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2012 Limit theorems for empirical processes based on dependent data
Patrizia Berti, Luca Pratelli, Pietro Rigo
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Electron. J. Probab. 17: 1-18 (2012). DOI: 10.1214/EJP.v17-1765

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.

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Patrizia Berti. Luca Pratelli. Pietro Rigo. "Limit theorems for empirical processes based on dependent data." Electron. J. Probab. 17 1 - 18, 2012. https://doi.org/10.1214/EJP.v17-1765

Information

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

zbMATH: 1246.60009
MathSciNet: MR2878788
Digital Object Identifier: 10.1214/EJP.v17-1765

Subjects:
Primary: 60B10
Secondary: 60F05, 60G09, 60G57

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