A note on the central limit theorems for dependent random variables

The paper contains a few limit theorems for stationary sequences $\{\xi_j\}$ under the condition $\rho(n)\to0$, where $\rho(n)$ denotes the maximal correlation coefficient between the past $\{\xi_j,j\le0\}$ and the future $\{\xi_j, j\ge n\}$ of the sequence $\{\xi_j\}$.