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With the natural filtration ${\mathcal{F}}_n=\sigma (Z_1,\dots ,Z_n)$, the sums form a martingale. Put $v_n={\sum }_{k\le n}{a}_k^2$. Independence and the characteristic function of a standard normal random variable giveIf $M_n\to M_{\infty }$ almost surely with a finite limit, dominated convergence gives convergence of these characteristic functions to $\mathbb{E}{e}^{iu{M}_{\infty }}$. If $v_n\to \infty$, that limit would be $1$ at zero and $0$ at every nonzero $u$. A characteristic function is continuous at zero: apply dominated convergence to $e^{iu{M}_{\infty }}$ as $u\to 0$. This contradiction provesThis is also sufficient by the L2-bounded martingale convergence theorem, since $\mathbb{E}{M}_n^2=v_n$. Thus the almost sure convergence criterion for an independent Gaussian series is an equivalence.