# 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 $$Z_2$$?

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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