Local Central Limit Theorems in Stochastic Geometry

Mathew D. Penrose (University of Bath)
Yuval Peres (Microsoft Research)

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.

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Pages: 2509-2544

Publication Date: December 3, 2011

DOI: 10.1214/EJP.v16-968

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