# $\lim \sup\{X_n\geq x\}$ vs $\{\lim \sup X_n \geq x\}$

Let $(X_n)$ (n is a natural number) be a sequence of real valued random variables. For any real number $x$, let's define: $E_x = \limsup \{ X_n \geq x\}$, $F_x = \{\limsup X_n \geq x\}$

If $x$ is fixed, is one of the following Relations valid (why?): $E_x \subset F_x$, $E_x \supset F_x$ or $E_x=F_x$

What if, for example, $a < b$ (for $E_a$, $F_b$)?

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You really need to show at least SOME of the things you tried to solve this question, otherwise the whole shebang is becoming quite ridiculous. – Did Mar 20 '13 at 18:21
I am failing to understand how this is related to stochastic analysis. . $F_x \subset E_x$ is false. Take a sequence $x_n$ converging up to $x$, but $x_n<x$ then this event belongs to $F_x$ $E_x \subset F_x$ is true. prove this by taking a sequence which exceeds $x$ infinitely often, it has a $\limsup$, by the definiton of $\limsup$... I will leave you to figure out the rest... – Lost1 Mar 20 '13 at 20:15
@Lost1 stochastic-analysis is inappropriate but beware that the $X_n$ are random variables, not numbers. (Also, it seems that your comment misapplies $\in$ for $\subset$.) – Did Mar 20 '13 at 20:17
@Did I corrected it but assuming these sequences are constructed on some probability space $\Omega$, we can surely look it per $\omega\in\Omega$ in each of these sets. Although it is possible they only differ up to a null set, (which I have not thought about more carefully) whether these events are included in each other does not differ from a real sequence? – Lost1 Mar 20 '13 at 20:20
Thank you guys! :) And in fact, if a < b, the same relation is correct as in the x-case, right? – JohnD Mar 20 '13 at 20:44

• $E_x\subset F_y$ is guaranteed if and only if $y\leqslant x$.
• $F_x\subset E_y$ is guaranteed if and only if $y\lt x$.