Suppose $ X $ is a random variable. Then there is an associate characteristic function $\phi_X(t)=E(e^{itX})$. Then we know that the moments can be expressed as derivatives of $\phi_X$, $$EX=\frac{d\phi_X}{idx}\biggr|_{t=0},\quad EX^2=\frac{d^2\phi_X}{i^2dx^2}\biggr|_{t=0} \text{ and so on}.$$
My question is, is there such a relationship between the autocorrelation function $$E(X(x)X(x'))$$ or higher order versions $$ E(X(x)X(x')X(x''))$$ and the characteristic function $\phi_X$?