Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. Consider events indexed by $m, n \in \mathbb N$: $ \ \ \ \ \ \ \ \ \ \ \ A_{1,n}, A_{2,n}, A_{3,n} ...$ are n-wise independent. $A_{m,1}: A_{1,1}, A_{2,1}, A_{3,1}, ...$ are 1-wise independent. $A_{m,2}: A_{1,2}, A_{2,2}, A_{3,2}, ...$ are 2-wise independent. $A_{m,3}: A_{1,3}, A_{2,3}, A_{3,3}, ...$ are 3-wise independent. $\vdots \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ddots$ Do we eventually have mutual independence if we let n increase? $\lim_n A_{m,n}$ does not necessarily exist, but we can define: $$CI_m := \liminf_n A_{m,n}$$ $$CS_m := \limsup_n A_{m,n}$$ 1. Are $CI_1, CI_2, ...$ independent? How about some subsequence $CI_{m^{*}}, CI_{m^{*}+1}, ...$ or $CI_{f(1)}, CI_{f(2)}$ for $f(m): \mathbb N \to \mathbb N$? 2. Are $CS_1, CS_2, ...$ independent? How about some subsequence $CS_{m^{*}}, CS_{m^{*}+1}, ...$ or $CS_{f(1)}, CS_{f(2)}$ for $f(m): \mathbb N \to \mathbb N$? --- **Column sums:** $\forall m \in \mathbb N, P(A_{m,1}) + P(A_{m,2}) + ... \le \infty$. If we can find an $m^{*} \in \mathbb N$ s.t. $A_{m^{*},1}, A_{m^{*},2}, ...$ are at least pairwise independent, then by the Borel 0-1 Law, we have that $P(\limsup_n A_{m,n}) = 0$ or $1$ for $m \ge m^{*}$. Hence, $CS_{m^{*}}, CS_{m^{*} + 1}, CS_{m^{*} + 2}, ...$ are independent. If $m^{*} = 1$, then $CS_1, CS_2, ...$ are independent. Is that right? How can we prove or disprove the existence of such a $m^{*}$? --- **Row sums:** Not sure if this helps, but meanwhile it seems that $\forall n \ge 2, P(A_{1,n}) + P(A_{2,n}) + ... \le \infty$ and the $A_{1,n}, A_{2,n}, ...$'s are at least pairwise independent. If so, then by the Borel 0-1 Law, we have that $P(\limsup_m A_{m,n}) = 0$ or $1$ for $n \ge 2$. Hence, we can define $$RI_n := \liminf_m A_{m,n}$$ $$RS_n := \limsup_m A_{m,n}$$ $RS_2, RS_3, RS_4, ...$ are independent.