Limit theorems for empirical processes based on dependent data

摘要

Let $(X_n)$ be any sequence of random variables adapted to a filtration $(mathcal{G}_n)$. Define $a_n(cdot)=Pigl(X_{n+1}incdotmidmathcal{G}_nigr)$, $b_n=rac){n}sum_{i=0}^{n-1}a_i$, and $mu_n=rac){n},sum_{i=1}^ndelta_{X_i}$. Convergence in distribution of the empirical processes $$ B_n=sqrt{n},(mu_n-b_n)quadext{and}quad C_n=sqrt{n},(mu_n-a_n)$$ is investigated under uniform distance. If $(X_n)$ is conditionally?identically distributed, 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.