Coupling for tail bounds

Question

Let $X_1, X_2$ be two random variables such that $\mathbf{E}X_1=\mathbf{E}X_2=0$ and $$\forall x\geq 0, \quad \mathbf{P}[X_1\geq x]\leq \mathbf{P}[X_2\geq x].$$ Can we construct a coupling $(Y_1,Y_2)$ such that $Y_1=^dX_1$, $Y_2=^dX_2$ and a.s. $$Y_1\mathbf{1}_{\{Y_1>x\}} \leq Y_2\mathbf{1}_{\{Y_2>x\}} \tag{1}$$ for all $x$ large enough? If not, can we construct a coupling such that $(1)$ is true at the limit $x\uparrow+\infty$?

Answer 1

Here is an algorithm to sample a random variable from a distribution function $F$:

(1) Sample $Z$ uniformly in $[0,1]$.
(2) Take $X=\inf\{x : F(x)\ge Z\}$.

It is a simple exercise to check that indeed, $X$ has distribution function $F$. Note that if $F$ was injective, we could simply take $X=F^{-1}(Z)$.

---

We can perform this construction in parallel to get a nice candidate coupling for your problem. Let $F_1$ and $F_2$ be the distribution functions of $X_1$ and $X_2$, respectively. Take $Z$ uniform in $[0,1]$, $$Y_1=\inf\{x : F_1(x)\ge Z\}$$ and $$Y_2=\inf\{x:F_2(x)\ge Z\}.$$ We already know that this is a coupling of $X_1$ and $X_2$. But by assumption $F_1(x)\ge F_2(x)$ for all $x\ge 0$, so that a.s. $$ Y_1 1_{Y_1>0}\le Y_2 1_{Y_2>0}. $$