# Stationary and ergodic r.v: relation between error and independent variable

In time series often hold the condition that a r.v. is stationary and ergodic, allowing the application of the law of large number. If in a model as: Y= a + bX + u where u is the error term, we assume that u is stationary and ergodic, this imply something about Y? also Y is stationary and ergodic?