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I want to show the following:

Let $X_1,X_2\dots$ be i.i.d. random variables. Let $\text{E}[|X|^p]=\infty$ for $p>0$. Show that $$P(\limsup\limits_{n\to\infty }\{|X_n|\geq n^{1/p}\})=1$$

What I thought so far:

When $$P(\limsup\limits_{n\to\infty }\{|X_n|^p\geq n\})=1$$ Then Borel-Cantelli says that the sum of all independent events in the probability space must be $$\sum\limits_{n=1}^\infty P\{|X_n|^p\geq n\}=\infty$$ otherwise $$P(\limsup\limits_{n\to\infty }\{|X_n|^p\geq n\})\neq 1$$ $$\sum\limits_{n=1}^\infty P\{|X_n|^p\geq n\}=\sum\limits_{n=1}^\infty \text{E}[\{\mathbb{1}_{|X_n|^p\geq n}\}]$$

And here I'm stucked. I have great troubles working with the limes superior of a sequence of random variables, I often don't know what to do, besides using somehow Borel-Cantelli.

Thank you for help

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I often don't know what to do, besides using somehow Borel-Cantelli

Well, more often than not, you don't need to use anything else:)

As for the question: $$ \sum_{n=0}^{\infty}P(|X_n|^p\geq n)=\sum_{n=0}^{\infty}P(|X_1|^p\geq n)\geq \int_0^{\infty}P(|X_1|^p\geq t)dt=E|X_1|^p=\infty. $$

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