Ergodic for the mean, but not ergodic stochastic process?

Question

Is there an example of a strictly stationary (zero mean, finite variance) stochastic process $(X_t\mid t\in \mathbb{N})$ that satisfies the conclusion of the ergodic theorem, i.e., the sample mean $\overline{X}_n\rightarrow_{a.s.} 0$, as $n\rightarrow \infty$, such that $X_s$ is uncorrelated with $X_t$, whenever $t\neq s$, but $(X_t\mid t\in \mathbb{N})$ is not ergodic (i.e., the underlying shift operator is not an ergodic transformation)? Any ideas? Many thanks!

Answer 1

Let $Z$ be a random variable with $P(Z=1) =P(Z=2)=1/2$.

Let $Y_1, Y_2, \dots$ be iid and independent of $Z$ with $P(Y_t = 1) = P(Y_t = -1) = 1/2$.

Set $X_t = Z Y_t$. Then $X_t$ is stationary, and by the strong law of large numbers, $\overline{X}_n = Z \overline{Y}_n \to Z E[Y_t] = 0$ almost surely. It is also uncorrelated, since for $s \ne t$ we have $$E[X_s X_t] = E[Z^2 Y_s Y_t] = E[Z^2] E[Y_s] E[Y_t] = 0 = E[X_s] E[X_t].$$ But $X_t$ is not ergodic; for example, the event $A = \{X_t = 2 \text{ i.o.}\}$ is shift invariant, but since $A = \{Z=2\}$ we have $P(A) = 1/2$.

Answer 2

Consider the process $X_n = (-1)^n X_0$ where $X_0$ is a random variable with distribution symmetric about $0$ (but $|X|$ not a.s. constant).