Let $X_n$ be an i.i.d. sequence of positive random variables with expectation 2 and variance 1.
What is the almost sure limit of $$\log(X_1 + X_2 + ... + X_n) - \log(n)$$ as $n \to \infty$
Would it be just $$\log \left( \frac{X_1 + X_2 + ... + X_n}{n} \right)$$ which by strong law of large numbers is then $$\log(\mu) = \log(2)$$