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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.

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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. 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 $t:=\epsilon>|m_1-m_2|$, $$P(|X_1-X_2|>t)\le P(|Y_1-Y_2|>t-|m_1-m_2|) \\ \le P(|Y_1|>(t-|m_1-m_2|)/2)+P(|Y_2|>(t-|m_1-m_2|)/2) \\ \le c_1 e^{-a_1(t-|m_1-m_2|)^2/4}+c_2 e^{-a_2(t-|m_1-m_2|)^2/4}.$$

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  • $\begingroup$ Thanks. I meant sub-Gaussian in the general sense, with different means $\mu_1 \ne \mu_2$, e.g as in the pair $P=\mathcal N(\mu_1,\sigma_1^2)$ and $P'=\mathcal N(\mu_2,\sigma_2^2)$. $\endgroup$
    – dohmatob
    Commented Apr 5, 2020 at 9:13
  • $\begingroup$ Was just about to write down the same addendum. Thanks! Any hope for the case $\epsilon \le |m_1-m_2|$ ? $\endgroup$
    – dohmatob
    Commented Apr 5, 2020 at 17:25
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    $\begingroup$ @dohmatob : Of course, no hope for that case: If the $X_j$'s are concentrated near the $m_j$'s, then $X_1-X_2$ is naturally concentrated near $m_1-m_2$ -- rather than near $0$. E.g., let the $X_j$'s be iid normal with means $m_1=-m_2=m$, where $m$ large compared with the standard deviation -- then you have almost no concentration for $X_1-X_2$ near $0$. $\endgroup$
    – Iosif Pinelis
    Commented Apr 5, 2020 at 20:30
  • $\begingroup$ Yes, indeed. Thanks again! $\endgroup$
    – dohmatob
    Commented Apr 5, 2020 at 20:41

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