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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
DOI: https://doi.org/10.1090/S0002-9939-1975-0407918-X
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 LĂ©vy-Khintchine formula. The LĂ©vy measures $M$ of this formula are characterized by the property that the integral of $\min [1,||x|{|^2}]$ with respect to $M$ is finite. This simple situation no longer holds in the Banach space $C = C[0,1]$ where integrability of $\min [1,||x||]$ is sufficient but integrability of $\min [1,||x|{|^2}]$ is neither necessary nor sufficient. Certain other conditions which are sufficient to imply that $M$ is the LĂ©vy measure of a distribution on $C$ can be obtained with the use of an integral formula of Garsia.


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Article copyright: © Copyright 1975 American Mathematical Society