Let $X, Y: \Omega \to \mathbb{R}$ be two random variables with respective cumulative distribution functions $F_X, F_Y$ respectively. I was wondering whether there's an inequality possible relating the two functions:
(1) $|X-Y|: \Omega \to \mathbb{R}$ and (2) $|F_X - F_Y|: \mathbb{R} \to [0,2]$. (Note that the domains of (1), (2) are different!) In both of the cases, you can replace the pointwise difference by some kind of norm (e.g. sup norm, $L^p$ norm etc.).
Now, I'm aware that: $F_X - F_Y \ne F_{X-Y}$, but still hoping that if $|F_X - F_Y|$ is small on $ K \subset \Omega$, then $|X-Y|$ will also be small on a neighborhood of $X(K)$, and vice versa. Or, at least a form of probabilistic inequality like:
If $|F_X-F_Y| \le \epsilon$ uniformly on a compact subinterval $I \in \mathbb{R},$ then there exists a $\delta \equiv \delta(\epsilon) > 0$ so that: $P[\{\omega \in X^{-1}(K): |X(\omega) - Y(\omega)| > \delta] \le \epsilon$
It's perhaps not possible in general, but is there a condition on $X, Y$ that'll make it possible? Thanks!!