I am wondering if I can find a decomposition of $Y$ that is absolutely continuous nto two i.i.d. random variables X' and X'' such that Y=X'-X'', where X' is also Lebesgue measure with an almost everywhere positive density w.r.t to the Lebesge mesure.
My main intent is to come up with two i.i.d. random variable, X' and X'' and $Y$ and Y'', such that \operatorname{\mathbb{Pr}}(m> Y'-Y'')=\operatorname{\mathbb{Pr}}(m>X'-X'') for $m \in (-b,b)$ for some $b$ small enough, while \operatorname{\mathbb{Pr}}(m+2> Y'-Y'')=\operatorname{\mathbb{Pr}}(m+1> X'-X''). I figured starting first by constructing a measure on the difference first that satisfies the above then decomposing it. Is this possible?
Thanks so much in advance for your much appreciated help.