2020/12/31 by Gwo Dong Lin, Xiaoling Dou
Mathematics · #Applied mathematics #Calculus (dental) #Differential Equations and Boundary Problems #Distribution (mathematics) #Fourier analysis #Fourier transform #Fractional Differential Equations Solutions #Fractional Fourier transform #Identity (music) #Integral equation #Laplace transform #Laplace transform applied to differential equations #Laplace–Stieltjes transform #Mathematical analysis #Mathematical functions and polynomials #Mathematics #Pure mathematics #Riemann–Stieltjes integral #Two-sided Laplace transform #Uniqueness #math.PR #math.ST #msc:30E05 #msc:46F12 #msc:60E05 #msc:62E10 #stat.ME #stat.TH
paper · pdf · doi:10.1080/02331888.2021.1893728
published as Statistics 2021 · 22 pages
openalex created_date 2020/12/07 · openalex publication_date 2021/03/04 · arxiv created 2021/03/07 · arxiv updated 2021/03/09 · openalex updated_date 2026/08/05
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.