# Relationship between autocorrelation function and characteristic function

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 $\phi_X$?

• $t$ in your question is used are argument of $\phi_X(t)$ and also as parameter/argument of $X(t)$. That is confusing. – drhab May 24 '18 at 11:33
• Why would you expect this? The characteristic function $\phi_X$ sees only the distribution of $X$ whereas the the autocorrelation function depends on the joint distribution of $(X(x),X(x'))$. – saz May 24 '18 at 15:20
• Does it mean that the autocorrelation is related to the characteristic function of products of random variables? – adamG May 24 '18 at 15:33