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I am trying to prove the following question:

Let $X_i$ be sequence of indepedent non-negative random variables with $E(X_i) = \infty$ and $X_i$ having different distributions.

Is it true that $$ \frac{X_1 + \cdots + X_n}{n} \to \infty \ a.s.? $$ I know that this is true when $X_i$ are i.i.d., but I have no idea how to proceed with $X_i$ having different distributions.

Any hint would be appreciated. Thank you!

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Actually we can even have $\frac {X_1+X_2+...+X_n} n \to 0$ almost surely!.

Let $(Y_n)$ be i.i.d non-negative with $EY_1=\infty$. Choose $a_n >0$ such that $P (\frac {Y_n} {a_n} >\frac 1 {2^{n}} )<\frac 1 {2^{n}}$. Use Borel Cantelli Lemma to conclude that $\frac {Y_n} {a_n} \to 0$ almost surely. Now put $X_n=\frac {Y_n} {a_n}$. Since $X_n \to 0$ almost surely it follows that the Cesaro averages $\frac {X_1+X_2+...+X_n} n$ also $\to 0$ almost surely.

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  • $\begingroup$ Hi, I'm a bit confused as to the introduction of $a_n$ here. Why not simply choose $X_n$ such that $P(X_n > 1/2^n) < 1/2^n$? $\endgroup$ Jul 1, 2020 at 6:06
  • $\begingroup$ @davidolohowski Your inequality is satisfied by $X_n=0$ but then the condition $EX_n=\infty$ is not satisfied. $\endgroup$ Jul 1, 2020 at 6:10

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