Let $Y_0, Y_1, Y_2, \dots$ be independent random variables. For $n \in \mathbb{N}$, define
$$X_n:=Y_0Y_1\cdots Y_n.$$
Is it true that $\bigcap\limits_n \sigma(X_{n+1},X_{n+2},\dots)=\bigcap\limits_n \sigma(Y_{n+1},Y_{n+2},\dots)?$
Otherwise, how to show that $\bigcap\limits_n \sigma(X_{n+1},X_{n+2},\dots)$ and $\sigma(Y_1,Y_2,\dots)$ are independent?
The question Prove random variable $Y_0$ and $\sigma$-algebra $\mathscr{R}$ are independent., about the symmetric Bernoulli case when $P(Y_n=1)=P(Y_n=-1)=\frac12$, is related.
