# Stochastic dominance, coupling

Let $X,Y$ be positive random variables on some probability space $\Omega$ such that $\mathbb{P}(X>x)\leq \mathbb{P}(Y>x)$. Can one remove a set of measure zero (with respect to both distributions) from $\Omega$ so that in the new probability space we could couple $X,Y$ so that we would have $Y=X+Z$, where $Z$ is some positive random variable?

• Did you try a maximal coupling of $X$ and $Y$? – Nawaf Bou-Rabee Jan 4 '18 at 17:44
• Not sure what you mean: take $[0,1]$ as your probability space and define $\tilde{X}(\omega) = F_X^{-1}(\omega)$, $\tilde{Y}(\omega) = F_Y^{-1}(\omega)$, where $F_X^{-1}$ denotes the (generalised) inverse of the distribution function $F_X(x) = \mathbb{P}(X < x)$, and similarly for $F_Y^{-1}$. – Mateusz Kwaśnicki Jan 4 '18 at 21:53