Assume that we have observations for a random process $X(t)$. We would like to express $X(t)$ in the following form:
$$X(t)\approx f(t) + g(t)\chi $$ where $f(t)$ and $g(t)$ are deterministic, and $\chi$ is a random variable which does not depend on time. We do not know anything about the form of $f$ and $g$. How can we do it only from the observations on $X(t)$? Is it possible?
This is similar to the following: Let $X_1, X_2, \cdots, $ be i.i.d. random variables with $E(X_i)=\mu$ and $SD(X_i)=\sigma$. Let $S_n=X_1+X_2+\cdots+X_n$. Then $$ S_n\approx \mu n+ \sigma \sqrt n Z$$ where $Z$ is standard normal. But in this set up, of course, we know how to compute the deterministic parts $\mu n$ and $\sigma\sqrt n$, and $Z$ follows from CLT.