Limit theorems for nonlinear functions of weakly dependent random fields

Asymptotic properties of nonlinear functions of weakly dependent random fields are studied. Special attention is paid to indicators arising in analysis of empirical distributions, non-parametric statistics and excursion sets. The optimal covariance estimate for indicator functions of positively or negatively associated random variables is obtained. This estimate improves the inequality by I. Bagai, B. Prakasa Rao and Hao Yu. We also establish a new moment estimate for sums of dependent random variables. This result permits to give wide conditions to guarantee the Donsker-Prokhorov type functional central limit theorem validity for random fields. The new estimates allow us to obtain several limit theorems for excursion set volumes. In particular, we prove a functional version of the theorem established in the recent paper by A. Bulinski, E. Spodarev and F. Timmermann. Limit theorems for functions of random measures are obtained as well. Generalizations of the results by S. Evans and Yu. Bakhtin are provided.