Applying Law of Large Numbers to Find Almost Sure Limit

Question

I am working on the following problem:

Let $\xi_{1},\xi_{2},...$ be a sequence of independent random variables such that $P(\xi_{i} = 2^i ) = \frac{1}{2^i}$ and $ P(\xi_{i} = 0) = 1 − \frac{1}{2^i}$ , $i ≥ 1$. Find the almost sure value of the limit $\lim_{n\to \infty}(\xi_{1}+...+\xi_{n})/n$.

This is clearly an application of the Law of Large Numbers, but how do I determine what the limiting value is?

Answer 1

Hint: by the (first) Borel-Cantelli lemma, we can find $\Omega'$ of probability one such that for any $\omega\in\Omega'$, there exists an integer $N(\omega)$ for which $\xi_i(\omega)\neq 2^i$ for any $i\geqslant N(\omega)$. What does it imply for the sequence $\left(\xi_{i} (\omega)\right)_{i\geqslant 1}$?