# $\limsup$ sequence independent $\mathcal{N}(0,\sigma^2)$

In my lecture notes there is the following application of Borel-Cantelli's 2nd lemma: Let $(X_n)_{n\geq 1}$ be a sequence of independent $\mathcal{N}(0,\sigma^2)$-distributed random variables, with $\sigma > 0$. From Borel-Cantelli 2, it follows that $$\text{P-a.s}\ \limsup_nX_n = \infty$$

I know that for events $(A_n)$, $\limsup_nA_n = \bigcap_{n\geq 1}\bigcup_{m\geq n} A_m$, but I can't figure out how can I apply the lemma to random variables instead events.

• The question is about a limsup of functions, namely, the function $Y=\limsup X_n$ defined pointwise by $$Y(\omega)=\limsup X_n(\omega).$$
– Did
Commented Jun 13, 2016 at 11:27

Since the $X_n$ have the same distribution we know for any real number $x$, that $$\sum_{n=1}^{\infty}P(X_n>x) = \infty.$$ Therefore, by independence and the second Borel-Cantelli lemma, it follows that $$P(\limsup\{\omega\in\Omega\mid X_n>x\}) = 1.$$ This in turn implies almost surely that infinitely many of the $X_n$ are bigger than $x$, and thus $$\limsup_{n\rightarrow\infty}X_n \ge x.$$ Since we chose $x\in\mathbb{R}$ arbitrary we conclude that the limit superior is infinite.