Note: I have previously asked this on CrossValidated here. Since I didn't get an answer to my question I'm posting it on this community.
My lecture notes on time series analysis give the following definition:
A stochastic process $(X_t)_{t\in\mathbb{Z}}$ is called autoregressive of order $p$ if it satisfies: $$X_t=\phi_1X_{t-1}+...+\phi_{t-p}X_{t-p}+W_t.$$ Where $W_t$ is independent from $(X_s)_{s<t}$, and $\phi_p \neq 0$.
Then it is stated that if the process is weak-sense stationary it follows that the $W_t$'s are identically distributed. How does this follow? Or, if incorrect, could you give me a counterexample?