# Law of Large Numbers when expectation is infinite and random variable not identical

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!

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