On the existence of uniformly consistent estimates

Abstract:

Let $\mathcal {M}$ be a family of probability measures on $(\mathfrak {X},\mathcal {A})$ and $U$ the uniform structure defined by vicinities of the form \[ \left \{ (P, Q):\sup \limits _{1 \leqslant i \leqslant K} | P^n(A_{i,n}) - Q^n(A_{i,n}) | < \varepsilon \right \},\] where ${P^n}$ is the product measure on $({\mathfrak {X}^n},{\mathcal {A}^n}),{A_{i,n}} \in {\mathcal {A}^n},\varepsilon > 0,n \wedge K \geqslant 1$. Let ${\phi ^ * }:(\mathcal {M},U) \to ({\phi ^ * }(\mathcal {M}),d)$, where \[ d\left (\phi ^*(P), \phi ^*(Q) \right ) = ||P - Q||_{L_1} = 2\sup \limits _{A \in \mathcal {A}} |P(A) - Q(A)|.\] We consider the case where the space of measures $M$ is ${L_1}$ separable and relate the existence of uniformly consistent estimates for ${\phi ^ * }(P)$ with uniform continuity of ${\phi ^ * }$ and ${L_1}$-total boundedness of $M$.