The Distribution of Filtered Gaussian White Noise Suppose I have some time series $s(t)$ which contains Gaussian white noise generated by a distribution $N(0,\sigma^2)$

Then I apply a filter to s(t) with a frequency response $H(\omega)$, giving me $s_H(t)$

What distribution does $s_H(t)$ have?

I'm particularly interested in cases where $s_H(t)$ is band limited Gaussian white noise.
If you filter a Gaussian random process with an LTI system, the output will also be Gaussian. You can make intuitive sense of this by considering that a linear combination (which is what filtering does) of jointly Gaussian random variables is a Gaussian random variable.

You can find an in-depth treatment of filtering random processes in this MIT OpenCourseWare document (section 7.4).