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

Mathematics Asked on January 1, 2022

Let $Phi$ be the CDF of the normal distribution, and let $u,v,ssimmathrm{Unif}[0,1]$ 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 $X_2:=Phi^{-1}(u+s – lfloor u+srfloor ),Y_2:=Phi^{-1}(v+s – lfloor v+srfloor )$, 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 $Z_2$?

One Answer

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.

Answered by Robert Israel on January 1, 2022

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