Hamburger moment problem and Maximum Entropy: On the existence conditions

The existence of Maximum Entropy solution for the reduced Hamburger moment problem is reconsidered. Existence conditions, previously appeared in literature, are revisited allowing an easy way to identifying the existence of Maximum Entropy solution. The obtained results suggest that, except for special sequences of moments unknown a priori, the Maximum Entropy solution for the non symmetric reduced Hamburger moment problem exists. For practical purposes, the replacing of the support R with a large enough finite interval finds a theoretical warranty. The symmetric case may be formulated as follows: once assigned the first 2M moments, if MaxEnt density does not exist (conclusion drawn uniquely from numerical evidence), MaxEnt density with the first 2M-2 moments exists. In such a case, even if the first 2M moments are known, we have to settle for a density which carries less information. Theoretical results are illustrated through some numerical examples.