# Is strict stationarity a sufficient condition for ergodicity?

For a given time series, is strict stationarity a sufficient condition for ergodicity? I am wondering if it isn't also sufficient for a time series to be weakly stationary because then the mean is a constant. I can't see why the variance shouldn't be allowed to vary over time in order for the time series to be ergodic.

No, it isn't. @MossMurderer gives a good example of a time series that is stationary (both weakly and strictly), but not ergodic. Let $\pi$ be the chain's stationary distribution, assume that $X_1 \sim \pi$ and suppose for $t > 1$ $X_t = X_{t-1}$. In other words, the transition kernel is $$K(x,A) = \delta_x(A).$$
This process is stationary (weakly if it has first and second moments), but not ergodic because $$||K^n(x,A) - \pi(A)||_{\text{TV}} =||\delta_x(A) - \pi(A)||_{\text{TV}}$$ doesn't decrease (or even change at all) as $n \to \infty$. No matter how many times you apply kernel $K$, you still have point mass at that starting point. To convince yourself of this observe that $$K^2(x,A) = \int K(x,dy)K(y,A) = \int \delta_x(dy)K(y,A) = K(x,A) = \delta_x(A).$$