Let $Z_t$ be an i.i.d. sequence with $\mathbb{E} \log(Z_t^2) < 0$.
I want to show that \begin{align} \sum_{j=0}^\infty Z_t^2 Z_{t-1}^2 \ldots Z^2_{t-j} < \infty\ \text{ a.s.} \end{align} using the Law of Large numbers. However I simply do not see how I should find back the average $\frac{1}{n}\sum_{i=1}^n Z_i$ to apply the LoL-numbers theorem. Any help is appreciated!
Using the strong law of large numbers for the i.i.d. sequence $\left(\log\left(Z_{t-j}^2\right)\right)_{j\geq 0}$ ($t$ is fixed), we derive that there exists $\Omega'\subset\Omega$ such that $\mathbb P\left(\Omega'\right)=1$ and for each $\omega\in\Omega'$, there exists an integer $J=J(\omega)$ such that for $j\geqslant J$, $$\frac 1j\sum_{i=1}^j \log\left(Z_{t-i}\left(\omega\right) ^2\right)\leqslant \frac 12 \mathbb E\left[\log\left(Z_{0}^2\right)\right]=:c\lt 0.$$ Therefore, for $j\geqslant J$, $$\prod_{i=1}^j \left(Z_{t-i}\left(\omega\right) ^2\right)\leqslant e^{cj}$$ and the wanted convergence follows.