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 \begin{equation*} B_n=\sqrt{n}\,(\mu_n-b_n)\quad\text{and}\quad C_n=\sqrt{n}\,(\mu_n-a_n) \end{equation*} is investigated under uniform distance. If $(X_n)$ is conditionally identically distributed (in the sense of \cite{BPR04}) 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.
Limit theorems for empirical processes based on dependent data
RIGO, PIETRO
2012-01-01
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 \begin{equation*} B_n=\sqrt{n}\,(\mu_n-b_n)\quad\text{and}\quad C_n=\sqrt{n}\,(\mu_n-a_n) \end{equation*} is investigated under uniform distance. If $(X_n)$ is conditionally identically distributed (in the sense of \cite{BPR04}) 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
