## Electronic Journal of Probability

### Limit theorems for empirical processes based on dependent data

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 9, 18 pp.

Dates
Accepted: 29 January 2012
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465062331

Digital Object Identifier
doi:10.1214/EJP.v17-1765

Mathematical Reviews number (MathSciNet)
MR2878788

Zentralblatt MATH identifier
1246.60009

Rights

#### Citation

Berti, Patrizia; Pratelli, Luca; Rigo, Pietro. Limit theorems for empirical processes based on dependent data. Electron. J. Probab. 17 (2012), paper no. 9, 18 pp. doi:10.1214/EJP.v17-1765. https://projecteuclid.org/euclid.ejp/1465062331

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