Indeterminate Stieltjes Moment Problem: Entropy Convergence

Abstract

The aim of this paper is to consider the indeterminate Stieltjes moment problem together with all its probability density functions that have the positive real or the entire real axis as support. As a consequence of the concavity of the entropy function in both cases, there is one such density that has the largest entropy: we call it fhmax, the largest entropy density. We will prove that the Jaynes maximum entropy density (MaxEnt), constrained by an increasing number of integer moments, converges in entropy to the largest entropy density fhmax. Note that this kind of convergence implies convergence almost everywhere, with remarkable consequences in real applications in terms of the reliability of the results obtained by the MaxEnt approximation of the underlying unknown distribution, both for the determinate and the indeterminate case.