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

2022/05/25 by Herdegen, Martin, Liang, Gechun, Shelley, Osian
#60-XX (Primary) 26A12 #93E20 (Secondary) #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR)

paper · doi:10.48550/arxiv.2205.13075

Abstract

The Laplace transforms of positive measures on 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.

Related

• Vague and basic convergence of signed measures
• Vague and weak convergence of signed measures
• Karamata's theorem for regularised Cauchy transforms
• Topology and convergence on the space of measure-valued functions
• On continuity of correspondences of probability measures in the category of Tychonov spaces