$(\dots, X_{t-1}, X_t)$ and $(X_{t+1}, X_{t+2}, \dots) $ are independent $\Rightarrow (X_t)$ are independent

Question

In some lecture notes in Time Series Analysis it was written: If the random vectors $(\dots, X_{t-1}, X_t)$ and $(X_{t+1}, X_{t+2}, \dots)$ are independent for all $t \in \mathbb{Z}$, then it is obvious that the random variables $X_t$ are independent. I am not sure how is this obvious. It seems true but I can not find a way to prove it. Any hints?

Answer 1

If $X,Y$ are independent and $f$ and $g$ are (measurable) functions then $f\left(X\right)$ and $g\left(Y\right)$ are independent.

Here define $f$ by $\left(\dots,X_{t-1},X_{t}\right)\mapsto X_{t}$ and $g$ by $\left(X_{t+1},X_{t+2},\ldots\right)\mapsto X_{t+1}$.

Likewise you can prove more generally that $X_t$ and $X_{t+k}$ are independent for $k>1$.