The use of heavy-tailed distributions is a valuable tool in developing robust Bayesian procedures, limiting the influence of outliers on posterior inference. In this paper, the behavior of the posterior density for a location-scale model is investigated when the sample contains outliers. L-exponentially varying functions are introduced in order to characterize the tails of the densities. Simple conditions on the tails of the likelihood, using L-exponentially varying functions, are established to determine the proportion of observations that can be rejected as outliers. It is shown that the posterior distribution converges in law to the posterior that would be obtained from the reduced sample, excluding the outliers, as they tend to plus or minus infinity, at any given rate
