  We consider the asymptotic properties of Approximate Bayesian Computation
(ABC) for the realistic case of summary statistics with heterogeneous rates of
convergence. We allow some statistics to converge faster than the ABC
tolerance, other statistics to converge slower, and cover the case where some
statistics do not converge at all. We give conditions for the ABC posterior to
converge, and provide an explicit representation of the shape of the ABC
posterior distribution in our general setting; in particular, we show how the
shape of the posterior depends on the number of slow statistics. We then
quantify the gain brought by the local linear post-processing step.
