Are Normal variables constructed by CDF inverse of uniform variables indepdent?

Question

Let $\Phi$ be the CDF of the normal distribution, and let $u,v,s\sim\mathrm{Unif}[0,1]$ be iid uniform variables, then $X_1:= \Phi^{-1}(u),Y_1:= \Phi^{-1}(v)$ will be independent normal variables, therefore $Z_1:=(X_1+Y_1)/\sqrt{2}$ will follow a normal Gaussian. Now if we shift $u,v$ by $s$ and define $X_2:=\Phi^{-1}(u+s - \lfloor u+s\rfloor ),Y_2:=\Phi^{-1}(v+s - \lfloor v+s\rfloor )$, where $\lfloor \cdot \rfloor $ stands for floor, $Z_2:=(X_2+Y_2)/\sqrt{2}$ will all analogously follow the normal Gaussian distribution. My question is, is $Z_1$ independent of $Z_2$?

Answer 1

As $s \to 0+$, $Z_2 \to Z_1$. So they should certainly not be independent if $s$ is sufficiently small. I would guess that they are dependent for all $s$, but it'll be a bit messy to prove.