This article is cited in 7 papers

On conditions for a probability distribution to be uniquely determined by its moments

E. B. Yarovaya^a, J. Stoyanov^b, K. K. Kostyashin<sup>c

a</sup> Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
^b Institute of Mathematics and Informatics, Bulgarian Academy of Sciences
^c Lomonosov Moscow State University

Abstract: We study the relationship between the well-known Carleman's condition guaranteeing that a probability distribution is uniquely determined by its moments, and a recent, easily checkable condition on the rate of growth of the moments. We use asymptotic methods in the theory of integrals and involve properties of the Lambert $W$-function to show that the quadratic growth rate of the ratios of consecutive moments as a sufficient condition for uniqueness is slightly more restrictive than Carleman's condition. We derive a series of statements, one of which shows that Carleman's condition does not imply Hardy's condition, although the inverse implication is true. Related topics are also discussed.

Keywords: random variables, moment problem, M-determinacy, Carleman's condition, rate of growth of the moments, Hardy's condition, Lambert $W$-function.

MSC: 60E05, 62E10, 44A60

Received: 16.05.2019
Revised: 09.07.2019
Accepted: 18.07.2019

DOI: 10.4213/tvp5304

 English version: Theory of Probability and its Applications, 2020, 64:4, 579–594

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