Strong approximation of density estimators from weakly dependent observations by density estimators from independent observations

We derive an approximation of a density estimator based on weakly dependent random vectors by a density estimator built from independent random vectors We construct on a su ciently rich probability space such a pairing of the random variables of both experiments that the set of observations fX Xng from the time series model is nearly the same as the set of observations fY Yng from the i i d model With a high probability all sets of the form fX Xng fY Yng a b ad bd contain no more than O f n Q bi ai g log n elements respectively Although this does not imply very much for parametric problems it has important implications in nonparametric statistics It yields a strong approximation of a kernel estimator of the stationary density by a kernel density estimator in the i i d model Moreover it is shown that such a strong approximation is also valid for the standard bootstrap and the smoothed bootstrap Using these results we derive simultaneous con dence bands as well as supremum type nonparametric tests based on reasoning for the i i d model Mathematics Subject Classi cation Primary G secondary G M