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
$X_1,\ldots,X_n}$ from the time series model is nearly the same as the set of
observations $Y_1,\ldots,Y_n}$ from the i.i.d. model. With a high probability,
all sets of the form
$({X_1,\ldots,X_n}\\Delta{Y_1,\ldots,Y_n})\bigcap([a_1,b_1]\times\ldots\times[a_d,b_d])$
contain no more than $O({[n^1/2 \prod(b_i-a_i)]+ 1} \log(n))$ 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.