UNIFORM CONVERGENCE OF RANDOM FUNCTIONS WITH APPLICATIONS TO STATISTICS
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. 1. Statement of the theorem.3 Let X1, , X", be a sequence of independent and identically distributed variables with values in an arbitrary space X. Let T be a compact topological space, and let f be a complex-valued function on T X X, measurable in x for each t e T. Let P be the common distribution of the Xi.