 formula_45, formula_46 and formula_47, then the Hill tail-index estimator is

where formula_49 is the formula_50-th order statistic of formula_51.This estimator converges in probability to formula_37, and is asymptotically normal provided formula_53 is restricted based on a higher order regular variation property 
.Consistency and asymptotic normality extend to a large class of dependent and heterogeneous sequences, irrespective of whether formula_54 is observed, or a computed residual or filtered data from a large class of models and estimators, including mis-specified models and models with errors that are dependent.