arXiv subjects

Osian Shelley

Publications and source records attributed to Osian Shelley.

2 recordsLinked to original sources

arXiv2023

A continuity theorem for generalised signed measures with an application to Karamata's Tauberian theorem

The Laplace transforms of positive measures on $\mathbb{R}_{+}$ converge if and only if their distribution functions converge at continuity points of the limiting measure. We extend this classical continuity theorem to the case of generalised signed Radon measures. The result for the signed case requires some additional conditions, which follow from recent results on vague convergence of signed Radon measures. As an application, we introduce a novel Tauberian condition for generalised signed Radon measures that extends Karamata's Tauberian theorem.

math.OC↗

arXiv2022

Vague and weak convergence of signed measures

Necessary and sufficient conditions for weak and vague convergence of measures are important for a diverse host of applications. This paper aims to give a comprehensive description of the relationship between the two modes of convergence when the measures are signed, which is largely absent from the literature. Furthermore, when the underlying space is $\mathbb{R}$, we study the relationship between vague convergence of signed measures and the pointwise convergence of their distribution functions.

math.FA↗