New local linear estimators are proposed for a wide class of nonparametric regression models. The estimators are uniformly consistent regardless of satisfying traditional conditions of dependence of design elements. The estimators are the solutions of a specially weighted least-squares method. The design can be fixed or random and does not need to meet classical regularity or independence conditions. As an application, several estimators are constructed for the mean of dense functional data. The theoretical results of the study are illustrated by simulations. An example of processing real medical data from the epidemiological cross-sectional study ESSE-RF is included. We compare the new estimators with the estimators best known for such studies.

The limit behavior is studied for the distributions of normalized U-and V-statistics of an arbitrary order with canonical (degenerate) kernels, based on samples of increasing sizes from a stationary sequence of observations satisfying φ-or α-mixing. The corresponding limit distributions are represented as infinite multilinear forms of a centered Gaussian sequence with a known covariance matrix.

Exponential inequalities are obtained for the distribution tails of canonical (degenerate) U- and V-statistics of an arbitrary order based on samples from a stationary sequence of observations satisfying ϕ-mixing.