In many statistical problems, we obtain functions of both the random variable and the parameters involved, from whose asymptotic behavior we may deduce the asymptotic behavior of certain estimates. In many of these cases, it is sufficient to demonstrate uniform convergence with probability one of these functions. In this paper, a set of sufficient conditions for this is given, and we show how these results may be applied to some statistical problems.
Herman Rubin. "Uniform Convergence of Random Functions with Applications to Statistics." Ann. Math. Statist. 27 (1) 200 - 203, March, 1956. https://doi.org/10.1214/aoms/1177728359