I'm stucked with this exercise.
Let $X_1,X_2,\ldots$ be i.i.d. random variables with $E(X_1)=0$ and $Var(X_1)=\infty$
Prove that$$P(\limsup\limits_{n\to\infty}\{|X_n|\geq \sqrt{n}\})=1$$
I need to show $$\sum\limits_{n=1}^\infty P(|X_n|\geq \sqrt{n})=1$$ so I can use Borel-Cantelli.
I know from the law of large numbers that
$$\sum\limits_{n=1}^\infty \frac{X_n}{n}=E(X_1)=0$$
So this means that $$\lim\limits_{n\to \infty}\frac{X_n}{c_n}=0\quad \text{or} \quad P(\lim\limits_{n\to \infty}\frac{X_n}{c_n}=0)=1$$
I don't know how to go further.
Best Answer
First of all, note that you have to show
$$\sum_{n \geq 1} \mathbb{P}(|X_n| \geq \sqrt{n}) = \color{red}{\infty} \tag{1}$$
in order to apply the Borel-Cantelli-lemma.
Suppose that $(1)$ does not hold true, i.e. $\sum_{n \geq 1} \mathbb{P}(|X_n| \geq \sqrt{n}) < \infty$. Since the random variables are identically distributed, this implies
$$\sum_{n \geq 1} \mathbb{P}(|X_1|^2 \geq n)< \infty. \tag{2}$$
Now recall that for any non-negative random variable $Y$ we have
$$\mathbb{E}(Y) \leq \sum_{n \geq 0} \mathbb{P}(Y \geq n)$$
(this is a direct consequence of the pointwise inequality $Y \leq \sum_{n \geq 0} 1_{\{Y \geq n\}}$). Applying this for $Y= |X_1|^2$, we conclude from $(2)$ that
$$\mathbb{E}(X_1^2) \leq \sum_{n \geq 0} \mathbb{P}(|X_1|^2 \geq n)<\infty$$
in contradiction to our assumption that $X_1$ has infinite variance.