Let $x_t = φx_{t−1} + e_t$ where, $|φ| < 1$ Assume $e_t$ is iid with mean $0$ and variance $\sigma^2$, has a continuous distribution, with a density function that is bounded on $R$.
Now I have to prove that it is strong mixing. The error term satisfies the density requirements to be able to show mixing, but I am stuck after that. I know that to show strong mixing you have to show that $α(F, G) := \sup |P(A ∩ B) − P(A)P(B)|, A ∈ F, B ∈ G;$ goes to $0$. However, I am having a very hard time figuring out how to start the proof. I know that this AR($1$) process is geometrically strong mixing but I am struggling to formally prove it.
Also, I am wondering if there is a clear relationship between covariance or correlation and showing mixing. Any help or hints would be appreciated!