On the Glivenko-Cantelli Theorem for Weighted Empiricals Based on Independent Random Variables

Abstract

For $X_1, \cdots, X_n$ independent real valued random variables and for $\alpha \in \lbrack 0, 1 \rbrack$, let $F_j(x) = \alpha P\lbrack X_j < x \rbrack + (1 - \alpha)P\lbrack X_j \leqq x \rbrack$ and $Y_j(x) = \alpha I_{\lbrack X_j < x \rbrack} + (1 - \alpha) I_{\lbrack X_j \leqq x \rbrack}$, where $I_A$ is the indicator function of the set $A$. For numbers $w_1, w_2, \cdots, w_n$, let $D_n = \sup_{x, \alpha} \max_{N \leqq n}|\sum^N_1 w_j(Y_j(x) - F_j(x))|$. We will obtain an exponential bound for $P\lbrack D_n \geqq a \rbrack$ and a rate for almost sure convergence of $D_n$. When $w_j \equiv 1$ the bound and the rate become, respectively, $4a \exp \{-2((a^2/n) - 1)\}$ and $O((n \log n)^{\frac{1}{2}})$.