According to Kolmogorov 0-1 law, if $ \left(X_{i}\right)_{i=1}^{\infty} $ are independent, then the tail sigma algebra is trivial. I want to construct such variables which are "almost independent": k wise independent, such that the tail sigma algebra is non-trivial.

We can look at $ X_{i}\sim U\left( \left\{ 0,1\right\} \right) $ where we set $ X_{a\left(k+1\right)}=\bigoplus_{i=1}^{k}X_{a\left(k+1\right)-i} $ in order to construct k-wise independent variables, but I can't find an event in their tail sigma algebra which is not trivial. Maybe this is not a good (enough) example of k-wise independent variables, but I can't think of a better one.