MaPhySto
The Danish National Research Foundation:
Network in Mathematical Physics and Stochastics

MPS-RR 2004-25
November 2004

# Bicontinuity of the Upsilon Transformations

by: Ole E. Barndorff-Nielsen, Steen Thorbjørnsen

## Abstract

In the papers [BT3] and [BT4], the authors introduced and studied one-to-one mappings $\Upsilon$ and $\Upsilon^\alpha$ ( $\alpa \in ]0, 1[$) from the class $\mathcal{I}\mathcal{D}(*)$ of infinitely divisible probability measures on $\mathbb{R}$ into itself. In particular it was proved that these mappings are continuous, when $\mathcal{I}\mathcal{D}(*)$ is endowed with the topology corresponding to weak convergence. In the present note we prove that the $\Upsilon$-mappings are homeomorphisms onto their ranges, which are closed subsets of $\mathcal{I}\mathcal{D}(*)$.

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