Title: Qualitative robustness for bootstrap approximations,
Abstract: An important property of statistical estimators is qualitative robustness,
that is small changes in the distribution of the data only result in small
chances of the distribution of the estimator. Moreover, in practice, the
distribution of the data is commonly unknown, therefore bootstrap
approximations can be used to approximate the distribution of the estimator.
Hence qualitative robustness of the statistical estimator under the bootstrap
approximation is a desirable property. Currently most theoretical
investigations on qualitative robustness assume independent and identically
distributed pairs of random variables. However, in practice this assumption is
not fulfilled. Therefore, we examine the qualitative robustness of bootstrap
approximations for non-i.i.d. random variables, for example $\alpha$-mixing and
weakly dependent processes. In the i.i.d. case qualitative robustness is
ensured via the continuity of the statistical operator, representing the
estimator, see Hampel (1971) and Cuevas and Romo (1993). We show, that
qualitative robustness of the bootstrap approximation is still ensured under
the assumption that the statistical operator is continuous and under an
additional assumption on the stochastic process. In particular, we require a
convergence condition of the empirical measure of the underlying process, the
so called Varadarajan property.