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.Comment: V1: 31 pages. V2: 45 pages, with new results added in Section 5 and
extra explanation added elsewher
