Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space.

Let events $A_{1,n}, A_{2,n}, A_{3,n}, ...$ be n-wise independent for $n \in \mathbb N$

Define $I_m := \liminf_n A_{m,n}, S_m := \limsup_n A_{m,n}$.

1. Are $I_1, I_2, ...$ independent? What about $I_2, I_3, ...$ ?

2. Are $S_1, S_2, ...$ independent?

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By the Borel 0-1 Law, we have that $P(\limsup_n A_{m,n}) = 0$ or $1$ for $m \ge 2$. Hence, $S_2, S_3, S_4, ...$ are independent.

I don't see why $S_1, S_2, S_3, ...$ would be independent.