Does anyone have an idea how to prove the following? It is a step in the proof of some theorem in a book about gaussian processes.

Let $f_n$ be an orthonormal sequence of gaussian variables. Consider $\sum_{n \geq 1} a_n f_n$ and assume that it is convergent in $L^2$. Show that it converges also almost everywhere.

The hint is that every orthonormal family of gaussian variables is automatically independent, but I have no idea how to proceed.

jointlyGaussian variables is independent, but it's easy to construct a nonindependent orthonormal sequence of random variables, each of which is Gaussian. $\endgroup$ – Mark Meckes Jul 22 '10 at 14:38