Asymptotic Stability of Empirical Processes and Related Functionals

Let $E$ be a space of observables in a sequence of trials $\xi_n$ 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 $\xi_n$ 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 $\tau$.