M-Estimators Converging to a Stable Limit

We discuss the asymptotic linearization of multivariate M-estimators, when the limit distribution is stable. We consider two different types of kernels: VC and bracketing. When applied to the case of normal limits, our work improves the known results to obtain the limit distribution of M-estimators. We give weak conditions for the asymptotic normality of M-estimators over differentiable kernels. To obtain these results, we present an inequality on empirical processes satisfying a bracketing condition with respect to a norm smaller than the L₂ norm.