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Random process - stationary and ergodicity

Stationarity does not imply ergodicity.
Consider a process x(t):
x(t) = 1 with probability ½
x(t) = -1 with probability ½.
Thus, E[x(t)] = 0. (Actually R_x(tau) =1. Thus, this process is also WSS)
But if we take the time-average of one sample path, it is either 1 or -1.

Ergodicity implies stationarity.
We show that non-stationary implies non-ergodic.
Suppose that E[x(t)] = f(t) (i.e., non-stationary)
But time average of a sample path will yield a number independent of t. Thus, x(t) cannot be ergodic.

Conclusion: ergodicity is a stronger condition.