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

Asymptotics of approximate Bayesian computation when summary statistics converge at heterogeneous rates