If I have i.i.d. r.v.'s $X_1, X_2,...$ then the tail sigma algebra is defined as $\mathcal{T}:=\cap_n\sigma_n$ where $\sigma_n:=\sigma(X_n,X_{n+1},...)$. From this we get very nice results such as the Kolmogorov 0-1 law. I was wondering if it makes sense to consider limsup and liminfs in this fashion: $\bigcup_n\bigcap_{k\ge n}\sigma_k$ and $\bigcap_n\bigcup_{k\ge n}\sigma_k$. Do these have a sensible meaning and are there similar laws such as the 0-1 law pertaining to each?
Generalizations of Tail Sigma Algebras
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probability-theory