Let $(X_n)_{n=1}^\infty$ be independent, $N(0,1)$-distributed random variables. Prove that $$ \limsup_{n \to\infty}{X_n \over \sqrt{2 \log(n)}} = 1 \ \text{almost surely}.$$
I am aware of the process involved in doing this, but I am just stuck on one bit.
I have defined $A_n = \{X_n \ge a\sqrt{2 \log(n)} \}$, for a real parameter $a>0$. I then wish to apply the Borel-Cantelli lemmas to the case $\sum_{n=1}^\infty \Bbb P(A_n) < \infty$ and $\sum_{n=1}^\infty \Bbb P(A_n) = \infty$. The aim is that these occur for $a > 1$ and $a = 1$, respectively. This then gives the required result, by Borel-Cantelli's lemmas. The issue is showing that the sum converges if and only if $a>1$.
Note: There is this question, here, but I don't want to use the inequalities that (s)he has used, as I did not know this before I read that question. However, that may be the only way...?
Any help would be most appreciated. I have only got as far writing the probability as an integral.
Update: I can do it now using the inequalities in the question linked above. Is this just the way that is has to be done, or is there another way? (If this is the only way, then perhaps it's best that I remove this question, as it is then of no benefit over the linked one, but not a duplicate.)
\limsup_{n \to\infty}. – Did Jan 07 '15 at 22:50\tolooks the same as\rightarrow, and is the more semantic choice when you mean "to" and not just some arrow pointing to the right. As for the question itself: you need good estimates on the tail probabilities $\mathbb P{X_n \ge t }$ in order to get anywhere. This answer gives asymptotic behaviour of this tail probability, which is not hard to derive and should be enough for your purpose. – Jan 10 '15 at 17:44