I am new to the subject of stationary and ergodic proccesses so this question might be a simple one.
Say we are given a real valued stationary and ergodic proccess $\{X_t\}_{t=1}^{\infty}$ (with the Borel sigma-algebra) does the cartesian product of $\{X_t\}$ with itself $(X_t,X_t)$ is also stationary and ergodic?