Classical empirical process theory for Vapnik-Cervonenkis classes deals mainly with sequences of independent variables. This paper extends the theory to stationary sequences of dependent variables. It establishes rates of convergence for $\beta$-mixing and $\phi$-mixing empirical processes indexed by classes of functions. The method of proof depends on a coupling of the dependent sequence with sequences of independent blocks, to which the classical theory can be applied. A uniform $O(n^{-s/(1+s)})$ rate of convergence over V-C classes is established for sequences whose mixing coefficients decay slightly faster than $O(n^{-s})$.
Publié le : 1994-01-14
Classification:
Empirical process,
$\beta$-mixing,
rate of convergence,
random metric entropy,
V-C class,
blocking,
60F05,
60F17,
60G10
@article{1176988849,
author = {Yu, Bin},
title = {Rates of Convergence for Empirical Processes of Stationary Mixing Sequences},
journal = {Ann. Probab.},
volume = {22},
number = {4},
year = {1994},
pages = { 94-116},
language = {en},
url = {http://dml.mathdoc.fr/item/1176988849}
}
Yu, Bin. Rates of Convergence for Empirical Processes of Stationary Mixing Sequences. Ann. Probab., Tome 22 (1994) no. 4, pp. 94-116. http://gdmltest.u-ga.fr/item/1176988849/