The limsup of events $A_1, A_2, …$ is $\limsup A_n = \bigcap_{m\geq1} \bigcup_{n\geq m} A_n$
Is there a limsup for random variables $X_1, X_2, …$? I've seen $\limsup X_n$ sometimes but it usually precedes "= 5" or "$\geq \alpha$" thus referring still to a $\limsup$ or events namely $\limsup (X_n = 5)$ and $\limsup (X_n \geq \alpha)$, respectively. I mean, $(\limsup X_n) = 5$ and $(\limsup X_n) \geq \alpha$ don't make sense, do they?
According to this, the limsup of random variables is the same:
$\limsup X_n = \bigcap_{m\geq1} \bigcup_{n\geq m} X_n$
What does $\bigcup_{n\geq m} X_n$ even mean? I was thinking the equation was supposed to be $\bigcup_{n\geq m} \sigma(X_n)$, but that's taken…
Best Answer
The liminf / limsup for a sequence of random variables is defined pointwise: For each realization $\omega$ consider the sequence $\{X_n(\omega)\}$as $n$ varies, and take the real-valued liminf / limsup.
In other words: $\limsup X_n$ is a random variable defined by: $$ (\limsup X_n)(\omega) := \limsup [X_n(\omega)]\;. $$
In the link you've cited, the $X_n$ are set-valued (see the example down the page), so those $X_n$'s are not random variables.