Let $P$ and $Q$ are sub-Gaussian distributions on $\mathbb R$, and $(X,X')$ be a coupling of $P$ and $Q$, i.e $(X,X') \sim \pi$ for some distribution on $\mathbb R^2$ with marginals $P$ and $Q$.



> 
> **Question.** Does $|X-X'|$ have any concentration properties ?
>  Can one reasonably bound $\mathbb P(|X-X'| > \epsilon)$ for $\epsilon > 0$ ?
> 
> 
> 


Thanks in advance for any help.



Tail bounds for the absolute difference of a coupled pair of sub-Gaussian random variables
For $X\_1:=X$, $X\_2:=X'$, some positive real $c\_1,c\_2,a\_1,a\_2$, and all positive real $t$ we have 
$$P(|X\_j|>t)\le c\_j e^{-a\_j t^2} \tag{1}$$
for $j=1,2$. So, for $t:=\epsilon>0$,
$$P(|X\_1-X\_2|>t)\le P(|X\_1|>t/2)+P(|X\_2|>t/2) \\ 
\le c\_1 e^{-a\_1 t^2/4}+c\_2 e^{-a\_2 t^2/4}.$$




---


The OP commented that the notion of a sub-Gaussian distribution was meant not in [the usual sense](https://en.wikipedia.org/wiki/Sub-Gaussian_distribution). Namely, according to that comment, condition (1) should be replaced by 
$$P(|Y\_j|>t)\le c\_j e^{-a\_j t^2}, \tag{2}$$
where 
$$Y\_j:=X\_j-m\_j$$
for some real $m\_j$. Then, for $t:=\epsilon>|m\_1-m\_2|$,
$$P(|X\_1-X\_2|>t)\le P(|Y\_1-Y\_2|>t-|m\_1-m\_2|) \\
\le P(|Y\_1|>(t-|m\_1-m\_2|)/2)+P(|Y\_2|>(t-|m\_1-m\_2|)/2) \\ 
\le c\_1 e^{-a\_1(t-|m\_1-m\_2|)^2/4}+c\_2 e^{-a\_2(t-|m\_1-m\_2|)^2/4}.$$


