The Problem of Moments: A Bunch of Classical Results with Some Novelties

Abstract

We summarize significant classical results on (in)determinacy of measures in terms of their finite positive integer order moments. Well known is the role of the smallest eigenvalues of Hankel matrices, starting from Hamburger’s results a century ago and ending with the great progress made only in recent times by C. Berg and collaborators. We describe here known results containing necessary and sufficient conditions for moment (in)determinacy in both Hamburger and Stieltjes moment problems. In our exposition, we follow an approach different from that commonly used. There are novelties well complementing the existing theory. Among them are: (a) to emphasize on the geometric interpretation of the indeterminacy conditions; (b) to exploit fine properties of the eigenvalues of perturbed symmetric matrices allowing to derive new lower bounds for the smallest eigenvalues of Hankel matrices (these bounds are used for concluding indeterminacy); (c) to provide new arguments to confirm classical results; (d) to give new numerical illustrations involving commonly used probability distributions.

Maximum Entropy Criterion for Moment Indeterminacy of Probability Densities

“…and hence, just as above, we conclude that f n+2 satisfies the entropy relation h f n+2 = h f n . Joining together ( 2) and ( 3) (with obvious extension to the Hamburger case) with f n+1 or f n+2 and referring to the monotone increasing of the entropy with respect to b, we conclude that indeed there are infinitely many densities f n+1 and f n+2 ∈ E n whose entropy spans the interval in (1), with this property holding for all g ∈ D n . Theorem 1 is proved.…”

“…The recent review paper [1] contains a comprehensive description of significant results based on 'uncheckable conditions', including two illustrations of how to use this kind of condition as an indication for a specific property of a distribution in terms of its moments. However, from the applied point of view, most useful are the results involving 'checkable conditions'.…”

“…Here are the well-known classical necessary and sufficient conditions for the (in)determinacy of µ in terms of the limits of the smallest eigenvalues of sequences of Hankel matrices. Our recent review paper [1] describes the whole spectrum, called 'a bunch', of the fundamental results, old and recent. The reader will find in [1] details about the great contributions of T. Stieltjes, H. Hamburger, N. Akhiezer, M. Krein, C. Berg, K. Schmüdgen, M. Putinar, B. Simon, and others.…”

“…Our recent review paper [1] describes the whole spectrum, called 'a bunch', of the fundamental results, old and recent. The reader will find in [1] details about the great contributions of T. Stieltjes, H. Hamburger, N. Akhiezer, M. Krein, C. Berg, K. Schmüdgen, M. Putinar, B. Simon, and others. Their works are widely known.…”

Moment Problems and Integral Equations

“…Earlier and recent results, as well as various aspects of the moment problem, have been discussed in [5,[8][9][10][11][12][13][14][15][16][17][18]. Significant results on the moment problem have been recently published in [19][20][21][22]. Article [23] provides interesting applications of Hahn-Banach-type results in Banach lattice frameworks to a direction without any connection with the moment problem.…”