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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...

BCLC
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1 Answers1

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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.

grand_chat
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  • 1 So the limsup of a sequence of random variables depends on omega? Like for omega1, the limsup is 1, for omega2, the limsup is -6, etc? 2 Oh thanks for pointing that out. So just to clarify the term "$limsup X_n$" is meaningless? – BCLC May 14 '15 at 17:59
  • Wait, the term $\limsup X_n$ is not meaningless. The limsup of a sequence of random variables is itself a random variable, and the above is a recipe for calculating it. Think of the limsup of a sequence of functions; it's also defined pointwise. – grand_chat May 14 '15 at 20:45
  • I'll assume the answer to 1 is yes. So while $(limsup X_n) = 5$ is not meaningless, it is false if $limsup X_n$ is not constant? So such things don't arise very often? I can't recall right now, but I usually see things like $P(\limsup X_n > 10)$ and usually it doesn't help to interpret as $P((\limsup X_n) > 10)$ but it usually does for $P(\limsup (X_n > 10))$. – BCLC May 14 '15 at 20:53
  • The answer to 1 is yes. And generally $limsup X_n$ is not constant. Limsups arise very often because they exist for any sequence of RVs, whereas lims do not. – grand_chat May 14 '15 at 20:59
  • Thanks grand_chat. Do you know any example involving P((lim supXn)>10) and not P(lim sup(Xn>10))? – BCLC May 14 '15 at 21:32
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    Not exactly involving 10, but the probability $P(\limsup X_n<\infty)$ is often of interest, whereas $P(\limsup (X_n<\infty))$ is not (since it's always equal to 1). – grand_chat May 14 '15 at 21:37
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    On the question of the interpretation of $P(\limsup X_n >10)$: it should be $P((\limsup X_n)>10)$, if there are no parentheses to clarify. The event ${\limsup X_n>10}$ is the set of outcomes $\omega$ where the sequence $X_n(\omega)$ has a limsup greater than 10. The other event $\limsup {X_n>10}$ is the set of outcomes $\omega$ where $X_n(\omega)$ exceeds 10 infinitely often. The two events are kind of talking about the same thing, which might be a source of confusion about which $\limsup$ is intended. – grand_chat May 14 '15 at 21:39
  • grand_chat, is one of those events myb a subset of the other? http://math.stackexchange.com/questions/1522359/what-is-the-difference-b-w-colorred-limsup-w-k-k-colorred-le-1 – BCLC Nov 10 '15 at 11:59