On the Convergence of Empiric Distribution Functions

Abstract

Let $\mu$ be a probability measure on the Borel sets of $k$-dimensional Euclidean space $E_k.$ Let ${X_n}, n = 1, 2, \cdots,$ be a sequence of $k$-dimensional independent random vectors, distributed according to $\mu.$ For each $n = 1, 2, \cdots$ let $\mu_n$ be the empiric distribution function corresponding to $X_1, \cdots, X_n,$ i.e., for every Borel set $A \epsilon E_k,$ we define $\mu_n(A)$ to be the proportion of observations among $X_1, \cdots, X_n$ which fall in $A.$ Let $\alpha$ be the class of Borel sets in $E_k$ defined below. The object of this paper is to prove that $P{\lim_{n\rightarrow\infty}} \sup_{A \epsilon \mathscr{a} \|\mu_n(A) - \mu(A)\| = 0} = 1.$