1
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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 $$ K^2(x,A) = \int K(x,dy)K(y,A) = \int \delta_x(dy)K(y,A) = K(x,A) = \delta_x(A). $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.