Electronic Journal of Probability

Local Central Limit Theorems in Stochastic Geometry

Mathew Penrose and Yuval Peres

Full-text: Open access

Abstract

We give a general local central limit theorem for the sum of two independent random variables, one of which satisfies a central limit theorem while the other satisfies a local central limit theorem with the same order variance. We apply this result to various quantities arising in stochastic geometry, including: size of the largest component for percolation on a box; number of components, number of edges, or number of isolated points, for random geometric graphs; covered volume for germ-grain coverage models; number of accepted points for finite-input random sequential adsorption; sum of nearest-neighbour distances for a random sample from a continuous multidimensional distribution.

Article information

Source
Electron. J. Probab., Volume 16 (2011), paper no. 91, 2509-2544.

Dates
Accepted: 3 December 2011
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464820260

Digital Object Identifier
doi:10.1214/EJP.v16-968

Mathematical Reviews number (MathSciNet)
MR2869414

Zentralblatt MATH identifier
1245.60032

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 05C80: Random graphs [See also 60B20]

Keywords
Local central limit theorem stochastic geometry percolation random geometric graph nearest neighbours

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Penrose, Mathew; Peres, Yuval. Local Central Limit Theorems in Stochastic Geometry. Electron. J. Probab. 16 (2011), paper no. 91, 2509--2544. doi:10.1214/EJP.v16-968. https://projecteuclid.org/euclid.ejp/1464820260


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