Hardy’s condition in the moment problem for probability distributions

The starting point of this article consists of two papers by G. H. Hardy, published in 1917 and 1918, in which the basic condition used by the present authors first appears. Translated into probabilistic terms, Hardy’s condition can be written as follows: $\mathbf{E}[e^{c\sqrt{X}}]<\infty$, where $X$ is a nonnegative random variable and $c>0$ a constant. Assuming this condition, it follows that all moments of $X$ are finite and the distribution of $X$ is uniquely determined by the moments (i.e., it is $M$-determinate). Moreover, Hardy’s condition is weaker than Cramer’s condition, which requires the existence of a moment generating function of $X$. Hardy’s condition allows the authors to prove that the constant $1/2$ (equal to the square root) is the best possible for $X$ to be $M$-determinate. They also describe the relationship between Hardy’s condition and properties of the moments of $X$, and establish a result concerning the moment determinacy of an arbitrary multivatiate distribution.