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aspiring88
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I am wondering if I can find a decomposition of [itex]Y[/itex] that is absolutely continuous nto two i.i.d. random variables [itex]X'[/itex] and [itex]X''[/itex] such that [itex]Y=X'-X''[/itex], where [itex]X'[/itex] 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, [itex]X'[/itex] and [itex]X''[/itex] and [itex]Y[/itex] and [itex]Y''[/itex], such that [itex]Pr(m> Y'-Y'')=Pr(m>X'-X'')[/itex] for [itex]m \in (-b,b)[/itex] for some [itex]b[/itex] small enough, while [itex]Pr(m+2> Y'-Y'')=Pr(m+1> X'-X'')[/itex]. 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.
Mod note: fixed LaTeX
My main intent is to come up with two i.i.d. random variable, [itex]X'[/itex] and [itex]X''[/itex] and [itex]Y[/itex] and [itex]Y''[/itex], such that [itex]Pr(m> Y'-Y'')=Pr(m>X'-X'')[/itex] for [itex]m \in (-b,b)[/itex] for some [itex]b[/itex] small enough, while [itex]Pr(m+2> Y'-Y'')=Pr(m+1> X'-X'')[/itex]. 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.
Mod note: fixed LaTeX
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