For every $t$, let $Y_t=\log(Z_t^2)$. Fix some $t$. The sequence $(Y_{t-k})_{k\geqslant0}$ is i.i.d. with $E[Y_t]\lt0$ hence the usual law of large numbers yields $\frac1j\sum\limits_{k=0}^{j-1}Y_{t-k}\to E[Y_1]$. Fix some negative $m\gt E[Y_1]$.

Then $\frac1j\sum\limits_{k=0}^{j-1}Y_{t-k}\leqslant m$ for every $j$ large enough, that is, for every $j\geqslant J$ where $J$ is random and almost surely finite. In particular, for every $j\geqslant0$, $\sum\limits_{k=0}^{j}Y_{t-k}\leqslant mj+X$, for some almost surely finite random $X$. This implies the pointwise convergence of the series since
$$
\sum_{j\geqslant0}\exp\left(\sum\limits_{k=0}^{j}Y_{t-k}\right)\leqslant\sum_{j\geqslant0}\mathrm e^X\mathrm e^{mj}=\mathrm e^X(1-\mathrm e^m)^{-1}.
$$
Note that the RHS above is almost surely finite since $m\lt0$ but is not (a priori) uniformly bounded since $X$ may be unbounded.