Criteria for Uniqueness and Non-Uniqueness of Distributions in Terms of Their Moments

Abstract

We are going to consider probability/statistical distributions on the real line with finite moments of all orders. Our goal is to discuss on the classical moment problem (originated by Chebyshev, Markov, Stieltjes): Do the moments determine the distribution uniquely? If "yes", then "when", if "no", then what follows? After mentioning briefly of results belonging to Carleman and Hausdorff, we concentrate on some recent developments involving the so-called Krein condition (1944) and its modifications. We are going to draw an almost complete picture of what is currently known as criteria for uniqueness and for non-uniqueness. Several new results, as well as, new proofs of known results will be presented. Interesting illustrations will be given for both absolutely continuous and discrete distributions (normal, log-normal, exponential, IG, etc.). Related topics, including some open questions, will also be discussed.