A central limit theorem and its applications to multicolor randomly reinforced urns

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