  We study the auto-correlation measures of invariant random point processes in
the hyperbolic plane which arise from various classes of aperiodic Delone sets.
More generally, we study auto-correlation measures for large classes of Delone
sets in (and even translation bounded measures on) arbitrary locally compact
homogeneous metric spaces. We then specialize to the case of weighted model
sets, in which we are able to derive more concrete formulas for the
auto-correlation. In the case of Riemannian symmetric spaces we also explain
how the auto-correlation of a weighted model set in a Riemannian symmetric
space can be identified with a (typically non-tempered) positive-definite
distribution on $\mathbb R^n$. This paves the way for a diffraction theory for
such model sets, which will be discussed in the sequel to the present article.
