Limits to classification and regression estimation from ergodic processes

We answer two open questions concerning the existence of universal schemes for classification and regression estimation from stationary ergodic proces ses. It is shown that no measurable procedure can produce weakly consistent regression estimates from every bivariate stationary ergodic process, even if the covariate and response variables are restricted to take values in t he unit interval. It is further shown that no measurable procedure can prod uce weakly consistent 'classification rules from every bivariate stationary ergodic process for which the response variable is binary valued. The resu lts of the paper are derived via reduction arguments and are based in part on recent work concerning density estimaton from ergodic processes.