Limit involving a sequence of non-integrable random variables

Question

Let $(\Omega,\mathcal{A},P)$ be a probability space. Moreover, let $(X_n)_{n \in \mathbb{N}}$ be a sequence of i.i.d. random variables that are not integrable. Next, fix $K>0$ and set $C_n:=\{|X_n| \geq nK\}$ for any $n \in \mathbb{N}$. Do we have $$P\left(\limsup_{n \to \infty} C_n \right)=1?$$

Answer 1

If $Y$ is a non0negative random variable then $$EY\leq 2\sum_{n=1}^{\infty} P\{Y\geq n\}$$: $$EY=\sum EYI_{n\leq Y <n+1}\geq \sum (n+1)P\{n\leq Y <n+1\}\leq \sum 2nP\{n\leq Y <n+1\}$$. Now $\sum_{n=1}^{\infty} P\{Y\geq n\}=\sum_n \sum_{j=n}^{\infty} P\{j\leq Y <j+1\}=\sum_j \sum_{n=1}^{j} P\{j\leq Y <j+1\}$. We have proved that this last quantity is $ \geq \frac 1 2EY$. Now we apply this to the given situation. $E|X_1|=\infty$ implies $\frac {E|X_1|} K=\infty$ which implies $\sum_{n=1}^{\infty} P\{|X_1|\geq nK\}=\infty$. Now apply Borel - Cantelli Lemma. We get $P\{\limsup |X_n|\geq nK\}=1$.