= Almost sure convergence criterion for an independent Gaussian series
{title2=$\sum_ka_kZ_k\text{ converges a.s.}\iff\sum_ka_k^2<\infty$}
For <independent> standard <Gaussian random variables>, the partial sums have <characteristic functions> $\exp(-u^2v_n/2)$, where $v_n=\sum_{k\leq n}a_k^2$. A finite almost-sure limit implies convergence of these <characteristic functions> to one continuous at zero; $v_n\to\infty$ would instead give zero at every nonzero $u$. Conversely, bounded $v_n$ makes the partial sums an <martingale> bounded in <L2 space>, which converges <almost surely> and in <L2 space>.
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