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!