Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $E_a$, $F_b$)?

share|improve this question
1  
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
add comment

1 Answer 1

For the sake of having an answer:

  • $E_x\subset F_y$ is guaranteed if and only if $y\leqslant x$.
  • $F_x\subset E_y$ is guaranteed if and only if $y\lt x$.
share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.