If we had independent events $E_1, E_2,...$ we'd just say that $\tau_{\{E_n\}}$, of which $\limsup E_n$ is an element, is $\mathbb P$-trivial by Kolmogorov 0-1 Law and thus is independent of anything.

For independent random variables $X_1, X_2, ...$, as an alternative to these:

Prove that limsup and liminf of an independent sequence are independent of finite number of terms

Showing independence of a limsup of an independent sequence

can we just say that $\limsup X_n$ is something like the random variable equivalent of $\mathbb P-$ trivial and thus independent of anything? What is the random variable equivalent of $\mathbb P-$ trivial?


See Part II of Kolmogorov 0-1 Law in Williams.

Just as $\limsup E_n$ is $\mathbb P-$trivial and thus independent of any event/s, $\limsup X_n$ is constant $\mathbb P-$a.s. and thus independent of any random variable/s or $\sigma-$algebra/s.


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