Let $(\Omega, \mathcal A, P)$ be a probability space. Let $X(\omega,t)$ be a stochastic process such that $X(\omega,.)$ is in $L^2[a,b]$ for every $\omega \in \Omega$. For simplicity, we can assume $a=0$, $b=1$. Let us take a complete ortonormal basis of $L^2[a,b]$, denoted $P_n(x)$ (for example the shifted Legendre polynomials for $L^2[0,1]$). The following integrals make sense for each $\omega \in \Omega$
$$Y_n(\omega)=\int_a^b X(\omega,t) P_n(t) \, dt$$
and define the "random" Fourier coefficients of $X$. If we know the distribution of $X$ for every $t \in [a,b]$, can we find the distribution of $Y_n$ ? What is the literature on this problem and what tools are available ?
You can suppose that $X(w,.)$ is continuous and non-decreasing for each $\omega \in \Omega$ if it simplifies the problem. Or add other hypotheses that do not make the problem trivial.
Intuitively, we feel that this integral is a Riemann sum for each $\omega$, which might hint that we must use a CLT to find the distribution of $Y_n$, but independence does not make sense for continuous time processes. Maybe a martingale CLT ? I never used one of those.
Edit: it would be also interesting to fix the distributions of the $Y_n$'s and try to retrieve the distributions of $X$ at each $t$.