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 sufficiently rich probability space, such a pairing of the random variables of both experiments that the set of observations X₁,…,Xₙ from the time series model is nearly the same as the set of observations Y₁,…,Yₙ from the i.i.d. model. With a high probability, all sets of the form (X₁,…,Xₙ\ΔY₁,…,Yₙ)([a₁,b₁]×…×[ad,bd]) contain no more than O([n¹/2 ∏(bᵢ-aᵢ)]+ 1 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 confidence bands as well as supremum­type nonparametric tests based on reasoning for the i.i.d. model.