Does stationarity of $X_t$ imply that $\lim_{j\to \infty} \mathbb{E}_t[X_{t+j}]$ exists?

Question

Let $X_t$ be univariate stochastic process. Is the following conjecture correct: if $X_t$ is strictly stationary (see e.g. https://en.wikipedia.org/wiki/Stationary_process), then $\lim_{j\to \infty} \mathbb{E}_t[X_{t+j}]$ exists and is constant.

Intuitively $\mathbb{E}_t[X_{t+j}]$ should approach its long run unconditional mean (which is constant due to stationarity). However, I am not sure how to prove this result. An example of a strictly stationary process for which this property is true would be the standard AR(1) process $X_t=\rho X_{t-1}+\epsilon_t$, where $-1<\rho<1$ and $\epsilon_t$ is Gaussian white noise.

Answer 1

While this is intuitively true, since most stationary processes that we consider have decaying dependence such that the limit in question is just the mean, the answer in general is false. Consider the Gaussian process for $\omega \in (0,1/2)$

$ X_t = Z_1 \sin(2 \pi \omega t)+ Z_2 \cos(2 \pi \omega t) $

Where $Z_i$ are iid standard normal random variables. It is not too hard to check that this is a stationary Gaussian process with autocovariance $\gamma(h)=cos(2 \pi \omega h)$, and the conditional mean is a simple function of $\gamma(h)$ and will not converge.

Answer 2

Here's a trivial example:

Consider $X_t \sim \textrm{Cauchy}(x_0, \gamma)$, and iid. It's clearly stationary and strictly stationary: no dependence on time is even possible. Nonetheless it fails to have $E_t[X_{t+j}]$ be defined (much less the limit) because $E[X_t]$ is always undefined.