Rates of weak convergence for images of measures by families of mappings (Q1612954)

Let \(E_1\) be a Banach space and let \(\mu_n\) and \(\mu\) be Borel measures on \(E_1\) such that \(\mu_n\to \mu\) weakly. Given a certain set \(\mathcal F\) of continuous functions from \(E_1\) into another Banach space \(E_2\), for each \(f\in \mathcal F\) it follows that \(\mu_n\circ f^{-1}\to \mu\circ f^{-1}\) weakly in \(E_2\) as well. An interesting question is as follows: Compare for \(f\in\mathcal F\) the speed of convergence (w.r.t. the Prokhorov distance \(\pi\)) of \(\mu_n\circ f^{-1}\) to \(\mu\circ f^{-1}\) with that of \(\pi(\mu_n,\mu)\) in dependence of properties of \(\mathcal F\). As shown by \textit{W. Whitt} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 29, 39-44 (1974; Zbl 0268.60004)] the speed of convergence of the image measures is (up to a constant) the same for \(\mathcal F\)'s bounded w.r.t.~the Hölder norm. The present paper treats the unbounded case, yet with some control about the Hölder continuity of functions \(f\) belonging to \(\mathcal F\).