Approximate Bayesian computation (ABC) is a widely used inference method in Bayesian statistics to bypass the point-wise computation of the likelihood. In this paper we develop theoretical bounds for the distance between the statistics used in ABC. We show that some versions of ABC are inherently robust to misspecification. The bounds are given in the form of oracle inequalities for a finite sample size. The dependence on the dimension of the parameter space and the number of statistics is made explicit. The results are shown to be amenable to oracle inequalities in parameter space. We apply our theoretical results to given prior distributions and data generating processes, including a non-parametric regression model. In a second part of the paper, we propose a sequential Monte Carlo (SMC) to sample from the pseudo-posterior, improving upon the state of the art samplers.