If two PSDs are related, are the PDFs of the time-series related?

Question

Suppose a power spectral density (PSD) is related to another PSD by a function $\alpha(\omega)$. $$S_a(\omega) = \alpha(\omega)S_b(\omega)$$ If we generate an ensemble of time-series $x_a(t), x_b(t)$ from each of these PSDs, using the inverse Fourier transform (rather IFFT) with randomized phases, will the probability density function(PDF) (or Cumulative distribution function(CDF)) of the two be related,$$P(x_a(t)) = H\times P(x_b(t))$$ for some condition on $\alpha(\omega)$. Can this be shown analytically/ semi-analytically?

Answer 1

See section 8.2 in this book.

$S_a$ is the cross-power spectral density of filtering a wide sense stationary process with power spectral density through a linear time invariant filter with transfer function $\alpha(w)$ (for appropriate $\alpha$).

Note that the PSD is only a second moment characterization of a random process -- the PSD doesn't capture higher order information (e.g. what the third moments of $x_a$ or $x_b$ are), so $H$ cannot be uniquely determined in general (in fact, you don't even know what the distribution of $x_a$ is, just given the power spectral densities) but if the process $x_a$ is Gaussian+WSS, $x_b$ will be jointly Gaussian with $x_a$ and you have a complete characterization of what $H$ is provided you know what the mean of $x_a$ is).