Before I go to the statement of my question I just want to say a few words about the personal background of this question. I have recently taken a course on stochastic differential equations without prior exposure to the cornerstones of this subject such as measure theoretic probability, real analysis and so on. In contrast with my expectations at the beginning of the course I got one of the highest grades in a class filled with math majors at graduate level. One way or the other I have to say heartfelt thanks to saz and Did and this seems like a place as good as any. There is no way I could have gotten this far in this subject without their help. I felt like this had to be acknowledged somehow. Anyhow, with the knowledge and the confidence I got from this course I decided to study a rigorous treatment of time series on my own. To my surprise and also disappointment to a certain extent, this undertaking is not the smooth sailing I expected it to be. So here is my question.
Let $(Z_t)_{t \in \mathbb{Z}}$ be an i.i.d sequence with $E[\log(Z_t^2)] < 0 $ for each $t \in \mathbb{Z}$. Show that $$\sum_{j=0}^{\infty}Z_t^2Z_{t-1}^2\cdots Z_{t-j}^2 < \infty \quad \text{ almost surely}$$ There is also a hint that I should consider the law of large numbers.
My previous attempt was based on the monotone convergence theorem, which was proved wrong with a counterexample given by d.k.o. So I have erased that attempt.
This problem wants you to use the (strong) law of large numbers (LLN) on the individual terms. Recall by the LLN and the definition of convergence to a limit "with probability 1" that: If $\{W_i\}_{i=1}^{\infty}$ is a general sequence of i.i.d. random variables and $E[W]$ is finite, then (with prob 1) for any $\epsilon>0$ we can find a positive integer $M_{\epsilon}$ such that: $$ \left|E[W]-\frac{1}{n}\sum_{i=1}^n W_i\right| \leq \epsilon \: \: \: \: \forall n \geq M_{\epsilon} $$ You can apply this to the $Z_t \cdots Z_{t-j}$ product. In this case, $t$ never changes, so you can define $W_0=\log(Z_t^2), W_1=\log(Z_{t-1}^2), W_2=\log(Z_{t-2}^2)$, and so on. Then you can fix a sufficiently small $\epsilon>0$ (how small should it be?) and compute bounds on: $$ term_j \equiv \prod_{i=0}^j Z_{t-i}^2 = e^{\frac{j+1}{j+1} \log\prod_{i=0}^jZ_{t-i}^2} = e^{(j+1)\left[\frac{1}{j+1}\sum_{i=0}^jW_i \right]}$$
Then argue that $\sum_{j=0}^{\infty} term_j < \infty$ (with prob 1).