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 $\mathrm{E}\, e^{t_1\xi + t_2\eta} = \mathrm{E}\, e^{t_1 \xi} \mathrm{E}\, e^{t_2 \eta}, \ \forall t_1, t_2 < 0$?