Limit theorems for empirical processes based on dependent data

Let $(X_n)$ be any sequence of random variables adapted to a filtration $(mathcalG_n)$. Define $a_n(cdot)=Pigl(X_n+1incdotmidmathcalG_nigr)$, $b_n=rac1nsum_i=0^n-1a_i$, and $mu_n=rac1n,sum_i=1^ndelta_X_i$. Convergence in distribution of the empirical processes eginequation* B_n=sqrtn,(mu_n-b_n)quad extandquad C_n=sqrtn,(mu_n-a_n) endequation* is investigated under uniform distance. If $(X_n)$ is conditionally identically distributed (in the sense of citeBPR04) convergence of $B_n$ and $C_n$ is studied according to Meyer-Zheng as well. Some CLTs, both uniform and non uniform, are proved. In addition, various examples and a characterization of conditionally identically distributed sequences are given.