Contributions to strong approximations in time series with applications in nonparametric statistics and functional limit theorems.

This thesis is concerned with applications in probability and statistics of approximation theorems for weakly dependent random vectors. The basic approach is to approximate partial sums of weakly dependent random vectors by corresponding partial sums of independent ones. In chapter 2 we apply such a general idea so as to obtain an almost sure invariance principle for partial sums of Rd-valued absolutely regular processes. In chapter 3 we apply the results of chapter 2 to obtain functional limit theorems for non-stationary fractionally differenced processes. Chapter 4 deals with applications of approximation theorems to nonparamatric estimation of density and regression functions under weakly dependent samples. We consider L1-consistency of kernel and histogram density estimates. Universal consistency of the partition estimates of the regression function is also studied. Finally in chapter 5 we consider necessary conditions for L1-consistency of kernel density estimates under weakly dependent samples as an application of a Poisson approximation theorem for sums of uniform mixing Bernoulli random variables.