Limit theorems for statistical functionals with applications to dimension estimation

Abstract

This thesis deals with two different types of limit theorems: classical limit theorems and almost sure limit theorems (ASLT).We prove classical limit theorems for a special statistical functional arising in dimension estimation. These results allow to construct a new estimation procedure for the information dimension of probability measures. This method is relatively simpler than other existing methods and its applicability is illustrated with two numerical simulations.We refine Berkes and Csaki's ASLT for U-statistics of independent identically distributed random variables. Their result for non-degenerate U-statistics is also extended to the case of convergence to stable laws. Furthermore we prove almost sure central limit theorems for non-degenerate U-statistics of stationary weakly dependent random variables.