Rates of convergence of Lp-estimators for a density with an infinity cusp

We study the asymptotics of L_p-estimators, p>0, as estimates of a parameter of location for data coming for a symmetric density with an infinity cusp at the center of symmetry of the distribution. In this situation, the data are more concentrated around the parameter of location than in usual cases. The maximum-likelihood estimator is not defined. The rates of convergence of the L_p-estimators in this situation depend on p and on the shape of the density. For some densities and small values of p, the L_p-estimator converges with a fast rate of convergence.