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let $\{X_i\}$ be a sequence of non-negative, i.i.d random variables with $E[X_1] = \infty$. Prove that $\lim\limits_{n \to \infty} \frac{X_1 + \cdots + X_n}{n} = \infty$ almost surely.

There is a similar question here.

What I thought: maybe a suitable application of 0-1 law would work?

Hamilton
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1 Answers1

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For all $p>0$, $$ \liminf_{n\to+\infty}\left(\frac{X_1+\cdots+X_n}{n}\right)\ge\liminf_{n\to+\infty}\left(\frac{X_1\wedge p+\cdots+X_n\wedge p}{n}\right)=\mathbb E[X_1\wedge p]\underset{p\to+\infty}{\longrightarrow}+\infty. $$

Will
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  • So in the last equation, you truncated the random variables so you can make use of the Strong law of large numbers?(That is $\liminf$ is just $\lim$), is that correct? – Hamilton Dec 06 '22 at 00:40
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    Yes the last equality is the usual Strong law of large numbers applied with the sequence $(X_n\wedge p)_{n\in\mathbb N}$, and indeed since the limit exists it is equal to the lim inf. – Will Dec 06 '22 at 01:04