# 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.