Reconstituting the distribution of the components of a random Vector from distributions of certain statistics

Let $(X_1,\dots,X_n)$ be a random vector with independent components. It is proven in this paper that, under certain restrictions, the distributions of the pair $S_1=\sup(a_1X_1,\dots,a_nX_n)$ and $S_2=\sup(b_1X_1,\dots,b_nX_n)$ univocally define the distribution function of the components $X_j$.