Let $X_1, X_2, …$ be independent random variables.
Define $$\mathscr{T}_n = \sigma(X_{n+1}, X_{n+2}, \ldots)$$ and $$\mathscr{T} = \bigcap_{n} \mathscr{T}_n,$$ the tail σ-algebra of $(X_1, X_2, \ldots)$.
Are $\sigma(X_1), \sigma(X_2), …$ independent of $\mathscr{T}$?
If so, why?
If not, why, and what about $$\sigma(X_1), \sigma(X_2), …, \sigma(X_k) \ \;\forall k \in \mathbb{N}\quad?$$
All I got so far is that if $X_1, X_2, \ldots$ were events instead of random variables, $X_1, X_2, \ldots, X_k \ \forall k \in \mathbb{N}$ would be independent of some events in $\mathscr{T}$ such as $\limsup X_n$.
Best Answer
[This answers the original edition of the question, which did not assume $X_1,X_2,X_3,\ldots$ are independent.]
No. Suppose $R\sim\mathrm{Uniform}(0,1)$ and $(Y_1,Y_2,Y_3,\ldots)\mid R\sim\mathrm{i.i.d. Bernoulli}(R)$.
Then the strong law of large numbers implies that $$ \Pr\left( \lim_{n\to\infty} \frac{Y_1+\cdots+Y_n} n= R \mid R\right) = 1. $$ So \begin{align} & \Pr\left( \lim_{n\to\infty} \frac{Y_1+\cdots+Y_n} n= R \right) \\[10pt] = {} & \operatorname{E} \left( \Pr\left( \lim_{n\to\infty} \frac{Y_1+\cdots+Y_n} n= R \mid R\right) \right) = \operatorname{E}(1) = 1. \end{align}
Let $X_n= (Y_1+\cdots+Y_n)/n$. The event that $\lim_{n\to\infty} X_n = r$ is in the tail sigma-algebra of $X_1,X_2,X_3,\ldots$. But $X_1$ is not independent of that event, since $\Pr(X_1=1\mid R=r)=r$.
If you want a tail event whose probability is positive, observe that $\lim\limits_{n\to\infty} X_n \overset{\text{a.s.}}= R \sim\mathrm{Uniform}(0,1)$, so $\Pr(X_1=1) = \operatorname{E}(\Pr(X_1=1\mid \lim\limits_{n\to\infty} X_n)) = \operatorname{E}(\lim\limits_{n\to\infty} X_n) = 1/2$, and find $\Pr(X_1=1\mid \lim\limits_{n\to\infty} X_n>1/2)$.