Ergodic stationary process
= Ergodic stationary process
A stationary path law is ergodic when every time-shift-invariant event has probability zero or one. The <Birkhoff ergodic theorem> then identifies integrable time averages with deterministic ensemble expectations. A shared random scale multiplying iid noise gives a stationary counterexample: it is uncorrelated across distinct times, but the path retains information about the random scale.