Proof clarification: Slepian’s inequality

Question

I was reading about Slepian’s inequality on here (Lemma 7.2.8 on Page 163). In the proof it says we can assume that $X$ and $Y$ are independent. But why is that the case?

I thought about showing that their components are uncorrelated, that is

$$\text{cov}(X_s,Y_t) =0,$$

which I know is equivalent to the independence of $X$ and $Y$.

However I think the reasoning must be different, as the proof is using Gaussian interpolation and the author already states there (see 7.2.1 Gaussian interpolation on page 161), that we can assume independence.

I hope someone could explain it to me.

Sincerely,

Orbit

Answer 1

If $X'$ and $Y'$ are independent copies of $X$ and $Y$, i.e. $X'\overset{d}{=}X$, $Y'\overset{d}{=}Y$, and $X'$ and $Y'$ are independent, showing that $\mathsf{E}f(X)\ge \mathsf{E}f(Y)$ is equivalent to showing that $\mathsf{E}f(X')\ge \mathsf{E}f(Y')$.