Let $(X_n)$ be a sequence of integrable real random variables, adapted to a filtration $(\mathcal{G}_n)$. Define \begin{equation*} C_n=\sqrt{n}\,\bigl\{\frac{1}{n} \sum_{k=1}^nX_k-E(X_{n+1}\mid\mathcal{G}_n)\bigr\}\quad\text{and}\quad D_n=\sqrt{n}\,\bigl\{E(X_{n+1}\mid\mathcal{G}_n)-Z\bigr\} \end{equation*} where $Z$ is the a.s. limit of $E(X_{n+1}\mid\mathcal{G}_n)$ (assumed to exist). Conditions for $(C_n,D_n)\longrightarrow\mathcal{N}(0,U)\times\mathcal{N}(0,V)$ stably are given, where $U,\,V$ are certain random variables. In particular, under such conditions, one obtains \begin{equation*} \sqrt{n}\,\bigl\{\frac{1}{n} \sum_{k=1}^nX_k-Z\bigr\}=C_n+D_n\longrightarrow\mathcal{N}(0,U+V)\quad\text{stably}. \end{equation*}This CLT has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced urns.
A central limit theorem and its applications to multicolor randomly reinforced urns
RIGO, PIETRO
2011-01-01
Abstract
Let $(X_n)$ be a sequence of integrable real random variables, adapted to a filtration $(\mathcal{G}_n)$. Define \begin{equation*} C_n=\sqrt{n}\,\bigl\{\frac{1}{n} \sum_{k=1}^nX_k-E(X_{n+1}\mid\mathcal{G}_n)\bigr\}\quad\text{and}\quad D_n=\sqrt{n}\,\bigl\{E(X_{n+1}\mid\mathcal{G}_n)-Z\bigr\} \end{equation*} where $Z$ is the a.s. limit of $E(X_{n+1}\mid\mathcal{G}_n)$ (assumed to exist). Conditions for $(C_n,D_n)\longrightarrow\mathcal{N}(0,U)\times\mathcal{N}(0,V)$ stably are given, where $U,\,V$ are certain random variables. In particular, under such conditions, one obtains \begin{equation*} \sqrt{n}\,\bigl\{\frac{1}{n} \sum_{k=1}^nX_k-Z\bigr\}=C_n+D_n\longrightarrow\mathcal{N}(0,U+V)\quad\text{stably}. \end{equation*}This CLT has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced urns.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
