Can we characterize the laws of n-uple real random variables $(X_1...X_n)$ so that for any random variable Y which has the same law as the sum of the $X_i$ there exists a n tuple of functions $(f_i)_i$ and a random vector W independent from the $X_i$ such that the vector $(f_i(Y,W))_i$ has the same law as the random vector $(X_1, ... , X_n)$ and the sum of the $f_i(Y,W)$ is equal to Y ? (Hence a splitting of Y)
I know that if X is a vector of random gaussian independent variables this is true, but is there any other examples and can we describe all the possible examples ? If anybody has a reference it would be great.
Thanks
