limit of average of independent, but not identically distributed r.v.

Question

Let $\{X_i\}$ be a collection of independent r.v., but with distribution dependent on index $i$, such that $P(X_i=2^i)=2^{-i}$ and $P(X_i=0)=1-2^{-i}$ for $i \in \mathbb{N}$.

What can I say about $\lim_{n\rightarrow \infty} \frac{\sum^n_{i=1}X_i}{n}$ ? And what type of convergence would it be?

I don't know any law of large numbers that I can use, since $\operatorname{Var} (X_i)=2^i-1$, which depends on $i$. The only intuition I got is that $X_i\rightarrow^p0$, so, I would expect the limit to converge, but even then I still have no intuition on the value of the limit. I would like to no how one could get an intuition for the value of the limit, and also if there's any helpful Law of Large numbers for this case.

Any help would be appreciated.

Answer 1

Define for an integer $i$ the event $A_i:=\{X_i=2^i\}$. The series $\sum_i\mathbb P(A_i)$ is convergent, hence by the Borel-Cantelli lemma, $\mathbb P\left(\limsup_{i\to \infty}A_i \right)=0$.

Since each $X_i$ takes only two values, this means that for almost every $\omega\in\Omega$, we can find an integer $N(\omega)$ such that if $i\geqslant N(\omega)$, then $X_i(\omega)=0$. Thus the averages converge to $0$ almost surely.