We consider a point process i + ξ <Subscript> i </Subscript>, where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${i\in \mathbb{Z}}$$</EquationSource> </InlineEquation> and the ξ <Subscript> i </Subscript>’s are i.i.d. random variables with compact support and variance σ <Superscript>2</Superscript>. This process, with a suitable rescaling of the distribution of ξ <Subscript> i </Subscript>’s, is well known to converge weakly, for large σ, to the Poisson...</subscript></superscript></subscript></equationsource></inlineequation></subscript>