Nonparametric Regression and Prediction with Dependent Errors (running Title: Regression and Prediction under Dependence)

We study minimax rates of convergence for nonparametric regression and prediction under a random design with dependent errors. It is shown that dependence among errors in general does not hurt a prediction of the next response. For estimating the regression function, however, dependence may damage the minimax rate of convergence. Under the assumption that the errors are independent of the explanatory variables, we show that minimax rates of convergence are determined in terms of the massiveness (characterized by metric entropy) of the function class assumed to contain the underlying regression function, and behavior of the covariance matrix of the errors. It is shown that the minimax risk is at the worse rate between two quantities: the minimax risk of the same function class but under the assumption of i.i.d. errors, and the minimax risk of estimating the mean of the regression function. Examples of function classes under diierent covariance structures including both short and long range dependences are given.