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Let $X$, $Y$ be mean preserving spreads (MPS) of the same random variable $Q$ and assume that $X =_d Y$ in distribution. Then, by the definition of MPS, there exist variables $Z$ and $Z'$ such that $Q + Z =_d Q+ Z'$ where $E[Z|Q]=E[Z'|Q]=0$. Does it follow that $Z=_d Z'$ conditional on Q? If not, under what condition could it be true?

Edit for definition of MPS: The random variable $X$ is a MPS of random variable $Q$ if there is $Z$ such that $X=_d Q+Z$ (equal in distribution) where $E[Z | Q]=0$.

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  • $\begingroup$ please define mean preserving spread $\endgroup$
    – kodlu
    Commented May 7, 2021 at 1:11
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    $\begingroup$ I just added the definition in case it wasn't clear already. $\endgroup$
    – Margot.
    Commented May 7, 2021 at 1:22

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