Uniform Strong Consistency of Rao-Blackwell Distribution Function Estimators

In the independent sampling model, Rao-Blackwell distribution function estimators $\tilde{F}_n(x)$ obtained by conditioning on sufficient statistics $T_n(X_1, \cdots, X_n)$ are considered. If for each $n \geqq 1, T_n$ is symmetric in $X_1,\cdots, X_n$ and $T_{n+1}$ is $\mathscr{B}(T_n, X_{n+1})$ measurable, it is shown that $\tilde{F}_n(x)$ converges strongly to the corresponding $F(x)$ and uniformly in $x$. This is a direct generalization of the Glivenko-Cantelli theorem.