Let $X(t)$ be a continuous-time, mean-zero, wide-sense stationary (covariance stationary) stochastic process, and $p(t)$ be an integrable function.
Is there a general formula for the distribution of $Y(s) = \int_{-\infty}^\infty X(s-t)p(t)dt$?
I know there are problems with the existence of these integrals, but for now let's assume $p(t)$ is nice enough so the integral exists.
So far I have learned to calculate the mean and covariance of $Y(s)$ using the Wiener–Khinchin theorem. My guess for calculating the distribution of $Y(s)$ is to use the Karhunen–Loève theorem to write $X(t)$ as summation, then see if I can integrate each term against $p(t)$.
Thanks in advance.