Vtyjkyuk

Let

• $(Omega,mathcal A),(D,mathcal D)$ and $(E,mathcal E)$ be measurable spaces


• $Z_1,Z_2,ldots:Omegato D$ be independent and identically distributed random variables


• $X_0:Omegato E$ be a random variable independent of $Z_1,Z_2,ldots$


• $varphi:Etimes Dto E$ be $(mathcal Eotimesmathcal D,mathcal E)$-measurable and $$X_n:=varphi(X_n-1,Z_n);;;textfor ninmathbb N$$

Fix $xin E$. How can we show that $X_0,ldots,X_n-1$ is independent of $varphi(x,Z_n)$ for all $ninmathbb N$?

Clearly, since $X_0$ is independent of $Z_1$, $X_0$ is independent of $varphi(x,Z_1)$. This follows from the fat that if $X,Y$ are independent and $f,g$ are measurable, then $fcirc X,gcirc Y$ are independent.

However, I don't see how I need to proceed for $n>1$. The problem is that $X$ being independent of $Y,Z$ doesn't imply independence of $X$ and $(Y,Z)$.

Each of the variables $X_0,X_1,...,X_n-1$ is measurable w.r..t $sigma X_0,Z_1,...,Z_n-1$ and $Z_n$ is independent of this sigma algebra. Hence $phi (x,Z_n)$ is independent of this sigma algebra which makes it independent of $X_0,X_1,...,X_n-1$.