Consider $x_t = \phi x_{t-1} + \epsilon_t$, $\epsilon_t \stackrel{iid}{\sim} N(0,1)$,$|\phi| \lt 1$,$x_1 \sim N(0, \frac{1}{1-\phi^2})$,then we have $x_t | x_1, ..., x_{t-1} \sim N(\phi x_{t-1},1)$ for $t = 2,...,n$. How can I prove $x_s$ and $x_t$ with $1 \le s \lt t \le n$ are conditionally independent given $\{x_{s+1},...,x_{t-1}\}$ if $t-s > 1$?
Thanks in advance!