Conditions for a decaying power spectral density

Question

I am currently working with the concept of the power spectral density $S_X(\omega)$ defined for a stochastic process $X_t$ as $$S_X(\omega):=\lim_{T\rightarrow\infty}\mathbb{E}\left[\frac{1}{T}\left|\int_0^T\exp(-i\omega t)X_tdt\right|^2\right]$$ Do you know of any condition that allows one to make the statement $S_X(\omega)\xrightarrow{\omega\rightarrow\infty}0$? If $X_t$ is a stationary process, then one can apply the Wiener Khinchin theorem and see whether the auto-correlation function delivers such a power spectral density. But what if the process is not stationary? I feel like demanding continuous paths plus some extra regulation on the behavior of $X_t$ for large $t$ should be enough but so far I haven't been able to put my finger on it.