Using a proposed generalisation of the pair distribution function for a gas of non-interacting particles obeying fractional exclusion statistics in arbitrary dimensionality, we derive the statistical correlations in the asymptotic limit of vanishing or low temperature. While Friedel-like oscillations are present in nearly all non-bosonic cases at T = 0, they are characterised by exponential damping at low temperature. We discuss the dependence of these features on dimensionality and on the value of the statistical parameter α.