In this paper the analysis of estimates operating on dependent data is considered. Special dependent data structures are considered and the analysis is made for three different choices of contamination and performance distance measures. For contamination and performance measures both being Levy (Hampel model), an analysis that is particularly oriented toward fast convergence of the estimate to a value that is stable (robust) inside the contaminated family is undertaken. The minimum sample size to satisfy certain performance is investigated and it is found that the problem reduces to finding continuous, absolutely bounded estimates with logarithms of their moment generating function slowly increasing with the absolute value of the argument for all data distributions considered.