A characterization and closure property of Thorin's class
Abstract
Thorin's class coincides with the smallest class of probability laws on ℝ that contains all gamma laws and their reflections, while being closed under convolution and weak convergence. In particular, all Thorin laws are self-decomposable. This paper proves an alternative characterization of Thorin's class in terms of what we call the hyperbolic Bochner-Bernstein property. This property corresponds to the notion of hyperbolically completely monotone Laplace transforms on the positive half line, but extended to the real-valued setting. Using the new characterization, it is shown that Thorin's class is closed under multiplication with independent random variables admitting hyperbolically completely monotone densities, which partly settles an open question raised by Bondesson and allows for proving that a large class of possibly skewed real-valued distributions is in Thorin's class. As particular examples, we apply the closure property to show that Johnson's S\textsubscript{U} distribution and the Pearson type IV distribution are in Thorin's class for all parameter choices.
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BibTeX
@misc{kloster2026characterization,
title = {{A characterization and closure property of Thorin's class}},
author = {Thomas K. Kloster},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01058},
eprint = {2611.01058},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01058},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A characterization and closure property of Thorin's class
AU - Thomas K. Kloster
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01058
UR - https://arxiv.org/abs/2611.01058
AB - Thorin's class coincides with the smallest class of probability laws on ℝ that contains all gamma laws and their reflections, while being closed under convolution and weak convergence. In particular, all Thorin laws are self-decomposable. This paper proves an alternative characterization of Thorin's class in terms of what we call the hyperbolic Bochner-Bernstein property. This property corresponds to the notion of hyperbolically completely monotone Laplace transforms on the positive half line, but extended to the real-valued setting. Using the new characterization, it is shown that Thorin's class is closed under multiplication with independent random variables admitting hyperbolically completely monotone densities, which partly settles an open question raised by Bondesson and allows for proving that a large class of possibly skewed real-valued distributions is in Thorin's class. As particular examples, we apply the closure property to show that Johnson's S\textsubscript{U} distribution and the Pearson type IV distribution are in Thorin's class for all parameter choices.
ER -