mixing and martingale properties of the vertex Markovian jump stationary process, the asymptotic expressions are obtained for the expectation and variance of the external part of the area of the convex hull inside the parabola. This is a continuation of results by H. Carnal in [2], where an asymptotic expression was found only for mean values of basic functionals of a convex hull. The asymptotic expression for the variance of the area of a convex hull was later obtained by J. Pardon [3] as no regularity conditions were imposed on the boundary of the support of a uniform distribution. The asymptotic expressions obtained here are used in the proofs of the central limit theorem for the area of the convex hull. Similar results were established in the studies by A. J. Cabo and P. Groeneboom [4] for the case as the initial distribution in a convex polygon is uniform.

Détail

Keywords: convex hull, random points
Mots-clés : Poisson point process.

Bibliographie

[1] P. Groeneboom, “Limit theorems for convex hulls”, Probab. Theor. Relat. Fields, 79:3 (1988), 327–368 | DOI | MR | Zbl

[2] H. Carnal, “Die konvexe Hülle von $n$ rotationssymmetrisch verteilten Punkten”, Z. Wahrscheinlichkeitstheorie Verw. Geb., 15 (1970), 168–176 | DOI | MR | Zbl

[3] J. Pardon, “Central limit theorems for uniform model random polygons”, Theor. Probab., 25:3 (2012), 823–833 | DOI | MR | Zbl

[4] A. J. Cabo, P. Groeneboom, “Limit theorems for functionals of convex hulls”, Probab. Theor. Relat. Fields, 100:1 (1994), 31–55 | DOI | MR | Zbl

[5] P. Groeneboom, “Convex hulls of uniform samples from a convex polygon”, Adv. Appl. Prob., 44:2 (2012), 330–342 | DOI | MR | Zbl

[6] C. Buchta, “Exact formulae for variances of functionals of convex hulls”, Adv. Appl. Prob., 45:4 (2013), 917–924 | DOI | MR | Zbl

[7] B. Efron, “The convex hull of a random set of points”, Biometrika, 52:3 (1965), 331–343 | DOI | MR | Zbl

[8] A. Reny, R. Sulanke, “Über die konvexe Hülle von zufälling gewahlten Punkten”, Z. Wahrscheinlichkeitstheorie. Verw. Geb., 2 (1963), 75–84 | DOI | MR | Zbl

[9] T. Hsing, “On the asymptotic distribution of the area outside a random convex hull in a disk”, Ann. Appl. Prob., 4:2 (1994), 478–493 | DOI | MR | Zbl

[10] I. Hueter, “The convex hull of a normal sample”, Adv. Appl. Prob., 26:4 (1994), 855–875 | DOI | MR | Zbl

[11] A. V. Nagaev, I. M. Khamdamov, Limit theorems for functionals of random convex hulls, Preprint, No 51, Institute of Mathematics, Academy of Sciences of Uzbekistan, Tashkent, 1991 (in Russian) | Zbl

[12] Theor. Prob. Appl., 47:3 (2003), 533–541 | DOI | DOI | MR | MR | Zbl

[13] 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

[14] I. M. Khamdamov, T. Kh. Adirov, “Martingale properties of vertex functionals generated by Poisson point processes”, Doklady Akad. Nauk Uzbekistan, 1 (2015), 9–11 (in Russian)

[15] I. M. Khamdamov, T. Kh. Adirov, “One of the properties of the convex hull generated by a Poisson point process”, Uzbek Math. J., 3 (2019), 60–63 | DOI | MR

[16] Wolters-Noordhoff Publishing Company, Groningen, 1971 | MR | Zbl