## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 4, Number 2 (1994), 478-493.

### On the Asymptotic Distribution of the Area Outside a Random Convex Hull in a Disk

#### Abstract

The asymptotic distribution of the area $V_n$ outside the convex hull of $n$ i.i.d. points uniformly distributed on the two-dimensional unit disk is studied. The asymptotic variance of $V_n$ is found to be of the order $n^{-5/3}$, and the asymptotic distribution of $V_n$ is shown to be normal. The results are obtained by carefully analyzing the strength of dependence between sample points at different locations close to the boundary of the unit disk.

#### Article information

**Source**

Ann. Appl. Probab., Volume 4, Number 2 (1994), 478-493.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005069

**Digital Object Identifier**

doi:10.1214/aoap/1177005069

**Mathematical Reviews number (MathSciNet)**

MR1272736

**Zentralblatt MATH identifier**

0806.60004

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Secondary: 60F05: Central limit and other weak theorems

**Keywords**

Asymptotic distribution convex hull point processes

#### Citation

Hsing, Tailen. On the Asymptotic Distribution of the Area Outside a Random Convex Hull in a Disk. Ann. Appl. Probab. 4 (1994), no. 2, 478--493. doi:10.1214/aoap/1177005069. https://projecteuclid.org/euclid.aoap/1177005069