Asymptotic Stability of Empirical Processes and Related Functionals, October 2017

Let E be a space of observables in a sequence of trials zeta_n and define m_n to be the empirical distributions of the outcomes. We discuss the almost sure convergence of the sequence m_n in terms of the psi -weak topology of measures, when the sequence zeta n is assumed to be stationary. In this respect, the limit variable is naturally described as a certain canonical conditional distribution. Then, given some functional tau defined on a space of laws, the consistency of the estimators tau(m_n) is investigated. Hence, a criterion for a refined notion of robustness, that applies when considering random measures, is provided in terms of the modulus of continuity of tau.