I'm trying to answer a question where I have an ergodic and covariance stationary process $\{x_t\}$, and without imposing further moment conditions need to prove $\frac{1}{n} \sum\limits_{t=1}^n x_t^2 \rightarrow_p \mathbb{E}[x_t^2]$.
So far I have since $g(x) = x^2$ is a Borel measurable transformation, $x_t^2$ is ergodic. But I need strict stationarity to utilize the Ergodic Theorem, or at least $\sum_{j=1}^{\infty}|cov(x_1,x_{j+1})| < \infty $.
Why is ergodicity in general not enough to prove that the first moment will be ergodic?