I have $\log X \sim Exp(\vartheta - 1)$ and I would like to show
$$ P \Big [ \Big |\frac{1}{\frac{1}{n} \sum_{i=1}^n \log X_i} - (\vartheta - 1) \Big | > \varepsilon \Big ] \rightarrow 0 \hspace{5 mm} \forall \vartheta (n \rightarrow \infty)$$
In the answer to this question it states that all moments of the exponential distribution that are necessary for the strong law of large numbers exist. Therefore $\frac{1}{\frac{1}{n} \sum_{i=1}^n \log X_i}$ converges almost surely towards $\frac{1}{\vartheta - 1}$.
Can someone explain to me why the moments need to exist for the law of large numbers? And how this proof works? Many thanks for your help!