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Assume $X_{1}, X_{2}, X_{3}, ...$ be pairwise independent random variables with $N(0,1)$ distribution. Prove:

$$ P \left( \limsup_{n \to \infty} \frac{X_{n}}{\sqrt{\log n}} = \sqrt{2} \right) = P \left( \liminf_{n \to \infty} \frac{X_{n}}{\sqrt{\log n}} = -\sqrt{2} \right) = 1 $$ and all points in $[-\sqrt{2}, \sqrt{2}]$ are accumulation points.

Martin Sleziak
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mathsstudentTUD
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  • To be honest I have no clue and would really appreciate any hint...By the way this problem is marked as very challenging on my assignment sheet... – mathsstudentTUD Nov 14 '17 at 22:09
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    Relevant: https://math.stackexchange.com/questions/987604/a-reference-for-a-gaussian-inequality-mathbbe-max-i-x-i – Clement C. Nov 14 '17 at 22:13
  • I know what $\lim$ is and what $\liminf$ is but what is $\limsup$? – Mr Pie Nov 14 '17 at 22:24
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    @user477343 if you know $\lim\inf$, you should know $\lim\sup$. It's the similar definition, but for the upper bound (supremum) instead of lower bound (infimum). – Clement C. Nov 15 '17 at 04:29
  • Now I am even more confused. What is meant by my Professor: $$P \left( \limsup_{n \to \infty} \frac{X_{n}}{\sqrt{\log n}} = \sqrt{2} \right) = P \left( \left{ \omega \in \Omega : \limsup_{n \to \infty} \frac{X_{n}(\omega)}{\sqrt{\log n}} = \sqrt{2} \right} \right) =1$$ or is it:

    $$P \left( \limsup_{n \to \infty} \frac{X_{n}}{\sqrt{\log n}} = \sqrt{2} \right) = P \left( \bigcap_{n=1}^{\infty} \bigcup_{ m \geq n }\left{ \omega \in \Omega : \frac{X_{n}(\omega)}{\sqrt{\log n}} = \sqrt{2} \right} \right) =1$$

     – mathsstudentTUD Nov 15 '17 at 11:09
  • ?? The first option, of course. (But what about adding your personal tries to solve this problem, to the question? You know, to avoid the feeling that you take the site for a free answering machine...) – Did Nov 15 '17 at 11:55
  • @Didier Regarding my question above my confusion was triggered by my tutor who insisted that the limsup was meant in terms of sets...As I said in another post I am quite new to probability theory. So for now please excuse me if I ask trivial questions...I am very thankful for this forum and all of you who help me...I did indeed spend thoughts about this problem but all of that led to nowhere, at least so far (Borel-Cantelli, SLLN, Chebyshev..). Thats why I posted this problem to get some idea in what direction to go...Again as I said I am trying my best and am thankful for all of you. – mathsstudentTUD Nov 15 '17 at 12:09

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