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.