An identity for two integral transforms applied to the uniqueness of a distribution via its Laplace–Stieltjes transform

It is well known that the Laplace–Stieltjes transform of a nonnegative random variable (or random vector) uniquely determines its distribution function. We extend this uniqueness theorem by using the Müntz–Szász Theorem and the identity for the Laplace–Stieltjes and Laplace–Carson transforms of a distribution function. The latter appears for the first time to the best of our knowledge. In particular, if X and Y are two nonnegative random variables with joint distribution H, then H can be characterized by a suitable set of countably many values of its bivariate Laplace–Stieltjes transform. The general high-dimensional case is also investigated. Besides, Lerch's uniqueness theorem for conventional Laplace transforms is extended as well. The identity can be used to simplify the calculation of Laplace–Stieltjes transforms when the underlying distributions have singular parts. Finally, some examples are given to illustrate the characterization results via the uniqueness theorem.