Local central limit theorems in stochastic geometry

TL;DR

This paper establishes a general local central limit theorem applicable to sums of independent variables, with applications in stochastic geometry such as percolation, random graphs, coverage models, and sequential adsorption.

Contribution

It introduces a new local central limit theorem for sums where one component satisfies a CLT and the other a local CLT, with applications to various stochastic geometry models.

Findings

Derived a general local CLT for sums of independent variables.

Applied the theorem to percolation, random graphs, and coverage models.

Provided concrete probabilistic estimates for geometric quantities.

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.