An Identity for Two Integral Transforms Applied to the Uniqueness of a Distribution via its Laplace-Stieltjes Transform

TL;DR

This paper extends the uniqueness theorem for Laplace-Stieltjes transforms of nonnegative distributions using a new integral identity, enabling better characterization and calculation of such distributions, including high-dimensional cases.

Contribution

It introduces a novel identity linking Laplace-Stieltjes and Laplace-Carson transforms, extending uniqueness results to multivariate distributions and singular parts.

Findings

Characterization of distributions via countably many Laplace-Stieltjes transform values

Extension of Lerch's theorem for conventional Laplace transforms

Simplified calculation of transforms with singular distribution parts

Abstract

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 Muntz-Szasz 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.