Does partial averaging allow moving increments in and out of an expectation?

Question

Given a Levy process $X$ at different points in time $s$ and $t$, and if I have an expression like this:

$$\mathbb{E}[X_t \cdot \mathbb{E}[X_s]]$$

I want to know if I can use partial averaging to say that this is equal to

$$\mathbb{E}[X_t\cdot X_s].$$

Is that correct?

Answer 1

let $X$ be a Levy process. Note that $\Bbb E[X_s]$ is then a real number. Therefore $$ \Bbb E[\Bbb E[X_s] \cdot X_t] = \Bbb E[X_s] \cdot \Bbb E[X_t]. $$

As mentioned in the comments, this is (in general) not equal to $\Bbb E[X_s \cdot X_t]$.