Let $\xi, \eta, \eta'$ be non-negative random variables such that:


* $\eta \stackrel{\mathcal{L}}{=} \eta'$,
* $\xi + \eta \stackrel{\mathcal{L}}{=} \xi + \eta'$,
* $\xi$ and $\eta'$ are independent.


Does this imply that $\xi$ and $\eta$ are independent? Can one construct a counter-example? Any sort of reference would be of great help, too.


In terms of Laplace functions, does $\mathrm{E}\, e^{t(\xi+\eta)} = \mathrm{E}\, e^{t\xi} \mathrm{E}\, e^{t\eta}, \forall t<0$ imply $\mathrm{E}\, e^{t\_1\xi + t\_2\eta} = \mathrm{E}\, e^{t\_1 \xi} \mathrm{E}\, e^{t\_2 \eta}, \ \forall t\_1, t\_2 < 0$? 



Sum of two independent random variables
No, $\xi$ and $\eta$ need not be independent. For example let $\xi$, $\eta$, and $\eta'$ all be uniformly distributed on $\{1,2,3\}$ with the joint distribution of $\xi$ and $\eta$ is given by the matrix
\[
P = \frac{1}{9}\begin{bmatrix}
1 & 2 & 0 \\ 0 & 1 & 2 \\ 2 & 0 & 1
\end{bmatrix},
\]
i.e. $\mathbb{P}(\xi = i, \eta = j) = P\_{ij}$. You can generate lots of discrete examples like this because your conditions are linear equations on the row sums, column sums, and anti-diagonal sums of $P$. Altogether this is only linearly many conditions on the quadratically many entries of $P$ (as a function of the size of the support of the variables).


