The Empirical Discrepancy Over Lower Layers and a Related Law of Large Numbers

Abstract

Let $\{X_k\}$ be a sequence of independent random variables which are centered at their means; let $\{T_k\}$ be an i.i.d. sequence of $\beta$-dimensional random vectors with common distribution $\mu$; and let $\{X_k\}$ and $\{T_k\}$ be independent. With $\mathscr{L}$ the collection of lower layers, a necessary and sufficient condition for the almost sure convergence of $\sup_{L \in \mathscr{L}}|\sum^n_{k = 1} \chi L (T_k)/n - \mu(L)|$ to zero is given. In addition, this condition on $\mu$ is shown to imply that $\sup_{L \in \mathscr{L}}|\sum^n_{k = 1} X_{k\chi L}(T_k)|/n \rightarrow 0$ a.s. provided the $X_k$ satisfy a first moment-like condition. Rates of convergence are also investigated.