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Limit theorems for empirical processes based on dependent data

   
@article{EJP1765,
	author = {Patrizia Berti and Luca Pratelli and Pietro Rigo},
	title = {Limit theorems for empirical processes based on dependent data},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {17},
	year = {2012},
	keywords = {conditional identity in distribution; empirical process; exchangeability; predictive measure; stable convergence},
	abstract = {
Let $(X_n)$ be any sequence of random variables adapted to a filtration $(\mathcal{G}_n)$. Define $a_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid\mathcal{G}_n\bigr)$, $b_n=\frac{1}{n}\sum_{i=0}^{n-1}a_i$, and $\mu_n=\frac{1}{n}\,\sum_{i=1}^n\delta_{X_i}$. Convergence in distribution of the empirical processes $$ B_n=\sqrt{n}\,(\mu_n-b_n)\quad\text{and}\quad C_n=\sqrt{n}\,(\mu_n-a_n)$$ 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.
},
	pages = {no. 9, 1-18},
	issn = {1083-6489},
	doi = {10.1214/EJP.v17-1765},    
        url = {http://ejp.ejpecp.org/article/view/1765}}