Let $A = \varepsilon_1 + \varepsilon_2$ and assume $\varepsilon_1$ and $\varepsilon_2$ are independent.
Suppose now that you are given $A$, but you want to recover $\varepsilon_1$ and $\varepsilon_2$. The solution is not unique, but I would like to at least recover some solution $\varepsilon_1^\prime$ and $\varepsilon_2^\prime$ such that $A = \varepsilon_1^\prime + \varepsilon_2^\prime$, and $\varepsilon_1^\prime$ and $\varepsilon_2^\prime$ are independent.
Does anyone know of a method that does this?
It is easy enough if one of the $\epsilon_i '$ can be almost surely constant
Otherwise it may not be possible: for example $A=0$ or $1$ each with probability $\frac12$ will not be decomposable into the sum of two independent random variables each with positive variance