Hardy’s Condition in the Moment Problem for Probability Distributions

In probabilistic terms Hardy’s condition is written as follows: E[ec X] <∞, where X is a nonnegative random variable and c> 0 a constant. If this holds, then all moments of X are finite and the distribution of X is uniquely determined by the mo-ments. This condition, based on two papers by G. H. Hardy (1917/1918), is weaker than Cramér’s condition requiring the existence of a moment generating function of X. We elaborate Hardy’s condition and show that the constant 12 (square root) is the best possible for the moment determinacy of X. We describe relationships between Hardy’s condition and properties of the moments ofX.We use this new con-dition to establish a result on the moment determinacy of an arbitrary multivariate distribution