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.
Publié le : 1981-04-14
Classification:
Glivenko-Cantelli theorem,
lower layers,
isotone regression,
maxima of partial sums,
60F15,
62G05
@article{1176994475,
author = {Wright, F. T.},
title = {The Empirical Discrepancy Over Lower Layers and a Related Law of Large Numbers},
journal = {Ann. Probab.},
volume = {9},
number = {6},
year = {1981},
pages = { 323-329},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994475}
}
Wright, F. T. The Empirical Discrepancy Over Lower Layers and a Related Law of Large Numbers. Ann. Probab., Tome 9 (1981) no. 6, pp. 323-329. http://gdmltest.u-ga.fr/item/1176994475/