Maximum entropy in the finite Stieltjes and Hamburger moment problem

Abstract

A necessary and sufficient condition for the existence of the maximum entropy (ME) function, defined in an infinite or semiinfinite interval, is provided. The conclusions reached show that, except in a few particular cases, the necessary and sufficient conditions for the existence of maximum entropy function are identical to the conditions for the solution of the moment problem when the first M+1 moments are assigned. Even if the conclusions reached are very similar to the Hausdorff case, the specificity of the Hamburger and Stieltjes cases demands a different handling. A sufficient condition for the entropy convergence of the resulting sequence of maximum entropy estimators to the entropy of the recovering function is also provided.