Fréchet-Shohat theorem: stronger modes of convergence for a class of absolutely continuous distributions

Using recent results from information theory, maximum entropy (briefly, MaxEnt) and convergence in entropy of MaxEnt densities, stronger modes of convergence than convergence in distribution are obtained for absolutely continuous distributions. As a first result, an alternative proof of the Fréchet–Shohat theorem is given. Moreover, due to the flexibility of the MaxEnt entropy formalism, the new proof is valid for Hamburger, Stieltjes and Hausdorff moment problems with support $\Real$, $\Real^{+]$, $[0, 1}$, respectively.