In particular, if the distribution of $X-Y$ is known, can the distribution of $X$ and $Y$ be determined up to some simple transformations e.g. translation and reflection?
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2$|Ee^{it(X-Y)}|=|Ee^{itX}|^{2}$. This might help, depending what $Ee^{it(X-Y)}$ is. – geetha290krm Jul 15 '23 at 12:23