Vtyjkyuk

Suppose $X,X',Y,Y'$ are independent random variables. We know that $X+Y oversetd= X'+Y'oversetd=Uniform[0,1]$ and $Xprec X'$ in the sense that $P(X> x)le P(X'>x)$ for any $xin mathbbR$. Is it true that $Y'prec Y$?

This is an answer to the earlier version of the question where $X+Y$ was not required to have uniform distribution. This is false. If this implication is true then the following must also be true: $X,X',Y,Y'$ independent, $X+Y overset d =X'+Y'$ and $Xoverset d=X'$ implies $Yoverset d=Y'$. It is well known that there exist characteristic functions $phi, phi_1,phi_2$ such that $phi (t) phi_1(t)=phi (t) phi_2(t)$ for all $t$ but $phi_1 neq phi_2$. Such an example is available in Volume 2 of Feller's book. [ See Curiosities ii) in Section 2 of the chapter on Characteristic Functions]. Now just consider independent random variables $X,X',Y,Y'$ such that $X$ and $X'$ have characteristic function $phi $, $Y$ has characteristic function $phi_1$ and $Y'$ has characteristic function $phi_2$ to get a contradiction.