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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 $$P(|X_j|>t)\le c_j e^{-a_j t^2}$$ for $j=1,2$. So, for $t:=\epsilon>0$, $$P(|X_1-X_2|>t)\le P(|X_1|>t/2)+P(|X_2|>t/2)\le c_1 e^{-a_1 t^2/4}+c_2 e^{-a_2 t^2/4}.$$