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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On infinitely divisible laws in $ C[0,1]$


Author: Aloisio Pessoa De Araujo
Journal: Proc. Amer. Math. Soc. 51 (1975), 179-185
MSC: Primary 60B05
Erratum: Proc. Amer. Math. Soc. 56 (1976), 393.
MathSciNet review: 0407918
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Abstract: In Euclidean spaces, or in a separable Hilbert space, the characteristic function of an infinitely divisible distribution has the familiar form given by the Lévy-Khintchine formula. The Lévy measures $ M$ of this formula are characterized by the property that the integral of $ \min [1,\vert\vert x\vert{\vert^2}]$ with respect to $ M$ is finite. This simple situation no longer holds in the Banach space $ C = C[0,1]$ where integrability of $ \min [1,\vert\vert x\vert\vert]$ is sufficient but integrability of $ \min [1,\vert\vert x\vert{\vert^2}]$ is neither necessary nor sufficient.

Certain other conditions which are sufficient to imply that $ M$ is the Lévy measure of a distribution on $ C$ can be obtained with the use of an integral formula of Garsia.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0407918-X
PII: S 0002-9939(1975)0407918-X
Article copyright: © Copyright 1975 American Mathematical Society