Theory of Probability &Its Applications, Volume 64, Issue 4, Page 579-594, January 2020.
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.

中文翻译：

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关于由概率矩唯一确定的概率分布的条件

概率论及其应用，第64卷，第4期，第579-594页，2020年1月。
我们研究了保证概率分布唯一地由其矩确定的著名卡尔曼条件与最近的，易于检查的条件之间的关系。关于瞬间的增长率。我们在积分理论中使用渐近方法，并涉及Lambert $ W $函数的性质，以证明连续矩比率的二次增长率作为唯一性的充分条件比Carleman条件更具限制性。我们推导出一系列的陈述，其中的一个表明尽管反蕴涵是正确的，但Carleman的条件并不暗示Hardy的条件。还讨论了相关主题。